Eulerian Graphs and Related Topics: Part 1, Volume 2 (Annals of Discrete Mathematics, Volume 50)

EULERIAN GRAPHS AND RELATED TOPICS Part 1, Volume 2 ANNALS OF DISCRETE MATHEMATICS General Editor: Peter L. HAMMER R...

Author: Herbert Fleischner

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EULERIAN GRAPHS AND RELATED TOPICS Part 1, Volume 2

ANNALS OF DISCRETE MATHEMATICS

General Editor: Peter L. HAMMER Rutgers University, New Brunswick, NJ, USA

Advisory Editors: C. BERGE, Universite de Paris, France R.L. GRAHAM, AT& T Bell Laboratories, NJ, USA M.A. HARRISON, University of California, Berkeley, CA, USA V: KLEE, University of Washington, Seattle, WA, USA J.H. VAN LINT; California Institute of Technology, Pasadena, CA, USA G.C. ROTA, Massachusetts Institute of Technology, Cambridge, MA, USA 7: TROTTER, Arizona State University, Tempe, AZ, USA

50

EULERIAN GRAPHS AND RELATED TOPICS Part 1, Volume 2

Herbert FLEISCHNER Institute for Information Processing Austrian Academy of Sciences Vienna, Austria

1991

NORTH-HOLLAND - AMSTERDAM

NEW YORK

OXFORD TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211,1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, NY 10010, USA Eulerian Graphs and Related Topics, Part 1, Volume 1 in the series Annals of Discrete Mathematics (no. 45) was published in 1990 (ISBN: 0 444 88395 9 )

Library of Congress Cataloging-in-Publication Data (Revised for vol. 2) Fleischner. Herbert. Eulerian graphs and related topics (Annals of discrete mathematics : 45 Includes bibliographieal references and index. 1. Eulerian graph theory. 1. Title. 11. Series: Annals of discrete mathematics ; 45. e s . QA166.19.F54 1990 511’3 90-7300 ISBN 0-444-88395-9 (pt. I , v. 1) ISBN 0-144-89110-2 (pt. I , v. 2)

ISBN: 0 444 891 10 2

0 ELSEVIER SCIENCE PUBLISHERS B.V., 1991 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers 8.V. /Academic Publishing Division, PO.Box 103, 7000 AC Amsterdam, The Netherlands. Special regulations for readers in the USA - This publication has been registered w i t h the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred t o the publisher.

NQ responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or f r o m any use or operation of any methods, products, instructions or ideas contained in the material herein.

PRINTED IN THE NETHERLANDS

To those ... t o those as well who err in seeking answers

and have the courage to say ‘we erred’. Therefore? also t o my parents.

This Page Intentionally Left Blank

vii

PREFACE This volume is the ‘second half’ of “Eulerian Graphs and Related Topics”, Part 1, and relies to some extent on results contained in the preceding volume. Both the author and the editors decided to present Part 1 in two volumes because the original manuscript would have exceeded a maximum of 600 pages. This division of Part 1 into two volumes thus makes it possible to include an Appendix containing corrections of errors in Volume 1 as well as exact references to certain articles which were available only as preprints at that time. The two volumes comprising Part 1 therefore embrace the theme of eulerian trails and covering walks. In addition to the people mentioned in Volume 1as having been of help in completing that book, I would like to express my gratitude to P.A. Catlin, C.J. Colbourn, I. Nishizeki (these three colleagues informed me of some errors), A. Sebii, and in particular to P. Rosenstiehl (whose valuable comments on some maze search algorithms are included) and t o P.D. Seymour who not only read Chapter VIII and found some errors and inaccuracies, but also gave me his recollection of how he discovered his famous 6-Flow Theorem (which I elaborated on extensively). Last but not least I wish to express my thanks to Ms. I. Hosch, secretary of the Institute for Information Processing, who gave w c a l help in preparing the final version of this volume, and to Ms. C. Lillie who read various addenda as well as the preface and the appendix from a linguistic point of view.

This Page Intentionally Left Blank

ix

CONTENTS

PREFACE.. . . . VIII.

. . . . . . . . . . . . . . . . . . .

VARIOUS TYPES OF CLOSED COVERING WALKS . . .

. .

. . . . . . . . . . . . . . . . .

vii

VIII.1

VIII.1.

Double Tracings .

VTII.2.

Value-True Walks and Integer Flows in Graphs . VIII.14

VIII.3.

The Chinese Postman Problem . . . . . . . . VIII.82

VIII.3.1.

The Chinese Postman Problem for Graphs

VIII.3.1.1.

Some Applications and Generalizations of the CPP VIII.93

.

VIII.1

. . . VIII.82

. . . . . . . . . . . . . . . . V.III.93 VIII.3.1.1.2. t-Joins, $-Cuts and Multicommodity Flows . . . VIII.96 VIII.3.1.1.1. Applications

VIII.3.1.1.3. Hamiltonian Walks, the Traveling Salesman and Their Relation to the Chinese Postman . . . . VIII.1109 VIII.3.2.

The Directed Postman Problem . . . . . . .

VIII.3.3.

The Mixed Postman Problem . . . . . . . .

VIII.3.4.

The Windy Postman Problem and Final Remarks VIII.135

VI 11.4.

Exercises . . . . . . . .

IX.

EULERIAN TRAILS - HOW MANY ?

IX. 1.

. . . As Many As . . . - Parity Results for Digraphs and (Mixed) Graphs . . . . . . . . . . . . . . IX.l An Application to Matrix Algebra . . . . . . . . IX.59 The Number is . . . A First Excursion Into Enumeration . . . . . . . IX.72

IX.1.1. IX.2.

VIII.114 VIII.124

. . . . . . . . . VIII.142 . . . . . IX.l

Contents

X

IX.2.1.

The Matrix Tree Theorems . . . . . . . . . . . I X .73

IX.2.2.

Enumeration of Eulerian Trails in Digraphs and Graphs . . .

IX .2.3.

On the Number of Eulerian Orientations . . . . . ix.90

IX .2.4.

Some Applications and Generalizations of the BEST-Theorem . . . . . . . . . . . . . 1x.99

IX.2.5.

Final Remarks

IX.3.

Exercises . . . . . . . . . . . . . . . . . . IX .108

X.

ALGORITHMS FOR EULERIAN TRAILS AND CYCLE DECOMPOSITIONS. MAZE SEARCH ALGORITHMS . . . . . . . . . X.l

x.l.

Algorithms for Eulerian Trails . . . . . . . . . . . X.1

x.2.

Algorithms for Cycle Decompositions

x.3.

Mazes . . . . . . . . . . . . . . . . . . . .

x.4.

Exercises . . . . . . . . . . . . . . . . . . . X.32

I X .79

. . . . . . . . . . . . . . . I X .106

. . . . . . X.14 X.17

Bibliography . . . . . . . . . . . . . . . . . . . . . . . .

A.1

Index . . . . . . . . . . . . . . . . . . . . . . . . . . .

B.l

Appendix: Corrections and Addenda to Volume 1 . . . . . . . . C.1

VIII. 1

Chapter VIII

VARIOUS TYPES OF CLOSED COVERING WALKS The starting point of this chapter is Euler’s observation that in any connected graph there exists a closed covering walk using every edge exactly twice (see paragraph 18 in Euler’s paper and Corollary V.8). In fact, there even exists a closed covering walk which uses every edge exactly once in each of the two possible directions (Corollary V.9).

In this chapter we shall investigate various types of closed covering walks, considering first such walks which use every edge exactly twice.

VIII.1. Double Tracings We introduce some notation that is needed to distinguish between the various types of closed covering walks which use every edge exactly twice. In doing this we follow [EGGL84a, SKIL85al.

Definition VIII.l. Recall that a closed covering walk in a graph using every edge exactly twice is called a double tracing. A double tracing is called bidirectional if it uses every edge (precisely once) in each of the two possible directions, while it is called retracting-free if no edge is passed the second time immediately after being passed the first time. On the other hand, if it contains a section e, v,e , we speak of a retracting of e at v. We speak of a strong double tracing if the double tracing is both bidirectional and retracting-free. We observe that the question of the existence of strong double tracings was first raised by Ore,[OFtE05la, p. 531. Note that [EGGL84a] uses the term retracing-free instead of retractingfree, while Sabidussi, who was the first to consider retracting-free double tracings, uses the term non-backtracking quasi-eulerian walk instead, [SABI77a].

VIII. Various Types of Closed Covering Walks

vIII.2

With the above notation at hand, Corollary V.9 says that every connected graph admits a bidirectional double tracing. As for retracting-free double tracings, we have the following result of Sabidussi, [SABI77a], which has also been proved in [EGGL84a, Theorem 91 (the authors of the latter paper apparently being unaware of Sabidussi’s earlier work).,) Its proof is a direct application of Corollary VI.5.

Corollary VIII.2. A connected graph G has a retracting-free double tracing if and only if G has no end-vertices. Proof. Since no double-tracing in a graph with end-vertices is retractingfree, it s d c e s to prove the existence of a retracting-free double tracing if G has no end-vertices. For this purpose, consider G,obtained from G by ‘doubling’every edge e E E(G). Consequently, G, is a connected eulerian graph with 6(G,) 2 4. Denoting by el the edge of E(G,) -E(G) resulting from the duplication of e E E(G), we obtain a system of transitions X , of G, by defining for every 2, E V(G,) = V ( G ) and where E*(G) = UVEV(C)

E: X,(v)= {{e’,ei}/e’

x,=

u

E Ez

C E*(G)},

Xl(4

VEV(G1)

Since S(G,) 2 4, an eulerian trail T,of G, exists that is compatible with X , (Corollary VI.5). By the very construction of G,,TI corresponds to a double tracing of G which is retracting-free precisely on account of the definition of X , (note that e’ and ei are not consecutive in T,).The corollary now follows. In fact, Corollary VIII.2 admits an interesting interpretation of a graph without end-vertices as a homomorphic image of a cycle. This interpretation is equivalent to Corollary VIII.2 and is, in fact, the indirect starting point of Sabidussi’spaper (see [SABI77a, Theorem 3.21 and the introduction of that paper). The proof of this result can be easily derived from the proof of Corollary VIII.2 and is therefore left as an exercise.

Corollary VIII.3. Let G be a connected graph without end-vertices and let C be a cycle of length 2 I E(G)I. Then there exists a homomorphism

In fact, this result was communicated to me by Sabidussi as early as July 1975.

VIII.l. Double Tracings such that Icp-’(e)

VIII.3

I= 2for everye E E(G)

and cp is locally one-to-one. We observe that for bidirectional double tracings (which may, however, have retractings) a similar result was established in [JAV07la, Theorem 2.11. In that instance, the application of cp will either result in G or in a detachment F of G (see Definition V.11) to which one applies another homomorphism $ such that $ ( F ) = G. Of course, Corollary VIII.2 does not tell us whether G has a strong double tracing. In fact, the next result, [TROY66a], shows that in general, we cannot expect to find a strong double tracing in graphs without endvertices. Theorem VIII.4. Suppose the graph G satisfies V,(G) = 8 for i 2 4, and I V.(G)I= 0 (mod 4). Then G has no strong double tracing. Instead of proving Theorem VIII.4 directly, we choose the more general approach developed recently by Thomassen, [THOM87a], from which Theorem VIII.4 will follow easily. It should be noted at this stage, however, that Theorem VIII.4 is the key to Proposition VI.45 (see the discussion preceding that proposition). 2 , We now discuss in detail the most important part of [THOM87a]. Consider a graph G and d e h e O+(v) for every v E V(G), either arbitrarily or its an abstraction of an embedding of G in some surface. Define a closed walk W by starting at v E V ( G )along e = vw,and continue at w along the edge f = wx,where f’ appears as the successor of e’ in O+(w). After the arrival at x continue along g , where g‘ succeeds f’ in O + ( x ) ;a.s.0. Stop after arriving at v along the edge h, where h’ is the predecessor of e’ in O+(v). In the next lemma we summarize some observations, the proof of which is left as an exercise. Lemma VIII.5. Let G and W be given as above. Every edge of G is used by W at most twice. If some edge el is used twice, it is used once in each direction. Moreover, if e2 = w1v2 is passed in W from v1 to v2, 2, This is why we state Troy’s result as a theorem and not as a corollary to Theorem VIII.9 below.

VIII.4

VIII. Various Types of Closed Covering Walks

then W,, the closed walk starting at 2 1 ~along e2 and defined by the same rule as W above, satisfies W, = W . Consequently, if the above W uses every edge twice, W is a bidirectional double tracing: it is even a strong double tracing if G has no end-vertices. Suppose now that some edge of G appears at most once in W . If W contains an edge used precisely once, we can call it eo = vowo and assume W passes e0 from w o to 2 1 ~ . If, however, W does not contain such an edge, let eo = uow0 be chosen arbitrarily in E(G) - E(W). In fact, in this case ( E ( W ) )is a component of G (Exercise VIII.3). In any case, we can define a new walk W’ starting at 2 1 ~along eo towards wo. It follows from Lemma VIII.5 that an edge appearing in both W and W’ is used precisely once by both W and W’ and in opposite directions. Continuing this procedure, one finally arrives at a system S of closed walks such that every edge is used precisely once in each direction. Observe that S is uniquely determined by the choice of O+(v);this follows from Lemma VIII.5. We note that for some W E S, W-l E S may hold as well (e.g., if G has a component which is a cycle), and that S depends on the definition of O+(v), TJ E V ( G ) . In particular, if these O+(v) have been obtained from an embedding of G on some surface F, S contains precisely the boundary walks of the faces of G in F.

Definition VIII.6. Let G,O+(v)for every

21

E V ( G ) ,and S be given

as above. An element of S is called an S-orbit, and we call S an orbit double cover. Moreover, if there exists a choice of O+(v), TJ E V ( G ) ,

such that the corresponding orbit double cover S contains precisely one S-orbit, we call G strictly upper embeddable. Next we prove some results needed t o establish the main result of [THOM87a]. In the sequel, the proofs presented here follow the lines of the proofs presented in that paper.

Lemma VIII.7. If G is a strictly upper embeddable graph without endvertices, it has a strong double tracing. Conversely, a graph G having a strong double tracing and satisfying K ( G ) = 0 for i > 3, is strictly upper embeddable. Proof. The first part of the lemma follows from Definition VIII.6 and the discussion preceding it. For the converse we first note that the assumption implies d(21) E (2,3} for every TJ E V ( G ) . Since O+(v) is uniquely determined if d(21) = 2, we assume w.1.o.g. that 21 is 3-valent. Denote E, = (e, f,g } and observe that I E, I= 3 since G has a strong double

VIII.l. Double Tracings

VIII.5

tracing W . Since each pair of elements of E, must appear in W because W is retracting-free, and because W is bidirectional, either W or W-’ must be either of the form

...e,v,f ,...,f , v , g ,..., g,v,e ... or of the form

...e,v,f ,...,g,v,e ,..., f , v , g ... . In either case, W or W-’ defines O+(v) := (el, f’,g’). That is, for W we obtain O+(v) E { ( e l , f’,g’), (e’, g’, f‘)}. Thus, for these O+(v) we have S = { W } as an orbit double cover; i.e., G is strictly upper embeddable. The lemma now follows. We conclude that in the special case of 3-regular graphs, strict upper embeddability and strong double tracings are equivalent concepts. However, in his paper Thomassen constructs for every r 2 3 an infinite number of r-regular, 3-connected graphs which are not upper embeddable. On the other hand, if 6(G) > 3 for the connected graph G, G has a strong double tracing (consequently, any connected eulerian graph other than a cycle has a strong double tracing). This follows immediately from the next lemma.

Lemma VIII.8. Every connected graph G has a bidirectional double tracing such that retractings occur only at 1- or 3-valent ~ e r t i c e s . ~ ) Proof. We proceed indirectly and choose G with minimal qG =I E ( G )I such that every bidirectional double tracing has a retracting at some vertex v E V(G)-(V,(G)UV,(G)). If V2(G)# 0, let G , be the graph homeomorphic to G with V,(G,) = 0. Applying the lemma to G, (qG1 < qG) and considering a corresponding double tracing W , of G , , one can readily extend W , to a bidirectional double tracing W of G such that W satisfies the conclusion of the lemma. V2(G)= 0 follows. Construct the connected eulerian digraph D by replacing in G every edge by an oriented digon. Consequently, for E V ( D ) = V ( G ) , d G ( v ) = id,(v) = od,(v); so, an end-vertex of G becomes a 2-valent vertex of The same result can be found in [SKIL85a, Theorem 11 but the proof presented there contains some flaws. - The proof presented here is independent of the proofs of both Skilton and Thomassen. 3,

VIII. Various Types of Closed Covering Walks

VIII.6

D,while a 3-valent vertex v of G becomes 6-valent in D. For such we transform, by a 2-fold application of the Splitting Lemma, D into the connected eulerian digraph D, where v has been replaced by three 2-valent vertices in D,. Doing this in D for all 3-valent vertices of G, we arrive at the connected eulerian digraph D, with dDl(z)2 8 for x E V(D,)- V,(D,). Define X,, a (partial) system of transitions of D,, on V(D1) - U D , )by {(a:,)’,

and

(a;,)’} E X,(z) if and only if a : ,

+ a=,, a,,,,

correspond to e E

E A:, a:,

E A,

E,.

By Corollary VI.15, D, has an eulerian trail T, compatible with X,. By construction, T, corresponds to a bidirectional double tracing W of G. W has no retractings at vertices with a valency of at least four because of the definition of X,. Retractings occur at vertices of valency 1, and at most along one edge incident with a 3-valent vertex for each such vertex (this can be secured by an appropriate application of the Splitting Lemma). The lemma now follows. We state the following theorem without proof. However, it is nothing but a rephrasing of [XUON79a, Theorem 21. 4,

Theorem VIII.9. A connected graph G is strictly upper embeddable if and only if G has a spanning tree T such that every component of G - T has an even number of edges. Theorem VIII.9 enables us to derive Theorem VIII.4 easily. For, as we have noted above, in the case of cubic graphs strong double tracings and strict upper embeddability are equivalent concepts. Consequently, this also holds true for every graph homeomorphic to a cubic graph. Thus, let us consider a cubic graph G with p = 2k, k E N. For every spanning tree T of G, G - T has q

3P - ( p - 1) = - ( p - 1) = k + 1

2

4, See also [BEHZ79a, Theorems 5.14, 5.15, 5.161, where the last of the quoted theorems is attributed to both Xuong and Jungermann. In fact, Thomassen introduces the concept of strict upper embeddability as a special case of upper embeddability as defined in [XUON79a]. We note [SCOT89a,SKOV89a] as recent articles on the theme of upper embeddability, with the latter containing several references to articles on the same theme.

VIII. 1. Double Tracings

VIII.7

edges. Thus, G - T has an odd number of edges if Ic is even. By Theorem VIII.9, G cannot have a strong double tracing if p 0 (mod4). However, Theorem VIII.9 (=[THOM87a, Theorem 2.11) is the key to the following result ([THOM87a, Theorem 3.31, the main result of that paper) which relates strong double tracings to special types of spanning trees.

Theorem VIII.10. The following statements are equivalent for a connected graph G without end-vertices.

1) G has a strong double tracing. 2) G contains a spanning tree T such that every component of G - T with an odd number of edges contains a vertex v with &(v) > 3. Proof. 1) implies 2). Consider a strong double tracing W of G, and let v be an arbitrary vertex of G. For every section e, v,f of W, e, f E E,, the function w, : E: + E: defined by ?r(e’) = f’ is a bijection and hence a permutation of E: which can be written as the product of disjoint (permutation) cycles, w, = wl... w r v , r , 2 1.

This way of expressing w, gives rise to a partition

P(E:) = {E:,i/i = 1,.. . ,r,} together with a cyclic ordering O t i of the elements of E,*i, where OVlj + -- wi

, i = 1,.. . , r , .

Consequently, if we replace every v E V(G) with r , vertices vl,. . . ,vrV and define the incidences by EGi := we obtain the graph Go in which the original W appears as a strong double tracing W, (note that S(G,) > 1 since W is a strong double tracing which implies that x, has no fixed points). Defining 0: := O z i , it now follows from the very construction and definition of Go,W,, { O z / i = 1,. . . ,r,, v E V(G)} respectively, that So, the orbit double cover of Go, satisfies the equation

VIII.8

VIII. Various Types of Closed Covering Walks

That is, Go is strictly upper embeddable. By Theorem VIII.9, Go has a spanning tree Tosuch that every component of Go - To has an even number of edges. Now consider G, := (E(To))G. If G, is a tree, then G = Go and G, = To, and 2) follows from Theorem VIII.9. Whence we can suppose G, has a cycle C. Since To is a tree, there exists v1 E V(C)which corresponds to at least two vertices of Go which in turn implies (together with S(Go)> 1) that dG(vl) > 3. Now we form G, := G, - e l , where el E E(C)n E V 1 and , repeat the above argument until we reach, for some i > 1, the acyclic graph T := Gj C G. By construction, T is a tree. A component H of G - T with an odd number of edges cannot be a component of Go - To and must, by construction, contain one of the edges e j E E v j 7where d G ( v j ) > 3 and 1 5 j 5 i - 1; i.e., v j E V ( H ) . This finishes the proof of the implication. 2) implies 1). Consider a spanning tree T of G as described in 2). If every component of G - T has an even number of edges, it follows immediately from Theorem VIII.9 and Lemma VIII.7 that G has a strong double tracing. In order to draw the same conclusion in the case of components with an odd number of edges in G - T , we add, for every such component H , a new vertex w H and two new edges e H , 1 , e H ,of 2 the form v H w H , for some fixed v H E V ( H ) with dG(vH) > 3. For the graph Go thus obtained, we extend T to a spanning tree T o of Go by adding w H and e H , ,to T for every H described above. Now, every component of Go -To has an even number of edges; whence we may conclude as above that Go has a strong double tracing W o .It follows from the definition of W othat W o must be of the form

where { i , j } = {1,2}. Thus, we can reduce W o to a bidirectional double tracing H Wo = . . ., e l vH1f,. . . ,9 , v H , h, . . . ; and doing this for each of the above H we arrive at W * ,a bidirectional double tracing of G. However, W * may not be retracting-free (e.g., if e = f ) . It follows from the construction of Go that W* can have retractings

VIII.l.

Double Tracings

vIII.9

only at the v H ,where d G ( v H )> 3. Proceeding now in a manner similar to the proof of Lemma VIII.8, we can modify W* so as to obtain a bidirectional double tracing W of G which no longer has retractings at any of these v H . Since the sections of W coincide with those of W * on V ( G )- ( v H / v HE V ( H ) ,q H 1(rnod2)}, it now follows that W is in fact a strong double tracing. This finishes the proof of the implication. The theorem now follows. Interestingly enough, there exists a polynomially bounded algorithm which either finds a tree T as stated in Theorem VIII.10 or decides that no such T exists, [THOM87a, Theorem 3.41. Looking at Theorem VIII.4 one is tempted to ask whether a connected 2 (mod4). cubic graph G has a strong double tracing provided pG Figure VIII.l shows that in general the answer is negative. In fact, any bidirectional double tracing of this graph starting at x towards y must have a retracting of ey. We note in passing that this example can serve as a base for constructing an infinite family of cubic graphs G that satisfy p G f 2(mod4), but have no strong double tracing (Exercise VIII.4). However, the connectivity of such G must be small. This follows from the next result which is an abbreviated and modified version of [THOM87a, Theorem 6.11. We state it without

G Figure VIII.l. A cubic graph G without strong double tracing, where pG 2 (mod 4). Theorem VIII.ll. A cubic 3-connected graph G with p , E 2 (mod4) has a strong double tracing if for every edge cut E,, of size 3, the (two) The proof of this theorem rests on a recursive construction of those strictly upper embeddable graphs G which satisfy 2 5 6(G) 5 A(G) 5 3, and on a recursive characterizationof cubic, cyclically 4-edge-connected graphs, [THOM87a, Theorems 4.2 and 5.21. 5,

VIII.10

VIII. Various Types of Closed Covering Walks

components GI, G, of G - E,, satisfy PG,

pG2

1(mod 4)

(note that G - E, cannot have more than two components since G is %connected). Now we can prove the following (equivalent) version of [THOM87a, Theorem 6-21which can be viewed as the most general counterpart to Theorem VIII. 4.

Theorem VIII.12. A simple graph G satisfying

S(G) > 2, n(G) > 1, X,(G)

>3

has a strong double tracing unless

Proof. Suppose A(G) < 4. Then b(G) = A(G) = 3, i.e., G is cubic. If n(G) = 2, then X(G) = X,(G) = 2 (see Lemma 111.38), a contradiction to the hypothesis X,(G) > 3. Hence G is %connected. In this case, the validity of the theorem follows immediately from Theorems VIII.4 and VIII.ll whence we may suppose A(G) 2 4. Consider a vertex v with d := d(v) 2 4. In order to construct a graph G, satisfying the hypothesis of the theorem and

we consider the maximal connected bridgeless subgraphs H I , . . . , H,, T 2 1, of G - v (note that G - v is connected but may have bridges); they are the components of (G - v) - {e/eis a bridge ofG - v}. Next we

define O+(v) = (ei,e;, . . . ,e&)according to the following rule: assuming first w.1.o.g. that for 1 5 k 5 s 5 T , Hk contains zk and Y k such that x k v , ykv E E,, denote ek = z k v , ek+L+l. - ykv, 1 5 k 5 s; then wedenote the remaining elements of E, arbitrarily with e j , where j $ {k 1 5 k 5 s}, s < j 5 d (note that s 2 2 if G - v has a bridge; this follows from b(G) > 2 and the fact that bc(G - v) is a non-trivial tree in this case). Observe that zk # yk since G is a simple graph.

+ [gJ/

Now introduce the vertices vi $ V(G - v), 1 5 i 5 d, and let vi be incident with ei,adjacent with wi-I and vi+I (putting vi-, = v d for i = 1

vIII.l. Double Tracings

vIII.ll

and vi+l = v1 for i = d). The graph Go thus obtained from G - v has the following properties: a) Go is a simple graph with 6(G,) > 2 (this follows from the very = 3, 1 5 i 5 d ) . construction of Go since dGo(vUi)

b) K(G,) > 1 because of K ( G )> 1 and the very construction of Go. c ) &(Go) > 3. Otherwise, consider E,, a cyclic edge cut of Go with I E, 15 3. We conclude that wivi+l, vjvj+l E Eo for some i and j , 1 5 i < j 5 d (putting vj+l = v1 for j = d ) ; this follows directly from d > 3, K(G) > 1 and Xc(G) > 3. Consequently, we can write the cycle C = ( { v n / n= 1,.. .,d } ) of Go in the form

C = Pi+i,j,vjvj+l,Pj+l,i, vivi+l, vi+l

7

where Pj+l,jand Pj+l,j are the components of C-{vjvj+l, Moreover, eo = ",yo € E, - {vivi+l,vjvj+l} is a bridge of both G - v and Go- (E, - { e o } ) . Because bc(G - v) is a non-trivial tree, it follows that G - v contains H , and H p such that

(i)

for every path in Go - (E, - { e , } ) , joining a vertex E of H , with a vertex of H p , 1 5 a < p 5 s ;

eo

(ii) both H , and H p are incident with precisely one bridge of G - v (note that G is a simple graph with 6(G) > 2); (iii) E(H,)

# 0 # E(Hp).

On the other hand, because of (ii), (iii) and the construction of Go, x,, y, E V(H,), x p , yp E V(Hp)exist such that v7 E V(Pi+l,j) if and only if v7+L$J E V(Pj+l,i),

where y = a,p. Note

(**I

V(C)= V(Pi+l,j)U V(Pj+l,j).

Now choose 'u6 E {v,, w , + ~ $ ~ } and ve E { u p , v P + L + l } such that (v6,

vE} G V(P,,,) for ( m ,n ) = (i

+ 1 , j )or (m,n ) = ( j + 1,i)

(see (**))

and consider the path P6,, joining vug and v, in Pm,n.It follows that z,vb E E(G,) - E, for z, = x, or z, = y,, and zpv, E E(G,) - E, for zp = xp or zp = yp. Consequently, := z,,

2ffv67 P6,e,

v,zp7 zp

VIII.12

WII. Various Types of Closed Covering Walks

is a path joining a vertex of Ha with a vertex of H p and not containing eo. This contradicts (i) and proves the validity of c ) .

If Go is a cubic graph with p

= 0 (mod4),then subdivide the edges 21~21, GP 7 and 2 1 ~ ~ 2 1 ~ + join ~ + , , the sublvision vertices by an edge, and denote this new graph by G , . Otherwise, let G , := Go. In any case, G , satisfies the hypothesis of the theorem, and if G , is cubic, pG, $ 0 (mod4). Moreover, G , satisfies (*). By induction, G , has a strong double tracing W, which readily can be transformed into a bidirectional double tracing W of G . If W is not a strong double tracing, retractings occur precisely at 21. Since d(v) > 3, W can be modified at v such that one obtains a strong double tracing of G (see the proof of Lemma VIII.8). The theorem now follows.

Theorems VIII.ll and VIII.12 are in a way best possible results. For, as Thomassen shows in his paper, there is an infinite number of cubic 3connected graphs G which have no strong double tracing although p G = 2(mod4). They contain, however, cyclic edge cuts Eo such that the components G , , G , of G - E , satisfy the congruence pGi 3 ( m o d 4 ) ,i = 1,2. In particular, if one replaces every vertex of a cyclically 4-edgeconnected cubic graph G with a triangle, the resulting graph has no strong double tracing. The study of a different type of double tracing had been posed in [WAGN70a, p.36, Problem 41, namely: consider Eo 5 E ( G ) , where G is a connected graph, and decide whether G has a double tracing W such that every eo E E, is used by W once in each direction, while every el E E(G)- E, is used twice in the same (not prescribed) direction. Call such W an E,-restricted double tracing. The solution of this problem is given by the following criterion, [VEST75a, Theorem 21. We present here a somewhat different proof.

Theorem VIII.13. Let G be a connected graph, and let E, 2 E ( G ) be chosen. G has an E,-restricted double tracing if and only if G - E, is eulerian. Proof. Suppose G has an E,-restricted double tracing W . Construct the digraph Dw from G by replacing every eo = zoyo E E, with two arcs of the form (%,,yo), (yo,zo), while every el = zlyl E E ( G ) - E, is replaced by either two arcs of the form (zl,yl) or two arcs of the form ( y , , ~ , ) , depending on whether W uses el from z1 to y, or from y1 to 5,. It follows immediately from the construction of D , that W corresponds to an eulerian trail Tw of D , and vice versa. That is, D,

VIII.1. Double Tracings

VIII.13

is an eulerian digraph. Since every eo E E, corresponds to an oriented digon C(eO) = (((xO, yo), (yo, x : O ) } ) Of D W , and since U e o E E o C(eO)is an eulerian subdigraph of D,, it follows that

is an eulerian digraph. By the definition of an eulerian digraph and by the construction of D, and D,, we have for every v E V ( G )

(note that e , E E, corresponds to two arcs with the same orientation). Hence G - E, is eulerian. Conversely suppose G - E, is eulerian. Let G , ,. . .,G,, k 2 1 , denote the components of G - E,,and let Tibe an eulerian trail of Gi, i = 1,.. . ,k. For every i = 1,.. . ,k, replace every e = xy E E(Gi) by two arcs of the form (z,y)or ( y , x ) depending on whether Ti passes e from x to y or from y to x. Denote by Dithe digraph thus obtained. Since for every v E V ( G i ) Ti , passes as many edges from v to elements of N ( v ) as it passes edges from elements of N ( v ) to V , it follows from the above construction that idD;(V) = OdD;(V) = dGi (V) for every v E V ( G i ) and i = 1,.. .,k. Thus, digraph, and therefore

k

D j is an eulerian

is an eulerian digraph. It follows from the very construction of D that any eulerian trail of D induces an E,-restricted double tracing W of G . The theorem now follows. In fact, it follows from the proof of Theorem VIII.13 that one can generalize this result to the extent that, in a certain covering walk W of G , every e E E, should be used me E N times by W in each direction, where me may vary as e varies, while every e E E(G) - E, should be used exactly m E N times by W in precisely one of the two possible directions (which,

VIII. 14

VIII. Various Types of Closed Covering Walks

however, is not prescribed). Calling such W an (Eo;me,m)-restricted covering walk we arrive at the next result, the proof of which is left as an exercise.

Corollary VIII.14. For a connected graph G, let E, C E(G), {rn,/e E E,} c N and rn E N be arbitrarily chosen. G has an (E,;rn,,rn)restricted covering walk if and only if G - E, is eulerian. We point out that instead of considering double tracings, or closed covering walks which use every edge precisely k times (possibly in different directions), one may ask whether a graph G has a V(G)-covering walk passing every vertex precisely k times. For k = 1 this is the question whether G is hamiltonian. Even if we let k > 1 be a fixed integer this problem remains NP-complete. However, if one asks whether there exists k E N such that G has a corresponding V(G)-covering walk, the problem can be solved in polynomial time (see [BROE88a]).

VIII.2. Value-True Walks and Integer Flows in Graphs K. Wagner’s problem, answered by Theorem VIII.13 and Corollary VIII.14, serves as the basis for a more general type of closed covering walks. They have been introduced in [FLE177a].

Definition VIII.15. Let G be a connected graph, and let an integer edge labeling cp : E(G) --f N U {0} be given. We call a closed covering walk W of G a value-true walk in G if for every e E E ( G ) , W uses e the same number of times in both directions provided cp(e) = 0, while W uses e exactly p(e) times in one direction and not at all in the other direction (where the direction is not prescribed) if cp(e) > 0. Moreover, for every i E N U (0) we denote Ei= {e E E(G)/p(e) = i} and call Ei the i-set (of cp).

Generalizing the proofs of Theorem VIII.13 and Corollary VIII.14, we arrive at the following sufficient condition for a graph to have a valuetrue walk (see also [FLEI77a, Theorem 31).

Corollary VIII.16. Let cp be a non-negative integer edge labeling for the connected graph G. If (Ei) is eulerian for every i-set Ei, i E N , there exists a value-true walk in G.

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.15

The proof of Corollary VIII.16 is left as an exercise. Note that Corollary VIII.16 reduces to Corollary VIII.14 if cp is a constant on E(G) - E,. Moreover, it should be noted that in Definition VIII.15 it would not have been a loss of generality, if we had only required that W uses e precisely once in each direction whenever cp(e) = 0. This can be seen from the fact that the addition/subtraction of digons to/from the eulerian digraph D, renders an eulerian digraph D,,where - in the case of subtraction - D, remains connected if D, is connected, as long as every e E E, is represented by a digon in D,. With this modification in mind, Corollary VIII.16 is reduced to Theorem V111.13, if cp(e) = 2 for every e E E(G) EO. The following criterion characterizes those functions cp which admit a value-true walk in a connected graph G . It also serves as the basis of some of the subsequent considerations.

Lemma VIII.17. Let cp be a non-negative integer edge labeling for the connected graph G. The following statements are equivalent.

1) G has a value-true walk. 2) An orientation

D of G - E, exists such that for every v E V(G) =

V(D)

where a, E A ( D ) corresponds to e E E(G) - E,.

Proof. If G has a value-true walk, one can, in principle, construct the digraph D , as in the proof of Theorem VIII.13, except that one replaces e = zy E E ( G ) - E, by cp(e) arcs of the form (z,y), (y, z) respectively, if W passes e from z t o y, y to z respectively, while for e = z y E E, one replaces e by as many arcs of the form (z,y) and (y, z) as W passes e in either direction. Similarly, W corresponds to an eulerian trail T, of D,, i.e., D , is eulerian. By construction, it follows for the digraph D E D , whose underlying graph is G - E,, that

for every v E V ( D )= V(D,) = V(G).

VIII. 16

VIII. Various Types of Closed Covering Walks

Conversely, suppose G - E, is the graph underlying a digraph D which satisfies the equation stated in 2). Construct the digraph D, by replacing every arc a, = (z,y) with p(e) arcs of the form (5, y), and for every zoyo E E, add an arc (",,yo) and an arc (yo,",). This construction yields for every ZI E V(D,) = V(D)

where A $ , A ; C A ( D ) . That is, D, is an eulerian digraph, and we conclude as before that an eulerian trail of D, corresponds to a valuetrue walk of G . The lemma now follows. Lemma VIII.17 allows us to characterize those non-negative integer edge labelings of G which admit a value-true walk, without resorting to digraphs.

Theorem VIII.18. Suppose for the non-negative integer edge labeling cp of the connected graph G that E ( G )- E, # 8. The following statements are equivalent. 1) G has a value-true walk.

(where 2) For every e = zy E E(G) there exist numbers s , ( ~ )and s, e(z) and e(y) denote the half-edges of e; put el = e(z), el' e(y) if z = Y) such that a) {se(z), S e ( y ) l = (1, -11, b) C secz)(~(e) = 0. e(z)EE;

Proof.

1) implies 2).

By Lemma VIII.17, there exists a digraph

D with G - E, as its underlying graph and satisfying CaeEAt p(e) cp(e) = 0. Consider a fixed z E V ( G ) . For e = zy E E,, define and s , ( ~ )arbitrarily subject to a); for e = zy E E(G) - E,, define s , ( ~ )= +1 if a , E A $ , and s , ( ~ )= -1 if a, E A;. Note that this includes the case where e is a loop; for, in this case A$ n A; 2 {a,}. This definition of se(%) and s , ( ~ )satisfies a). Observing that p(e) = 0 if e E E,, we may conclude that the above equation can be rewritten in the form

Ca,EA; s,(%)

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.17

The implication now follows. 2) implies 1). Equation b ) reduces to

Now replace every edge of the form e = xy by an arc of the form a, = (x,y) if s,(,) = +1 (hence s , ( ~ ) = -1 by a)); replace it by the arc a , = (y,x) if s,(,) = -1. In the case of a loop e = xy this means that (x,x) should be viewed as being oriented from e(x)’ to e(x)” if s , ( , ) ~ = +l. Doing this for every x E V(G) we can rewrite the above equation as

where A , A ( D ) for the digraph D whose underlying graph is G - E,. By Lemma VIII.17, G has a value-true walk. This finishes the proof of the second implication. The theorem now follows. Unfortunately, as simple as the statements of Lemma VIII.17 and Theorem VIII.18 appear, they reveal the difficulty one faces in deciding for a given (non-negative) integer labeling of the connected graph G whether G admits a value-true walk. For, it follows from the results just quoted, that at every vertex v E V(G) one necessarily has to partition A := E,* - {e(v)/v(e) = 0) into two classes A‘, A - A‘ such that

aEA’

aEA-A‘

where A‘ contains precisely those a = e(v) for which se(,,) = 1, say, if G has a value-true walk. For s := ‘p the preceding sentence is precisely the partition problem as quoted in [GARE79a, [SP12]]; and this problem is, according to Garey and Johnson, NP-complete. On the other hand, an obvious algorithm for deciding the existence of D, (see Lemma VIII.17.2)) consists in the construction of 29l-l digraphs D,, where q1 =I A(D,) I=I E(G)- E,,1, and which correspond to all orientations of G- E,,, where for each such orientation and its inverse orientation, one satisfies the equation of Lemma VIII.17.2) if and only if the other

VIII.18

VIII. Various Types of Closed Covering Walks

does (this is why we have q1 - 1 instead of q1 as an exponent). At each step of the construction of these digraphs D,,one has to decide whether the digraph obtained satisfies this equation. Moreover, even if we solve the corresponding partition problem at every v E V ( G ) ,this does not guarantee the existence of a value-true walk. This is demonstrated by the graph K4,p of Figure VIII.2 (=[FLEI77a, Figure 11). In fact, the partition problem for K4,phas a unique solution for every vertex: { { 1,2}, (3)). This and the fact that one can prescribe one of the arcs for constructing D, (see above) makes it simple to show that K4,(phas no value-true walk. For, if w.1.o.g. (u,w) E A(Dl), then (w, v), (w,t ) E A ( D , ) follows of necessity. But then we must also have (w, t ) E A(Dl). On the other hand, I A , I> 1 necessarily implies A t = { ( t , ~ ) a} ,contradiction to (v,t),(w,t) E A ( D , ) . Hence K4,(phas no value-true walk.

3

Figure VIII.2. The tetrahedron with an integer edge labeling 'p such that the partition problem has a unique solution for every z E V(K4,+,);but K4+ has no value-true walk. We introduce the following notation based on the equivalences expressed by Lemma VIII.17 and Theorem VIII.18. For the sake of brevity we denote 'p(A0)= C a . E A o 'p(e), where A, A ( D ) , and a, corresponds to e E E ( G ) in the graph G underlying D.

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.19

Definition VIII.19. For a (not necessarily connected) graph G and cp : E(G)+ N U {0}, we call cp an integer flow (or simply flow) in G if an orientation D of G exists such that for every 21 E V(D)

In particular, we call a flow cp a k-flow if maz,EE(C)cp(e)< k. For cp : E ( G ) + N U {0}, call cp a (mod k)-flow in G if 0

5 cp(e) < k for every

e E E(G)

and if an orientation D of G exists such that for every 21 E V ( D )= V ( G )

For cp being a ((mod k)-)flow, we say cp is a partial ((mod k)-)flow if E, # 0; otherwise, cp is called a nowhere-zero ((mod k)-)flow.

Remark VIII.20. 1) The term partial flow (see, e.g., [YOUN83a]) has been introduced in order to distinguish between E, # 0 and E, = 0. Most authors simply speak of flows instead of partial flows. 2) In the case of a k-flow, some authors define cp : E (G) + (0, fl,. . . , f ( k - 1)) (see, e.g., [SEYM81a781d]).The definition given here, however, is not a real restriction. For if cp is a k-flow of G in this more general sense, and if in an orientation D of G a , E A ( D ) is such that cp(e) < 0, the digraph D, obtained from D by replacing a , by an a x of opposite orientation, and p1 defined by cp,(f) = cp(f) for f # e and cpl(e) = -cp(e), also satisfy equation (1) in Definition VIII.19. On the other hand, and by the same token, the other definition of a k-flow is independent of a particular orientation; i.e., if (1) is fulfilled for cp in some orientation D of G , one obtains for any other orientation D, of D a k-flow p1 with I cpl(e) I=] cp(e) 1, e E E(G). However, the concept of a flow (k-flow) as defined above, is, by Lemma VIII.17, equivalent to the definition of a < k), provided G value-true walk (value-true walk with maz,EE(G)cp(e) is connected. This equivalence also implies that a graph with at least one bridge cannot have a nowhere-zero k-flow for any k E N . 3) By Definition VIII.19, every k-flow is also a (k 1)-flow. 4) In view of the first part of 2) and the definition of a (modk)-flow cp it follows that for an orientation D of G corresponding to cp, D, := ( D -

+

vIII.20

VIII. Various Types of Closed Covering Walks

{(x,y)}) U {(y, x)} is an orientation of D corresponding to the (rnodk)flow cpl defined by (PlIE(G)-{zy)

= cplE(C)-{q/}

%

Pl(4

= k - cp(xy)(modk) -

5 ) If p is a k-flow or a (rnodk)-flow of G and if D is an orientation of G corresponding to cp, then D R , the inverse orientation of D,also

corresponds to cp since the reversal of all arcs of D leaves (1) and (2) unaltered. Note that a graph G has a nowhere-zero 2-flow if and only if G is eulerian. For cubic graphs G one has an interesting relation between %flows, valuetrue walks in G and a structural property of G (see also [JAEG79a, Proposition 21).

Theorem VIII.21. Suppose for a connected cubic graph G with edge labeling cp that p(e) E {1,2} for an arbitrary e E E(G).G has a valuetrue walk (respectively, p is a nowhere-zero %flow) if and only if a) { e E E(G)/p(e) = 2) is a 1-factor of G7

b) G is bipartite. Proof. Suppose G has a value-true walk. Construct D with underlying graph G = G - E, according to Lemma VIII.17. We have for every v E V ( D ) and Ai,,:= { a e E A,/p(e) = i}, i = 1,2, that

and

either AS = A l , v , A , = A 2,v

Or

A$ = A2,v7 A, =

(*>

This follows from the assumption concerning G and p and from Lemma VIII.17: i.e., every vertex of G is incident with precisely one edge having label 2. This proves a). In D define a vertex partition {V,, V,} by

This vertex partition is well defined because of I A, I= 1, v E V ( D ) . If there is a , = (z,y) E A(D) with p(e) = 1 and {z,9) G Vp7P E (0,i}, at both x and y the arcs af(+)and a f ( v )with cp(f(z))= 2 = cp(f(y)) satisfy

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.21

s

A , A; either I a f ( + ) 7 a f ( v ) } c A: A: (if P = O), Or I a f ( + ) 7 a f ( v ) } (if p = i). In any case, one has a violation of the equation of Lemma VIII.17.2) at one of the vertices z,y. This contradiction and E, = 8 imply that G is indeed bipartite. Conversely, suppose G satisfies a) and b). Since G is connected, the = 0) is uniquely devertex bipartition {V,,y](i.e., E((V')) = E((y)) termined. Construct D by replacing every edge e satisfying cp(e) = 2 by the corresponding arc a , oriented from V, to K, while the edges f with ~ ( f= ) 1are replaced by the corresponding arc a oriented from V, to V,. By a) and the definition of D, at every E V(D5 we have fulfilled equations (*) (see the first part of the proof). This is tantamount to saying, however, that D satisfies Lemma VIII.17.2). Thus G has a value-true walk (respectively, a nowhere-zero %flow). This finishes the proof of the theorem.

Of course, one can drop in Theorem VIII.21 the hypothesis of G being connected, if one cares for %flows only.

In fact, Lemma VIII.17 and Theorems VIII.18 and VIII.21 lead us to an alternate characterization of those G and cp which admit a value-true walk. W.1.o.g. we assume E, = 8. Corollary VIII.22. For a connected graph G with positive integer edge labeling cp the following statements are equivalent. 1) G has a value-true walk (respectively, cp is a nowhere-zero flow). 2) A bipartite graph H exists with vertex bipartition {V,, V,) and with G such that integer edge labeling c p H , and a homomorphism $ : H

-,

a) ~!,t acts bijectively between E ( H ) and V ( G ) ,i = 1,2;

E(G),and between V, and

b) cpH(e)= cp($(e)) for every e E E ( H ) ; c ) The function : V ( H )+ N , defined by a(v) := CeEEw(II) cpH(e), satisfies a ( v l ) = a ( v 2 )if $(TI,) = +(v2), v1 E V,,v2 E V,. We leave the proof of Corollary VIII.22 as an exercise since it can be deduced easily from the aforementioned results and their proofs.

As we have said above, a pfold solution of the NP-complete partition problem is a necessary condition for deciding whether G has a value-true

vIII.22

VIII. Various Types of Closed Covering Walks

walk with respect to the non-negative integer edge labeling ‘p (respectively, whether cp is a flow). Another condition which we call Seymour’s Cut Condition, also proves useful in this context. We say G and ‘p : E(G) + N satisfy this condition if and only if for every edge cut

ES a)

’p(~,)

:=

C p(e) = o (mod21

,

eEEs

b)

1

maz{’p(e)le E

Ed Ip ( E d

Corollary VIII.23. Every nowhere-zero flow

‘p

*

in the connected graph

G, i.e., every cp : E(G)t N admitting a value-true walk in G, fulfills Seymour’s Cut Condition. Proof. For a value-true walk W of G construct D according to Lemma VIII.17.2); its underlying graph is G since ‘p(e) > 0 for every e E E(G). Construct D, from D as in the proof of Lemma VIII.17. Since D, is an eulerian digraph it follows that for every arc cut A, of D,, a vertex partition {Vo, K} exists such that in D, U+(V0)

= u - ( Y ) = a+(V;:) = .--(Vo) .

This is tantamount to writing in D

Consequently, by denoting E, the edge cut in G corresponding to A, we obtain

Thus part a) of Seymour’s Cut Condition is fulfilled. Assuming w.1.o.g. u f E A+(Vo)for some f having maximal p(f) with respect to E,, we obtain from the preceding equations

The corollary now follows.

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.23

Remark VIII.24.

Seymour’s Cut Condition, first formulated in [SEYM79a9(3.5)],was introduced in the context of cycle covers in planar graphs rather than for the sake of studying certain closed covering walks. While it is an easy to prove necessary condition for the existence of value-true walks, it is an absolutely non-trivial necessary and s a c i e n t condition for the existence of value-true cycle covers (i.e., cycle covers S in which every edge e belongs to precisely cp(e) elements of S) in planar bridgeless graphs and in the context of compatible cycle decompositions. We mention these facts as a justification for the introduction of this condition at this point (Corollary VIII.23)) but also because of its relevance in the context of the next result. There, an oriented cycle cover of the graph G is a cycle cover whose elements can be oriented in such a way that any edge is given the same orientation in all the cycles it belongs to.

Corollary VIII.25. A connected graph G with positive integer labeling has a value-true walk if and only if G has a value-true oriented cycle cover. Proof. Firstly, construct the digraph D, for a value-true walk W of G (see the proof of Lemma VIII.17). Since D, is eulerian it has a cycle decomposition S, (Theorem IV.8). Precisely because cp(e) # 0 and on account of the construction of D,, it follows that every cycle of S, corresponds to a cycle of G. Hence S, corresponds to a cycle cover S of G using every edge p(e) times. Thus, S is a value-true cycle cover. Moreover, S turns into an oriented cycle cover, if every element of S is oriented in accordance with the orientation of the corresponding element of s,. Secondly, if G has a value-true oriented cycle cover S, one can associate with S a digraph D, which is obtained from G by replacing e = xy with cp(e) arcs a,, all of which are oriented from z to y or from y to z in accordance with the orientation of the elements of S containing e . Thus S corresponds to a cycle decomposition S , of D,, i.e., D, is eulerian (Theorem IV.8); it is also connected since G is connected and p(e) # 0, e E E ( G ) . Consequently D, has an eulerian trail To which passes a l l of the above arcs of the form a, from x to y or all of them from y to z. That is, To corresponds to a closed covering walk W using every e E E(G) cp(e) times in the same direction; i.e., W is a value-true walk of G. Corollary VIII.25 follows. Again, Corollary VIII.25 renders a true statement, if one drops the hypothesis of G being connected and considers nowhere-zero flows instead of

VIII.24

VIII. Various Types of Closed Covering Walks

value-true walks. The above formulation of Corollary VIII.25, however, demonstrates its f i a t i o n with Theorem IV.8. In view of Corollaries VIII.23 and VIII.25 we can say that if a graph G with positive integer edge labeling cp has a value-true oriented cycle cover, it satisfies Seymour’s Cut Condition. This is true, even if G is disconnected since this condition is vacuously fulfilled if E, = 0. On the other hand, because of what has been said concerning K4,pof Figure VIII.2 and because of Corollary VIII.25, IT4,+,has no value-true oriented cycle cover although it satisfies Seymour’s Cut Condition. However, K4+ has a value-true cycle cover (see Remark VIII.24). If one considers, correspondingly, value-true walks in digraphs D , where cp : A ( D ) --t N , then Lemma VIII.17.2) and its proof show that the problem reduces to the examination whether the equation in that lemma is fulfilled at every vertex (actually it suffices to perform this examination at any p - 1 vertices; for, if that equation holds for each of these p - 1 vertices, it must hold for the p t h vertex as well). And, similarly, Corollary VI11.25 and its proof show, that in the case of a weakly connected digraph, value-true walks and value-true oriented cycle covers are equivalent concepts. As for value-true walks in mixed graphs, one encounters basically the same difficulties as in the case of graphs (see the discussion following the proof of Theorem VIII.18). However, the results established for graphs so far, hold similarly true in principle for mixed graphs. We leave it as an exercise to rephrase these results correspondingly.

On the basis of the difficulties connected with the problem of deciding the existence of value-true walks, flows respectively, for given cp : E ( G ) -+ N U (0) (see the discussion following Theorem VIII.18 and preceding Definition VIII.19), it should come as no surprise that this problem has attracted little attention. Apart from the two papers cited there seems to be only one other article, [SCHM79a], dealing with the issue. In fact, R. Schmidt’s starting point is - implicitly - the consideration of such non-negative integer edge labelings cp of G for which the partition problem has a solution at every vertex (in which case he calls cp starconforming), respectively, for every minimal edge cut (in this case he calls cp bond-conforming).6) Consequently, any flow cp : E(G) N U ---f

Since E, is the disjoint union of certain minimal edge cuts if G has no loops, it follows that every bond-conforming cp is also star-conforming. However, R.Schmidt fails to mention this fact. 6,

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.25

(0) is star-conforming as well as bond-conforming ([SCHM79a, Satz 11; see also the above discussion on the partition problem and the proof of Corollary VIII.23). The questions considered in this article are basically the following: 1)

2)

Which connected graphs G admit a nowhere-zero flow ? Which connected graphs have the property that every starconforming (bond-conforming) cp is a flow ?

In order to answer these questions, observe first that if e is a bridge of G , cp(e) = 0 must hold true for every flow cp, whence we may assume X(G) 2 2. In this case, we can prove the following folklore result. The proof presented here differs from the proof in [SCHM79a]. However, the following proof will be of relevance to the discussion of the Directed Postman Problem.

Proposition VIII.26. For every connected bridgeless graph G there exists a nowhere-zero flow cp, i.e., cp admits a value-true walk in G. Proof. By Lemma 111.22, a strongly connected digraph D exists with underlying graph G. Define diff(v) := o d D ( v ) - i d D ( v ) . If diff(v) = 0 for every v E V ( D ) , D is eulerian. Defining cp(e) = 1for every e E E ( G ) ,we may conclude that any eulerian trail T of D corresponds to a value-true walk W in G (W is an eulerian trail in this case). Whence we may suppose diff(v) # 0 for at least one v E V ( D )= V ( G ) .Since CvEv(D) diff(v) = 0 for every digraph, we may conclude that there are v , w E V ( D ) such that diff(v) < 0, difl(w) > 0. Since D is strongly connected, a path P ( v , w ) runs from v to w in D. We double the arcs along this path and call the digraph thus obtained D,.Of course, D, is strongly connected because of D C D,. Moreover, for C ( D ) := Cvcv(D) I d i f f ( v )1, we have C ( D , ) < C ( D ) by construction. Applying induction, we conclude for D, that a (strongly) connected eulerian digraph D, is obtained from D, by duplicating arcs along properly chosen paths of D,. Since D, had been obtained from D by such an operation, we may conclude that D, 3 D has been obtained from D by replacing certain arcs by X(a,) arcs of the same orientation. For every itrc a, E D define

VIII.26

VIII. Various Types of Closed Covering Walks

It follows from the construction of D, D,respectively, that every eulerian trail T of D, corresponds to a value-true walk in D with respect to ' p D and hence to a value-true walk of G with respect to cp. Also by construction, p(e) > 0 for every e E E(G). Thus, 'p is a nowhere-zero flow in G. This finishes the proof.

As for the second question, R. Schmidt answers it completely; but not surprisingly, these classes of graphs (called star-faithful, bond-fait hful respectively) are rather narrow classes of graphs. We present these results without proof in abbreviated form, [SCHM79a, Satz 2, Satz 31. Denote by G(A, n), n 2 3, the graph which has a cycle decomposition SA = {Ci/i = 1,.. . ,n } into triangles such that I V(C,)n V(Cj) 15 1 for i # j, 1 5 i,j 5 n, and the intersection graph I(SA)is a cycle. Note that G(A, n) is uniquely determined (up to isomorphisms) for every n E N - {1,2} (the graph H, of Figure VI.31 corresponds to G(A,3)). Furthermore, denote by GA,, the graph consisting of two disjoint triangles and one edge joining them. Theorem VIII.27. Let G be a connected simple graph.

1) G is star-faithful if and only if G contains no subdivision of either of the graphs GA,A, G ( A , 3), K4 as a subgraph. 2) G is bond-faithful if and only if G contains no subdivision of either K4 or G(A, n), n 2 3, as a subgraph. Noting that G(A, n) contains a subdivision of GAIA for every n 2 4, it follows that every star-faithful graph is also bond-faithful (this conclusion, however, follows already from the definition of being star-, bond-faithful since a bond-conforming 'p is also star-conforming). Moreover, the graphs G in Theorem VIII.27.1) are precisely those which have a vertex 21 belonging to all cycles of G. To show that a bond-faithful graph need not be star-faithful, we may consider the graph Go of Figure VIII.3, where cp is defined by the edge-labeling of Go. Note that 'p is star-conforming but not bond-conforming since the partition problem has no solution for { 1,3}, where we consider the edge cut consisting of the two edges which do not belong to any triangle of Go. Hence Go is not star-faithful (in fact, it contains GA,, as a subgraph), but it is bond-faithful since it contains no subdivision of K4 nor of G(A, n), n 2 3, as a subgraph. Looking at the function cp as defined in in the proof of Proposition VIII.26

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.27

1

Go Figure V111.3. Go with cp : E(Go)-+ N . cp is starconforming but not bond-conforming. Thus Go is not starfaithful; but it is bond-faithful indeed. one can only say that

For,

c

diff(U)>O

=

uEV(D)

c

diff(u)
di#(v) - 2

&iff(.)

= O(mod2)

d i f f( u )
since C u E V D ) di#(v) = 0. Moreover, C(D,) = C(D)- 2 since di#(z) remains unc anged in constructing D, from D provided x # v,w, and I diffDlz I=I diffDz 1 -1 for z E {v,w}. Thus, by introducing additional arcs corresponding to the arcs of P ( v , w ) , every arc of D corresponds either to 1 or 2 arcs of D , . Repeating this argument we may conclude that every arc of D corresponds to at most i C ( D ) additional arcs in D,. The upper bound for cp given in [SCHM79a] is in general much larger than + C ( D )+ 1. However, we are interested in the number

6

P(G,CP) := m'npmazeEE(c)cp(e)

7

where the minimum is taken over all nowhere-zero flows cp and where G is bridgeless. In this context, we have the following conjecture.

Conjecture VIII.28 (Tutte's 5-Flow Conjecture). For every bridgeless graph G, p(G, cp) 5; i.e., every bridgeless graph has a nowhere-zero

<

5-flow.

VIII. Various Types of Closed Covering Walks

VIII.28

Note that by Remark VIII.20.3), if G has a k-flow, 1 5 k a 5-flow.

5 5 , it also has

This conjecture, put forward in [TUTT54a, Conjecture 111, arose originally in the context of coloring problems. At present, it is considered one of the most outstanding unsolved problems in graph theory and has, therefore, attracted a lot of attention (for a thorough account on flows in graphs and its various applications, see Jaeger’s survey article [JAEG88a]). Before dealing with various results on k-flows, k 2 3, we show that the distinction between k-flows and (mod k)-flows is a matter of practicality rather than of essence (see [TUTT49a, Theorem IV], [YOUN83a, Proposition 31). The following proof proceeds along the lines of the latter reference.

Proposition VIII.29. Let G be a graph and let k be a positive integer. The following statements are equivalent. 1 ) G has a (nowhere-zero) k-flow with O-set E,,.

2) G has a (nowhere-zero) (mod k)-flow with O-set Eo. Proof. Since a (nowhere-zero) k-flow is a (nowhere-zero) (mod k)-flow by definition. we assume that G has a (nowhere-zero) (mod k)-flow ‘p. We assume w.1.0.g. that Eo = 8 (otherwise, the following arguments can be applied to G - Eo instead). Now, G has an orientation D such that for every v E V ( D )= V ( G )

ac€ A ;

a.E&

Assuming w.1.o.g. that G is connected we now proceed in a way similar to the proof of Proposition VIII.26. If diff,(v) = 0 for every v E V ( G ) , ‘p is a (nowhere-zero) k-flow with D as an orientation of G corresponding t o p. Whence we may assume

c

C,(D) :=

Idiff,(v)l> 0 .

vEV(G)

Since

CvEv(D) dZ#,(v)

= 0 for every cp : E(G) + N (by analogy to

d i f f ( v )= 0 in the proof of Proposition VIII.26), we may conclude that C v C V (D )

difl,(z)

> 0,

difl,(y)

<0

for some z,y E V ( D ) .

(*I

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.29

For a fixed z with difl,(z) > 0, consider R, := {v E V ( D ) / 3 P ( x , v ) } ; z E R, follows. We claim that there is y E R, with di#,(y) < 0. Suppose no such y E R, exists; then R, C V ( D )by (*). By definition of R, it follows that A,, the cut set separating (R,)Dfrom ( V ( D )- R , ) D , contains no arc (u,v) with u E R,. Now consider the tie-up D(A,) of D with respect to A, (see Figure IV.2). Since G is connected D ( A , ) has precisely two components D,, D,, with the notation chosen w.1.o.g. in such a way that R, U { z , } = V(D,). By the above, idD,(z,) = 0. Label the arcs a,,, E A$ corresponding to a , E A, (where e E S for the corresponding cut set S of G) with cp(e). Consequently, difl,(v) 2 0 for every v E V(D,) by assumption and because difl,(x) > 0, difl,(zl) > 0. This contradicts CVEV(DI) d$,(v) = 0. Hence y with dzfl,(y) < 0 exists in R,. Now let P ( x , y) be a fixed path joining x and y in D,and define := (D- A ( P ( x ,Y)))

Do

cpl(e)=

For every

ZI

{k

cpw

- p(e)

u {."/a

E A(%

9)))

7

if a, E A(D,) n A ( D ) , otherwise.

E V(D,) - V ( P ( x y,)),

dZfl,,(v)

= di#,(v)

.

The same equation holds, however, for w E V ( P ( x y)) , - {x,y} since for a,+ E A ( P ( x , y ) )n A$ and a,- E A ( P ( x ,y)) n A ,

Consequently,

c,, (Do) = q m - 2k < C,(D) (note that difl,(x) = X,k, difl,(y) = -X,k for certain X,,X, E N). It follows from the definition of p1 and Do that cpl is a (nowhere-zero)

VIII. Various Types of Closed Covering Walks

VIII.30

(mod k)-flow in G. If CP1(Do) = 0, cpl is a (nowhere-zero) k-flow in G; otherwise, apply induction to Do and cp, to obtain a (nowhere-zero) k-flow in G. Note that this transition from a (modk)-flow to a k-flow does not alter Eo if Eo # 8, provided one sets k - cp(e) = 0 for cp(e) = 0 in the above definition of 9,. The proposition now follows.

In fact, the above transformation of (D, cp) into (Do ,cpl) and CP1( D o )= C J D ) - 2k show that it takes $Cv(D) transformations of this type to obtain a nowhere-zero k-flow from a nowhere-zero (mod k)-flow. Also, Proposition VIII.29 permits us to consider arbitrarily in the ensuing discussion either k-flows or (mod k)-flows. Noting that the value of k in the k-flow constructed in the proof of Proposition VIII.26 depends, in fact, on the strongly connected orientation D of G chosen (see also the discussion preceding Conjecture VIII.28), we may reinterpret Tutte’s 5-Flow Conjecture in the following equivalent form: every connected bridgeless graph G has a strongly connected orientation D which can be transformed into an eulerian digraph D, by adding to every arc of the form (x,y) at most three arcs of the form (x,y) (and without adding any arc of the form (u,v) if (u,v) 6 A ( D ) ) . But what led to the formulation of Conjecture VIII.28 ? First of all, we take note of the following equivalence. Proposition ‘VIII.30. The following statements are equivalent. 1) Tutte’s 5-Flow Conjecture. 2) Every 3-regular 2-connected graph has a nowhere-zero 5-flow.

Proof. Since 2) is nothing but a special case of 1) it suffices to show that 2) implies 1).Suppose to the contrary that 2) is true, whereas there exists a bridgeless graph G admitting no nowhere-zero &flow. Choosing G with minimal p , qG and observing that G has a nowhere-zero k-flow if and only if each component of G has such a flow, we may conclude that G is connected. By the choice of G we may conclude that V2(G)= 0; otherwise construct for v E V,(G)and E, = {xu,yv} the graph H = (G - u) U {ezy}, ezy 6 E(G),ezy E E,(H) n E,(H). H is bridgeless and pH q H < pG qG; thus H has a nowhere-zero 5 - f l 0 ~‘ p H which is transformed into a nowhere-zero 5-flow cp of G by letting

+

+

+

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.31

(a corresponding orientation D of G is obtained by subdividing ueZy in D,, a corresponding orientation of H). This contradiction proves V2(G) = 8. By assumption and the choice of G, it follows that A(G) > 3. Apply the Splitting Lemma to some v E V(G) with d(v) > 3; w.1.o.g. G,,, is connected and bridgeless. For the graph H homeomorphic to G,,, and satisfying Q ( H ) = 0 we have p H q H < pG qG. By the same argument used above, we see that a nowhere-zero 5-flow of H can be transformed into such flow p,,, of G,,,. However, since E(G1,,) = E(G), we may conclude that v,,~is the more a nowhere-zero &flow cp of G, where a corresponding orientation D of G is obtained from a corresponding orientation D,,, of G,,, by identifying v,,, and v in D,,,. This h a l contradiction proves Proposition VIII.30.

+

+

In view of Proposition VIII.30 and Theorem VIII.21 one is led to ask which cubic graphs admit a nowhere-zero 4-flow. The answer is provided by the following result (see, e.g., [TUTT49a, Theorem I], [MINT67a], [JAEG76a, 79al). We present a proof similar to the one given in [MINT67a].

Theorem VIII.31. For a cubic graph G, the following statements are equivalent. 1) G has a l-factorization.

2) G has a nowhere-zero 4-flow. Proof. In view of Proposition VIII.29 and Remark VIII.20.4) it suf6ces to construct and to use (modk)-flows. Thus we shall construct orientations of G whose arcs carry the labels l and 2 only. Now suppose G has a l-factorization {L,, L,, L,}.(L,UL,) is a bipartite 2-factor Q of G, i.e., a set of disjoint even cycles spanning G. Let {V,, V,} be a bipartition of V ( Q ) (i.e., E((V,)) = E((V,)) = 0 ) and orient the elements of E(Q) from V, to V,. The digraph D, thus obtained has V, as its set of sources and V, as its set of sinks. Orient every e = xy E L, arbitrarily either from x to y or from y to x, and denote by A, the set of arcs corresponding to L,. Define cp : E(G) + (1,2} by cp(e) = 1 t+ e E E ( Q ) , cp(e) = 2

eE

t)

L, .

In D = DQ U A, we have for every v E V ( D )= V ( G )

VIII. Various Types of Closed Covering Walks

VIII .32

This follows immediately from the definitions of D and cp. Hence G has a (mod4)-flow with values in { 1,2). Finally suppose G has a nowhere-zero (mod4)-flow 9,.Let D, be an orientation of G corresponding to 'p,. If cpl(e) = 3 for some e E E ( G ) , then replace a, in D, by its oppositely oriented arc a:. Whence we can consider D = (Dl - { a e / P l ( e ) = 31) u { a f / ~ l ( e )= 3) and define cp : E ( G ) + {1,2} by cpl(4 E {172)

+

4 4 = PI(.>,

cpl(4 = 3

cp(4 = 1

*

It now follows from Remark VIII.20.4) that D classifies cp as a (mod4)flow of G. Consequently, for the i-sets Eiof this cp, i = 1,2, we have

IE, nEJ= 2, IE2nE,I= 1 for every v E V ( G ) . That is, Q := { e E E(G)/cp(e) = 1) is a 2-factor of G. Moreover, by definition of a (mod4)-flow and since d(v) = 3

{e,f}=E,nE,*({ae,af}~A~or{ae,af}~A;) This in turn implies that the components of Q are even cycles (for, the above equivalence implies that DQ has sources and sinks only, i.e., is bipartite). So we can write E(Q) = L, U L,, where L1 and L, are 1factors of G , which together with L, := { e E E(G)/cp(e)= 2 ) yields a 1-factorization {L,,L,, L,} of G. The theorem now follows. Consequently, in arbitrary bridgeless cubic graphs one cannot hope for a nowhere-zero k-flow with k < 5 . On the other hand, the Petersen graph has a nowhere-zero 5-flow although it is not 1-factorable (see Figure VII I.4). Because of Remark VIII.20.4) and Proposition VIII.29 a cubic graph G has a nowhere-zero 5-flow if and only if there exists cp : E ( G ) --t {1,2} and an orientation D of G such that dig (v) E { 0 , f 5 } for every v E V ( G ) ( 0 ) . The construction of a (mod57-flow of P5 as demonstrated by Figure VIII.4 is in line with ( 0 ) which in turn leads to the following equivalence.

Theorem VIII.32. A cubic graph G has a nowhere-zero &flow if and only if G can be decomposed into bipartite factors Gjwith (vertex) bipartition {Vl,j,t'&}, i = 1 , 2 , such that for j = 1 , 2 zE

y,, n vj,2

if and only if

d G z ( z )= 2

.

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.33

1

Figure VIII.4. An orientation of the Petersen graph P5 and an arc labeling describing a nowhere-zero (mod5)-flow of P5. Proof. For a nowhere-zero &flow of G consider, equivalently, D and 'p as described in the above statement (0). Denote by Gjthe subgraph of G induced by Ei, the i-sets of 'p, i = 1 , 2 . Since G is cubic it follows that d G i ( v ) # 0 for every v E V ( G ) and i = 1,2; i.e., Gi is a factor of G for i = 1,2. Considering Di, the subdigraph of D with G j as its underlying graph, we conclude from (0)that i d D i ( v )= 0 if and Only if

OdDi(V)

#0

for every v E V ( D i ) ,i = 1 , 2 . That is, every vertex of D;is either a source or a sink of Diwhich implies that Gi is bipartite. Define a vertex bipartition of G j by

i = 1,2. That is, Vl,i (V2,+) is the set of sources (sinks) of D;.Observing now that by ( 0 ) a source (sink) of D . is also a source (sink) of D,,{ j ,k} = 1,2, if and only if di#,(v) = 5 (-5j, it follows that x E V&

n Vm,2 if and only if

dGz(x) = 2, m = 1 , 2 .

VIII.34

VIII. Various Types of Closed Covering Walks

Conversely, let G i , i = 1,2, be given with the properties stated in the theorem. Define an orientation Di of Gi by

Since E(G) = E ( G , ) U E(G,) and E(G,) n E(G2)= 0, an orientation D of G is given by D := D, U D,. Now define cp(e) = i

if and only if

e E E(G,)

Because of Proposition VIII.29 it suffices to show that D and cp represent a (mod5)-flow of G . In order to see this, consider first x E n for any j E { 1,2}. It follows that x is a source of D if j = 1 , respectively a sink of D if j = 2; this follows from the definition of Di, i = 1,2. Moreover, since by assumption d G 2 ( x ) = 2 (= dGl(x) 1) in this case, we conclude difl,(x) = 5 . ( - 1 ) j - l 0 (mods) .

y,l y,?

+

Finally suppose x E Vl,l n V2,2or x E V2,1n V1,,. This and the definition of D j , i = 1,2, imply that z is a source of D j if and only if x is a sink of D,, { j , I c } = {1,2}. This and dG1(x)= 2 = d G 2 ( x ) 1 yield

+

difl,(x) = 0 if

x E y,, nv,,,,

Theorem VIII.32 now follows from

{j,k} = {1,2}

.

(0).

We note that in Theorem VIII.32, if one G j consists of cycles, G, is decomposable into two 1-factors of G , while G, is a 1-factor of G, { j , k} = { 1 , 2 } . That is, in this case G admits even a nowhere-zero 4-flow (Theorem VIII.31). Another equivalence with Tutte's 5-Flow Conjecture has been established in [JAEG83b]. There, Jaeger associates with every orientation D of a cubic graph G a vector

where V ( G )= {q, . . . ,wn}, and defines

B(G) = { z D / D is an orientation of G } . We state the main result of [JAEG83b] without proof.

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.35

Theorem VIII.33. For a non-bipartite cubic graph G the following statements are equivalent.

1) G has a nowhere-zero &flow. 2) There exist zD1,x D 2 ,z D 3E B(G)such that (-l)izDj, i = 1,2, j = 1,2,3, are six coplanar points in general position.

However, it should be noted that, as Jaeger points out, the requirement that G not be bipartite is necessary in order to avoid the existence of z D 1 , z DE 2 B(G)such that (-l)”zDj,1 5 i , j 5 2, are four colinear points. However, because of Remark VIII.20.3) and Theorem VIII.21 this requirement is not essential with regard to Conjecture VIII.28. The first big step towards settling Conjecture VIII.28 was made by Jaeger who showed in 1975 that every bridgeless graph admits a nowhere-zero 8-flow (announced in [JAEG76b] and explicitly proved in [JAEG79a]), thus proving that every bridgeless graph has a nowhere-zero k-flow for some k > 0 (see [TUTT54a, Conjecture I]). The same result was obtained independently and simultaneously by Kilpatrick (see [JAEG87a]). However, a few years later Seymour was able to improve Jaeger’s result, [SEYMSla]. We shall proceed along the lines of Seymour’s proof (except for some unessential modifications which are due to the different - but equivalent - definitions of a flow), but in order to make the proof more transparent we fist prove two lemmas (which, in fact, occupy the largest part of Seymour’s proof - see [SEYM8la, Proof of (3.1) and (3.2)]).

c

Consider E E(G)for a graph G. For k E N let (E)k defined by the following properties:

E(G)be

l) 2) No cycle C of G satisfies 0

<

3) Subject to 1) and 2 ) , I ( E ) , I is minimal.

15 IC;

E = E(G) satisfies 1) and 2). Moreover, for any E,E’,E” E(G) satisfying E E‘ and E El’, if El, El’ have property 2) (with El, E“ respectively, in place of E”’ = E‘ n El‘ 2 E also has property 2) since

c

c

IE(C)-E1”121E(C)-FI

for any choiceof

FE{E’,E’’}

,

with equality holding if E(C)fl E‘ = E(C)fl E”. Consequently (E)k is uniquely determined.

VIII.36

VIII. Various Types of Closed Covering Walks

w e dso note that the transition from E to (E)k is a closure operation since

E c (E)k,F = ( F ) , and (E)k ( F ) ,

if F = ( E ) k if E c F c E ( G ) .

F ; and For, if F = ( E ) k , F already satisfies 2), and in any event F E F implies E E (F)k by l), whereas the above consideration (with E' = (E)k, E" = (F)k) and 3 ) imply (E)k n ( F ) k = (E)k. Recall that for a flow cp, Ei denotes the i-set of cp, i E N U (0).

c

Lemma VIII.34. In a simple 3-connected graph G, totally disjoint cycles C,, . . . ,C,,T 2 1, exist, such that (U:=, E(C,)), = E(G). Proof. Let C, be a cycle of G; denote F, = E(C,)and consider (Fl)2. Suppose G, := ((Fl)2) is disconnected. Let Gcl c G, be the component containing C, and denote Go = G, - Gcl. Since E(GcI) c (F1)2,a cycle C C G exists such that 1 51 E(C)- E(GcI) 15 2. However, 1 #I E(C)-(F,), I# 2 by definition of (Fl)2.It follows that E ( C ) (F1)2 and thus E(C)nE(GcI) = 0 since Gcl nGo = 0. This implies that E(C) is either a loop or a pair of multiple edges. However, this is impossible since G is a simple graph by hypothesis. Consequently, G, is connected. Now let T- E N be as large as possible such that there are totally disjoint cycles C,, . . . ,C, in G having the property that G , := ((U:='=, E(Ci))2) is connected. We want to show that G , = G. Supposing V(G,) c V ( G )we consider Go := ( V ( G )- V(G,)) # 0. u E V(G,) cannot be adjacent to distinct vertices w,,w2 E V(G,): otherwise, since G, is connected by construction, a path P(w,, w q )c G, exists and thus a cycle Co in G satisfying

This contradicts the definition of E(G,)= (Uk, E ( C i ) ) 2 .It thus follows that every vertex of Go is adjacent to at most one vertex in G, which yields 6(Go) 2 2 since 6(G) 2 K ( G )2 3 and G is simple by hypothesis. Since Go has no end-vertices, it means that Go contains a 2-connected )2 3 as G is simple. Since at most one vertex of B, end-block B,;I V ( B o 1 belongs to Go - B, , it follows from K ( G )2 3 that z,, x 2 E V(B,) exist such that 'Bo

(xi) = 'Go (xi), = 172

7

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.37

and zi is adjacent to yi E V(G,), a = 1,2. Since every vertex of Go is adjacent to at most one vertex of G,, we conclude from K(G)2 3 that w.l.0.g.

On the other hand, B, contains a cycle C,+,with z,, x2 E V(C,+,), and

by definition of Go. Considering paths P ( z , G, we construct a cycle Cr,owith

z2) c

C,+,and P(y,, y , ) c

Since

I E(Cr,o)- (E(Gr)U E(Cr+l))I= we conclude from (1) that {C17. . . C,,,} is a set of cycles such that {el, e 2 } C (UI=',' E ( C i ) ) 2 . Define G,+, := (

2 T

+ 1 totally disjoint

( UE~( C~i ) ) ~ 2 and ) note that

is a connected subgraph of G,+, (G, c G,+l since UG, Cic UIzt Ci) and thus lies in a component H , of G,+,. We conclude as in the case of G, that Ho = G,+, since G is simple. Hence G,?, is connected and contradicts the maximality of T . Thus V(Go)= 8, i.e., V(G,)= V(G). Moreover, if q E E(G) - E(G,),

is a cycle of G for some path P,(y, z)

G,, and

O
,

contradicting the definition of G,. Hence G, = G. This finishes the proof of the lemma.

VIII. Various Types of Closed Covering Walks

VIII.38

Lemma VIII.35. Let G be a graph, and let k E N . Suppose for some F E E(G) that (F),= E(G). Then G has a (k 1)-flow cp such that F 2 E,.

+

Proof. We proceed by induction on nF :=I E(G) - F I. For nF = 0, cp : E(G) t (0) is a (k 1)-flow which is nowhere-zero on E(G) - F = 0, for every k E N (in this case, any orientation of G corresponds to cp).

+

Suppose nF > 0; then F c (F),= E(G). Consequently, by property 2) of the definition of ( F ) , a cycle C C G exists such that

0
.

(1)

Thus F C F‘ := F U E(C);this and ( F ) k = E(G) imply (F’)k = E(G). Since nFl < nF we may conclude by induction that G admits a (k 1)flow cp with F’ 2 E,. By definition of F’ we have

+

E(C) - F = F‘ - F E E(C) .

(2)

+

If (F’ - F)n E, = 0, then F _> E, and cp is a (k 1)-flow as required. Whence we can assume (F’ - F)n E, # 8. Consider an orientation D of G corresponding to cp. Suppose first that C corresponds to a cycle of D. (F’ - F)n E, # 0 and (1),(2) yield the existence of some n E (1,. . .,k} such that cp(e) # n for every e E E ( C ) - F . (3) Define the integer-valued function cp, : E(G) t {-n, 0) by cpn(e) = -n

if

eE

E(C), cp,(e)

=0

otherwise

(4)

(note that cpn is an integer flow in G). It follows from the choice of n and (3),(4) that the integer-valued function cpo := cp cpn satisfies

+

-k 5 cpo(e) 5 k for every e E E(G),

#0

for every e E E(C) - F, cpo(A:) = cpo(A;) for every 2) E V ( D )= V(G). cpo(e)

(5) (6)

(7)

For, po(A:) = cp(A$)if 21 $ V ( C ) ,while cpo(A:) = cp(A$)-n = cp(A;)n = cpo(A;) if 21 E V ( C ) .Finally, define

VIII.2. Value-'Rue Walks and Integer Flows in Graphs

VIII.39

and let

0 5 cpF(e)5 k

for every e E

E(G) ,

(9)

while (7) and the definition of DF yield

where A;,D, (A,,,) denotes the set of arcs incident from (to) 21 in DF. For V ~ ( A , ~#, cpo(A$) ~ , ) implies E V(C)and 'po(e)< 0 for some a, E A, c A ( D ) which in turn implies for a, E A: and {a, p } = {+, -}

+

It follows from (9) and (10) that ' p F is a (k 1)-flow in G. Moreover, for the O-set EO,Fof ' p F , we conclude from (4), the definition of 'po, as well as from (6) and (8) that

This and F' = F U E(C)2 E,, imply F

2 Eo F .

If C does not correspond to a cycle of D , we transform 'p into a (mod (k 1))-flow with a corresponding orientation of G in which some cycle corresponds to C (see Remark VIII.20.2) and 4)). Whence we can assume that p itself is such a (mod (k 1))-flow, and proceeding as above we obtain the ( m o d ( k 1))-flow cpF with F _> EO,F (note however, that (7),(10) and the corresponding equations connected with them, are replaced with congruences mod(Ic 1)). By Proposition VIII.29, ' p F can be transformed into a (k 1)-flow as required. The lemma now follows.

+

+

+

+

+

Theorem VIII.36. (Seymour's Six-Flow Theorem). For every bridgeless graph G, p(G, 'p) 5 6. Proof. Proceeding indirectly, assume G to be a counterexample with minimal pG q G . Using an argument analogous to the one employed

+

VIII.40

VIII. Various Types of Closed Covering Walks

in proving Proposition VIII.30 (with 6-flow in place of 5-flow) we may conclude that G is a 2-connected cubic graph. Suppose X(G) = 2. Consider a 2-cut E, = {zlyl,z,y,} such that z1,z2E V(G,) and yl,y2 E V(G,), where Gl,G, are the components of G - E,. Let H be the graph obtained from G by identifying z1 and y1 to become z E V(H) - V(G); d H ( z )= 4. H being bridgeless and pH q H < pG 4- qG imply that H has a nowhere-zero 6-flow c p H . Denote by DH a correspond- {zlyl}) is a cut in H ing orientation of H. E f := {z2y2} U (Ezl,G separating H, := (V(Gl)-{zl}) c H fromH2 := ((V(G2)-{yl})U{z}). Denoting by A: the arc-cut in D H corresponding to E f we can write A: = A+ U A - , where A+ ( A - ) contains precisely those elements of A? which are incident from (to) vertices in V(Hl) (and thus to (from) vertices in V(H2)). Since ‘ p H is a nowhere-zero 6-flow it follows that

+

(PH(A+) = cpH(A-)

and

< (PH(z2y2) <

*

(*>

and cp(zlY1) = cp(”2Y2)

*

Thus 0 < cp(zlyl) < 6. Moreover, viewing A(DH) as the arc set of an orientation of G-{z,y,} and observing that for A:, := {a,,,afl}, ALl := { u e a , u f a } the , definition of p and (*) imply diflp(zl) = -diflp(yl) E {fcp(z2y2)} with respect to A:, , A’yl, we now conclude from (*) that either A(DH) U {(zl,yl)} or A(D,) U {(yl,zl)} is the arc set of an orientation of G corresponding to the 6-flow cp in G. This contradiction to the choice of G implies X(G) > 2. Since G is 3-regular, &(G) = X(G) = 3; consequently, G is a simple 3-connected cubic graph. Applying now Lemma VIII.34 to G we find totally disjoint cycles C,, . . . ,C, such that (UI=l E ( C i ) ) , = E(G). By Lemma VIII.35 a 3flow cp exists in G such that Uy=l E(C,)2 Eo (this means, by the way, that Eo is a set of independent edges). Consider an orientation D of G corresponding to ‘p. Let Do C D be the subdigraph corresponding to UE1Ciand let D, be an eulerian orientation of this eulerian subgraph of G (note Cj n Cj= 8, l 5 i < j 5 r ) .

VIII.2. Value-True Walks and Integer Flows in Graphs Define cp3 : E(G) (p3(e) =

VIII.41

(0,f 3 } by

{

3 if a e ~ D 0 n D , -3 if a, E D ~ , u4, D ~{ i, , j } = {0,1} 0 otherwise.

Observe that by definition of cp and (p3, and for (po := cp+cp3 (we associate

D with

yo),

0 = diflV(u) = diflVJ(u)= diffV,(u) for every 21 E V(G) (**) and cpo(e) E {-3, -2, -1,1,2,3,4,5} for every e E E(G). (* * *)

On the other hand, in order to construct from yo a nowhere-zero k-flow ( p F with minimal k it follows from (* * *) that k = 6. Define ( p F and a corresponding orientation DF of G as in the proof of Lemma VIII.35 (see (8) and (9)). Equation (10) of that lemma now holds because of (**). Hence ‘pF is a nowhere-zero 6-flow indeed. The Six-Flow Theorem now follows. Having studied Seymour’s proof several times I could still not detect the cognitive process behind it. Finally, I asked him to explain how he found this proof. On the basis of his reply I can state his ideas as follows. a) A minimal counterexample G must be 3-regular and 3-connected, and thus simple. (This idea is ‘natural’ in this context - see also the proof of Proposition VIII.30).

b) For an arbitrary graph G to have a nowhere-zero 6-flow it suffices to find an eulerian subgraph G, such that G has a %flow $ with 0set Eo,+ E(G,).Noting that 6 = 2.3 we observe that this idea also applies when dealing with 2k-flows in general. For if we take an eulerian orientation D, of G, and set ‘p, f k for ’p, : E(G,) + N , then ‘pl is a nowhere-zero (k + 1)-flow of G,. Applied to the case of cubic graphs, G, is a set of totally disjoint cycles, S, = {Cl,. . . ,C,},T 2 1.

“We might as well choose C,, . . . ,C, so that no circuit is disjoint f r o m c,U. . .Uc, -for if there is such a circuit, c,,, say, then c,,. . . ,c,+,is a more promising list than C,, . . . ,C, (i.e., is bound to work if C,,. . . ,C, W O T ~ S(circuit )” = cycle). c)

d) “Given an aTbitTU?y “maximal” family c,,. . . ,c,, when is there a Z3-ji!0w non-zero on the complement? Don’t know, so simplify.” (Z3flow is basically the same as 3-flow, non-zero = nowhere-zero, and by complement is meant E(G) - U;=, E ( C i ) ) .

VIII.42

VIII. Various Types of Closed Covering Walks

“What i f r = l ? i.e., C, is a circuit meeting all circuits. Now when is there a Z,-flow non-zero o n E(G) - E(C,)? Always. The proof of this suggests the closure argument as well. This is the key idea.” (The quotations are taken from P.D. Seymour’s reply quoted above).

e)

Well, how to prove the above “Always”? Observe that since G is cubic, G - U;=, Ci is a binary forest T (i.e., d T ( v ) E {1,3} for v E V ( T ) = V ( G ) ) .Now we have two ideas at our disposal. el) T can be thought of as a subgraph of a bipartite cubic graph G’ which has a nowhere-zero %flow cp‘ by Theorem VIII.21. The restriction cpT := p’IT can be extended onto G to yield a %flow $ of G with Eo,+ E(C,) (Possibly, one has to invoke Proposition VIII.29). However, this argument will no longer work in this short form if IS,l > 1. e,) However, T being a binary forest implies the following: either there is a chord el of a cycle of G , := C,, where e E E(T,) and Tl := T , or else Tl contains two end-vertices vl,w1 (which therefore belong t o G,) such that d, (v,, w,)= 2. That is, the end-vertices v,, 20, of T, can be joined in T, by a path P, of length 2. Set H , = (el)or H , = P, depending on which case prevails, or choose arbitrarily H , E { ( e , ) ,P,}, respectively. Set G, = G, U H , and observe that G - G, is also a binary forest T,.

Consequently, we can define an increasing sequence G , ,G,, . . . , G, = G such that G, C Gi+l, Giis connected, G,+l -Gj = Hi is a path of length 1 or 2 in Ti, and the end-vertices of Hilie in G,.

f ) Now, if cp is a partial %flow of G, = G , and if j is the smallest index for which 0 # Eo,cp E ( G j ) ,and if j > 1, then one can define a (partial) %flow cpo such that Eo,cp E(Gj-,) (see the proof of Lemma VIII.35, ( 3 ) - ( 6 ) ,for k = 2). In this manner one finally obtains a (partial) %flow of G whose O-set lies in the basic set E(G,). g) In fact, the last paragraph of e , ) and f ) suggest a generalization for k-flows in arbitrary graphs, k 2 3. First, one only needs a basic set F such that the increasing sequence G,, G,, . . . ,G , (with G , = ( F ) and G , = G ) satisfies JE(Gi+,)- E(Gj)I 5 k, in order to obtain Lemma VIII.35. That is, e,) and f ) suggest the definition of ( F ) , as well. h) However, since one aims at a nowhere-zero 6-flow the choice of the basic set F has to be more specific; namely, ( F ) has to be eulerian which means for cubic G that ( F ) is a 2-regular subgraph of G. This leads to Lemma VIII.34 which is stronger than necessary (note that G is just simple and 3-connected in the hypothesis of that lemma).

VI1I.Z. Value-Ttue Walks and Integer Flows in Graphs

VIII.43

We note that Younger used the preceding proof to develop a polynomial time algorithm for constructing a nowhere-zero 6-flow, [YOUN83a]. A. Bouchet also uses the Six-Flow Theorem to show the following, [BOUC82c]: let G be an arbitrary graph, and let n E N be such that n f O(modm),m = 2,3,5. Let H be obtained from G b y replacing every vertex v e V(G) with n vertices vl, ...,v,, e V(H), and viwj E E(H), 1 5 i,j 5 n , if and only if vw e E(G). If G can be embedded in a surface 3 such that every face boundary is triangular, H can be embedded in a surface Fl such that every face boundary of H is triangular; and either both 3 and .Fl are orientable or both are nonorientable. Thus Bouchet was able to generalize some of his own work (e.g. [BOUC78a]), but also the work of others (see the references in [BOUC82c]). He notes, however, that the validity of Conjecture VIII.28 would improve his result inasmuch as the restriction n f 0 (mod5) could be deleted. We also take note of an interesting detail pointed out to me by F. Jaeger: as P.D. Seymour observes in [SEYM8la, 5.11, an alternate proof of the Six-Flow Theorem can be obtained by using Jaeger’s approach to prove the Eight-Flow Theorem. In this case, E, c E(G) (corresponding to the %flow cp in G - see the proof of Theorem VIII.36) does not separate G. As Jaeger states, “Every 3-edge-connected graph has a connected spanning subgraph which has a nowhere-zero 3-fEoW”. Calling this subgraph H , this means that cp induces a value-true walk W on H using every edge of H once or twice in the same direction. Since V(H) = V(G), W covers V(G); hence W is a closed V(G)-covering walk. It can be extended to a closed covering walk W , of G using every e E E(G)at most twice and using every e E E, precisely once in each direction. We shall deal with closed V(G)-covering walks in section VIII.3.1 when we relate the Chinese Postman Problem to the Hamiltonian Walk Problem. Having seen that not every 2-connected cubic graph admits a nowherezero 4-flow (see Theorem VIII.31) while it does admit a nowhere-zero 6-flow (Theorem VIII.36), Tutte’s Five-Flow Conjecture reminds one of the situation that long confronted one in dealing with the Four-ColorProblem. There it was known at a very early stage that not every planar 2-connected cubic graph G is 3-face-colorable, while it was easy to prove that such G has a 5-face-coloring (see Theorems 111.66 and III.65a). We return to the consideration of &flows, (mod 5)-flows respectively, which are equivalent to &flows (Proposition VIII.29), and their rephras-

VIII.44

VIII. Various Types of Closed Covering Walks

ing ( 0 ) which led to Theorem VIII.32. We shall make use of the following lemma in which Ei,, denotes the i-set of the flow cp.

L e m m a VIII.37. Let cp : E(G) + {1,2) be a nowhere-zero (mod5)flow in the cubic graph G, and let D, be a corresponding orientation of G . Consider an arbitrary 21 E V ( G ) . A nowhere-zero (mod5)-flow $ : E(G) + {1,2} and an orientation D, of G corresponding to $ exist such that odD,(v) = 0. Proof. If 21 is a sink of D,, then choose $ = cp and D, = D,. If w is a source of D,, then choose $ = cp and D, = Dt (see Remark VIII.20.5)); 21 is a sink of D,. Whence suppose that 21 is neither source nor sink of D,. Let D; := ({a, E A(D,)/e E E;,,}), i = 1,2; define $ : E ( G ) + {1,2} by $ ( e ) = 2 4 e ) (mod31 7 (1) i.e.7 El,, = EZ,,,EZ,, = El,,. Moreover, define

11, is a (mod5)-flow with D, as a corresponding orientation of G ($ is nowhere-zero by definition): for we have by (1) and (2)

where (3) characterizes the transformation of the sources and sinks of D, into vertices satisfying $(A$) = $ ( A ; ) , while (4)describes the transformation of the vertices of D, satisfying cp(A$) = cp(A;) into the sources and sinks of D,. Since (3) and (4) describe the transformation of all vertices of V ( G )= V(D,) = V ( D , ) , $ is a (mod5)-flow in G, and D, is an orientation of G corresponding to $. Moreover, because of the assumption concerning the chosen vertex w, it follows from (4) that 21 is either a source or a sink of D,. By Remark VIII.20.5), either the pair ($,D,) or the pair ($,D$) will be as required. This finishes the proof. The proof of Lemma VIII.37 shows that this lemma remains valid if one assumes Eo,cp # 0,provided E, n Eo,,E 10,E,}. For in this case Eo,, =

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.45

Eo,$,and the orientation of the elements of Eo,, = Eo,+.can be arbitrary (Of course, for E, C Eo,, we always have di#,(z) = dzfl+(z)= 0). But if 8 # Ei,, n E, # E,,then

For vertices of this type we have difl,(z) = d i f l + ( z ) = 0 as well, but 0 # cp(A$) = cp(A,) which means that v is neither source nor sink in

D*We shall also need the following lemma whose proof is left as an exercise.

Lemma VIII.38. Let G be a cubic graph, e = zy E E(G) and let G, be the graph resulting from G by splitting e in one of two possible ways (see Figure 111.14). Denote {e',e"} = E(Gl) - E(G). Suppose G, has a nowhere-zero (mod 5)-flow cpl : E(G,) -+ {1,2}, and consider a corresponding orientation DP1of G, and the two possible orientations D',D" of G corresponding to DP1 (the orientation D'" = D' n D" of G - {zy} is induced by D,J. If z or y is a source in D' or D",or if cpl(e') # cpl(et'), cpl can be extended to a nowhere-zero (mod5)-flow 'p of G such that either D' or D" corresponds to cp. Now we proceed to prove the following result. Theorem VIII.39. Let G be a connected, bridgeless graph which is 2-cell embedded in some surface 3 with Euler characteristic x ( T ) 2 0. Then G has a nowhere-zero 5-flow.

Proof. Suppose the theorem to be false and choose a counterexample G subject to the conditions that a(G) = 2qG - 2pG is minimal with minimal V,(G) and maximal x ( F ) . As in the proof of Proposition VIII.30 we conclude that G is a 2-connected cubic graph; however, the application of the Splitting Lemma has to be more specific: for ZI E V(G), d(v) > 3, let el, e,, ed be chosen according to O+(v) = (ei, eh,. . .,e&), where O+(v) is given by the embedding of G on 3, and where w.1.o.g. el and e2 belong to different blocks of G if is a cut vertex. Of course, if G,,, is connected and bridgeless, for example, it may not be 2-cell embedded in T . This does not matter, however, since G,,, can be 2-cell embedded in a surface Fl with x(Tl)> x ( F ) (see [BEIN78a, p.201, where the authors refer to Kagno and Youngs). For the remainder of the proof we may assume, therefore, whenever a derived graph is not 2-cell embedded in T , it admits a nowhere-zero 5-flow (since the above argument guarantees a

VIII.46

VIII. Various Types of Closed Covering Walks

%cell embedding on a surface F, with x(F,) > x ( F ) ) . Thus and because a(G,,,) < o(G), we can conclude that G,,, has a nowhere-zero &flow which can be interpreted as a nowhere-zero &flow in G since E(G, 2) = E ( G ) . Whence G has to be 2-connected and cubic since V,(G) = can be assumed.

8

In order to see that G has to be 3-connected we f i s t proceed as in the proof of Theorem VIII.36. However, for H obtained from G by identifying the vertices x1 and y1 (where xlyl E E,, a 2-cut of G ) , we have a ( H ) = a(G). On the other hand, splitting in G e = xlyl into two edges e‘, errwhich do not cross each other in .F (see Figure 111.14) renders a 2-connected graph G,: for G, is homeomorphic to H, obtained from H by splitting z E V ( H )- V ( G )in one of two possible ways (since we do not split ‘crosswise’ in order to obtain an embedding of H, on 3), while H2 obtained by splitting z in the other possible way contains z2y2 as a bridge, where E , = {x1y1,x2y2} C E(G). We have a(G1)< a(G),and G , is (possibly not 2-cell) embedded in F . In any case, since G , is 2-connected it admits a nowhere-zero 5-flow by assumption which can be readily transformed into a nowhere-zero 5-flow (PH of H. Now one transforms ( p H into a nowhere-zero 5-flow cp of G using the same arguments as in the proof of Theorem VIII.36 (with 5flow in place of 6-flow). Consequently, G is 3-connected. In particular, G contains no multiple edges, and thus every face boundary has at least 3-edges. Corollary 111.57 implies for the number f i of faces of G having precisely i edges in their boundary, that

fi

#0

for some i E {3,4,5,6}

.

(1)

In order to obtain a final contradiction we show, step by step, that f i = 0, 3 5 i 5 6. In what follows, let xj be the vertices of an i-gonal face boundary bd(F), 3 5 i 5 6, 1 5 j 5 i, in counterclockwise cyclic order = zl; and denote e j = xjzj+, E E(bd(F)),1 5 j 5 i, putting furthermore, let g j = yjxj E EZj- E(bd(F)),1 5 j 5 i. Moreover, in the ensuing discussion we shall consider nowhere-zero (mod 5)-flows p H : E ( H ) 4 (1,Z) which is no loss of generality by statement ( 0 ) preceding Theorem VIII.32. Also, for i = 3,4,5 let G , denote the graph obtained from G by contracting the i-gonal face F onto a vertex z E V(G,)-V(G) such that O+(z) = (g;, 96,.. . ,g:).

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.47

a) Suppose G contains a triangular face F. By assumption and Lemma VIII.37, H := G, has a nowhere-zero (rnod5)-flow cpH : E ( H ) + { 1 , 2 ) such that odDnx = 0 in some orientation DH of H corresponding to c p H . We distinguish the cases

(the case cpH(g3) = 1 , symmetric to case (i)).

vH(g1)

= cpH(g2) = 2 being, for our purposes,

In both cases we have two ways of extending cpH to obtain cp : E(G)t { l )2)- we fist cp IE(G)-E(bd(F))= ( P H . In case (i) define either

as a corresponding orientation of G, or

with

D, :=

22), ( z 3 , z2), (z3, z l ) ) )

being as required in this case (see Figure VIII.5).

Figure VIII.5. The two possibilities of extending cpH to cp if c p H ( 9 1 ) = 1 , c p H ( 9 2 ) = c p H ( g 3 ) = 2.

VIII. Various Types of Closed Covering Walks

VIII.48

In case (ii)define either

or c p h ) = 2,

4e2)

= 'p(e3) = 1

and

D,

:= MD,)

u ((52, 4 ( 2 3 , 22), (%,4>

to obtain cp and D as required (see Figure VIII.6). Note that in each of the cases ( i )and (ri),

can be extended t o 'p such that one can prescribe which one of the arcs (xl, x 2 ) ,(x2,xl) belongs t o D, without affecting the orientation an ( A (D H))D , * (2)

'pH

Figure VIII.6. The two possibilities of extending if (PH(91) = 2, (PH(92) = 1, 'PH(93) = 2*

'pH

to

'p

In all possible cases we have obtained nowhere-zero (mod 5)-flows from such flows ' p H in H . We conclude from the choice of G that

'p f3

in G =0

must hold.

b) If G has a quadrangular face F , either ( G F ) 1 , or 2 (GF)1,4,where the indices 1 , 2 and 1 , 4 correspond to g1,g2 and gq respectively, will by the Splitting Lemma be connected and bridgeless since G, is such a graph (note that n(GF) 2 2 since K ( G )= 3). We assume w.1.o.g. X(GF)1,22 2; this is equivalent to saying that H obtained from G by splitting el in

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.49

such a way that no crossing edges arise, is a 2 - c o ~ e c t e d3-regular graph. Note that H contains the digon (z3, z4). In any case, it follows from our basic assumption that H has a nowhere-zero (mod 5)-flow c p , (where, by Lemma VIII.37, odD,x4 = 0 in a corresponding orientation D, of H ) which, by Lemma VIII.38, can be extended to such a flow cp in G unless cpH(e’)= pH(e”),where e‘, err E E ( H )- E ( G ) are the two edges obtained from the splitting of el. W.l.o.g., e’ E E ( ( z 3 z4)). ,

If cpH(e’) # cpH(e3), cpH(g4) = 2 = cpH(g3) because of the choice of OdDnx4= 0. Define

and retain

D,

as an orientation of H corresponding to cph (note that (z3,z4)in this case). Thus, we have a

D, contains two arcs of the form

nowhere-zero (mod 5)-fEow cph which, b y Lemma VIII.38, can be extended . to such a flow in G since &(e‘) # & ( e “ ) (3)

Assuming now cpH(e’)= cpH(e3)we conclude that

By defining in this case

and

% := (DH - { a e 3 1 ) u {a: 1 it follows that cph is also a nowhere-zero (mod5)-flow in H satisfying & ( e l ) # ph(e”). We conclude from (3) and Lemma VIII.38 that cp : E(G) {1,2} as required exists (see also Figure VIII.7). Having again constructed cp as required in all possible cases, f4 = 0 follows. --f

c ) Suppose G has a pentagonal face F. Then G, is connected and bridge-

less and hence the Splitting Lemma implies w.1.o.g. that (GF)1,2 is connected and bridgeless. As in case b) we conclude that H obtained from

VIII. Various Types of Closed Covering Walks

VIII.50

Figure VIII.7. The case ‘pH(e’)= cpH(e”)= cpH(e3)= 2 being transformed into the case & ( e l ) # &(e”) which can be extended to ‘p : E(G) + {1,2}. G by appropriately splitting el is a 3-regular 2-connected graph. Note that bd(F) C G corresponds to the boundary of a triangular face A in H and that (GF)1,2 can be obtained from H by contracting A onto the vertex x E V(G,) - V(G) and by subdividing yly2 E E ( H ) with the vertex x1,2 E V((GF)l,2) - W,).

H has a nowhere-zero (mod 5)-flow cpH such that by (2), for a corresponding DH (x5, z3)

E A(DH) if and only if (313 32) f A(DH)*

(4)

(we assume cpH to be obtained from a (mod5)-flow cpo of Go := ((GF)1,2such that x is a source in a corresponding orientation Do of Go).

21,2)U {y1y2}

That is, either x1 or x 2 is a source in D’ 3 {(xl, x 2 ) } or D” 3 {(z2, xl)}, where D”‘ := D’ - {(xl, x 2 ) } = D” - {(x2,x1)} is the orientation of G - {x1x2} induced by DH (see Figure VIII.8). By (4) and Lemma VIII.38 we may conclude that G has a nowhere-zero (mod5)-flow cp such D”} (observe that 2 5 cpH(y1y2) c p H ( x 5 x 3 )5 4 implies that D, E {D’, 1 5 ’p(x1z2)5 2). Consequently, f5 = 0 must hold.

+

Having shown that fi = 0, i = 3,4,5, we obtain

VIII.2. Value-True Walks and Integer Flows in Graphs

YS

VIII.51

Y,

d Y, *Y2

y4

Dti

yv3 yv x4

2

x4

Figure VIII.8. Extending p H and D, (corresponding to H ) to and D, (corresponding to G) in the case PO(g3) = 'PO(g.5) = 2, where ' P H I H - - E ( h ) = 'PO. 'PO(g4) = If 'p(z,y,) = (p0(y1y2) = 1, we then define 'p(xlz2) = 2 and replace the mc joining z1and x 2 with its respective inverse.

VIII. Various Types of Closed Covering Walks

VIII.52

from which we conclude that x ( F ) = 0 and f = f6. Suppose X,(G) = 3 and consider S, C E ( G ) , a cyclic 3-cut of G (Note that G is nonhamiltonian: otherwise, it has a nowhere-zero 4-flow since it is 1-factorable; thus it also has a nowhere-zero &flow, contradicting the choice of G. Since G has a 2-factor, it thus has two totally disjoint cycles). Let B,,B, be the two components of G - S,. K ( B ; )= 2 since S, is a set of independent edges, i = 1,2. W.1.o.g. PB1

+ qB1

+qBz

-

Among the Bl’s corresponding to cyclic 3-cuts let B, be chosen such that PBo

+ qBo = min{pi?l + qB3}

7

the minimum taken over all cyclic 3-cuts S, of G. Denote by So the uniquely determined 3-cut corresponding to B, .

If X,(G) 2 4,denote B, = G. d) For a hexagonal face F with bd(F) C B, we form

G, := G - {e2i-1/2 = 1,2,3}, G, := G - {e,;/i = 1,2,3} . We claim that

both G , and G, are 2-connected. First of all, neither G, nor G, can be disconnected; otherwise, the addition of a certain edge from the set of deleted edges would render a disconnected graph contradicting X(G) = 3 (note that in this case, at most two pairs of adjacent 2-valent vertices belong to the same component). Suppose X(G,) = 1, and let e = 2/21 be a bridge of G,. Since 6(G,) = 2 every end-block of G, is 2-connected (and there are at least two end-blocks of G,). Since adjacent pairs of 2-valent vertices belong to the same block of G,, there must be an end-block B,say, such that

W.l.o.g., el = Z,Z, E E(B). Consequently, B is an end-block of the graph G, U {e4} which still has a bridge. Assuming now w.1.o.g. that u E V(B) ( B must be incident with a bridge of G, since K ( B )2 2, K(G,)= 1, and A(G,) = 3), we conclude that S, := {e, e,, e6} is a cyclic 3-cut of G: for,

G - s, = (G,u {e4H - {el

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.53

has B as one of its components, and e2,e6 are independent edges; hence S, separates G in a non-trivial way and A,(G) = 3. Now bd(F) satisfies

bd(F) c B,, bd(F) n B

# 0 # b d ( ~n) (G - ( B u s,))

.

We want to show that either

So n E ( B ) = 0 or So n E(G - ( B US,)) = 0

.

Observe that the choice of B, and F , and ( 5 ) and (6) imply

0 # Bo n B # B, which is tantamount to saying that

B n B , # 0 # B n (G - (B, u so)) . (9) and (6) imply (G - ( B u SC)>n B,

# 0 # (G - ( B u s,>>fl (G - P

o

u So))

B, is impossible by (9) and the choice of B,). (note that G - ( B US,) However, because of (10) we have two edge sets in G, {e}

u So n E ( B ) and

{e} u So n E(G - ( B US,))

,

(11)

where the former separates B n (G - (B,USo)) from B nB, and the latter separates (G - ( B U S,)) n (G - (B, U So)) from (G - ( B U S,)) nB,. On the other hand, because of (9) and (lo), {e2, e 6 } is a 2-cut in B, implying that the sets in (11) are separating edge sets in G . Since I So I= 3 and since these sets have precisely e in common it follows that G has a 2-cut (if we assume the falsity of (7)), contradicting X(G) = 3. The validity of (7) now follows.

On the other hand, So n E ( B ) = 0 implies B c B, since bd(F) U B is a 2-connected subgraph of G - So, and bd(F) C B,; this contradiction to the choice of B, and (7) imply So c E ( B ) U { e } . Since G* := bd(F) U (G - ( B U S,)) is 2-connected and E(G*) n So = 0, we now have G-(BUS,)CG*GBo

.

VIII.54

VIII. Various Types of Closed Covering Walks

That is, for the 3-cut S, of G, the component B, := G - ( B U S,) of G - S, satisfies pBl qgl < p B o qBo. This contradiction to the choice of B, finally proves (*). Consequently we consider the 2-connected cubic graph H , homeomorphic to G j for a fixed j E {1,2}; w.1.o.g. j = 2.

+

+

By the choice of G and by the proof of Lemma VIII.37 we may conclude the existence of a nowhere-zero (rnod5)-flow ’p, in H , (with Do as a corresponding orientation of H,) such that for hi = y2i-1y2i E E ( H o )E ( G ) ,i = 1,2,3,

G j being 2-connected and (12) imply that the cubic graph H obtained from G by splitting e3 and e5 is 2-connected and has a nowherezero (mod5)-flow ‘ P H with ‘PH I E ( H ) - ( E z 1 U E z 2 ) = ‘ P O I E ( H o ) - { h l } (thus 7 and where = 2; 2 = 2, 3)7 ‘ P H ( g 1 ) = ’ P H ( 9 2 ) = ‘pff(h,) = the edge ei # el joining x1 and x2 satisfies

This can be achieved regardless of the value ’ p H ( g 1 ) = ‘ p H ( g 2 ) ,while ‘ p H ( e l ) E {1,2} depends on pH(gl) = y H ( g 2 and ) the orientation of H {el,ei} induced by Do (see the definition of q~hand D H , D h respectively, in case b) as well as Figure VIII.7). Because of (12) and (13), a twofold application of Lemma VIII.38 yields an extension of ( p H to a nowhere-zero (mod 5)-flow ‘p in G satisfying

and p ( e , ) E {1,2}, k E {3,5}, depending on the orientation D- of G { e 3 ,e5} which is induced by the orientation of H - { e l , ei} corresponding to ‘pH (see Figure VIII.9). Having assumed that G does not admit any nowhere-zero (mod 5)-flow, and having shown X(G) = 3, (1)and fi = 0 for i = 3,4,5 (cases a),b),c)), it follows that fs # 0. In case d), however, we have shown that a nowherezero (rnod5)-flow ’p, in a smaller cubic graph can be extended to such a

VIII.2. Value-True Walks and Integer Flows in Graphs

H

VIII.55

G

Figure VIII.9. Extending ‘po in Ho to p H in H to cp in G. The values p ( e k ) ,k = 1,3,5 are deleted because they have to be determined with the help of the corresponding digraphs Do,DH respectively. flow p in G . This is the final contradiction to the choice of G. Theorem VIII.39 now follows. However, there are only four surfaces with non-negative Euler characteristic: (I) the 3-dimensional sphere S, (= plane, x ( F ) = 2); (11) the torus (the orientable surface with x ( F ) = 0); (111) the projective plane (= cross cap, x ( F ) = 1); and (IV) the Klein bottle (the non-orientable surface with x ( F ) = 0). For the plane, there is a more general theorem saying that a plane graph has a k-face-coloring if and only if it has a nowhere-zero k-flow (we shall prove this theorem in the chapter on colorings; see [TUTT54a, p.831). So the existence of a nowhere-zero 5-flow in planar graphs follows from the 5-Color-Theorem (Theorem III.65a), and the special cases k = 3,4 are dealt with in Theorem VIII.21 and VIII.31, respectively. In the case of the projective plane, Theorem VIII.39 has already been proved in [STEI84a]. Both here and in the planar case, one does not have to deal with hexagonal faces, as one can see from the discussion between the cases c ) and d).7) 7,

According to F. Jaeger (oral communication), the two cases where

x ( F ) = 0 have been settled recently (although he could not provide me with

VIII.56

VIII. Various Types of Closed Covering Walks

Theorem VIII.40. Every minimal counterexample G to Tutte’s Five5 and g(G) 7. Flow Conjecture is a snark with X,(G)

>

>

(By definition, a snark is a cubic graph satisfying x’(G) > 3 and X,(G) 4, whereas a weak mark only satisfies X,(G) 2 3).

2

We note that from the various results on flows and their proofs presented here, we can only conclude the weaker form of Theorem VIII.40 in which ‘snark’ is replaced with ‘weak snark’ and 5 is replaced with 3. Note that the proof of X(G) = 3 in Theorem VIII.39 only uses a 2-cell embedding of G in some surface F,and that the consideration of the cases a)-d) can serve just as well for showing that a counterexample to Conjecture VIII.28 must have girth at least 7 (for we did not really make use of the fact that bd(F) was an i-gonal face boundary, i = 3,4,5,6). We leave it as an exercise to check the details (Exercise VIII.ll).

On the other hand, if G is a cubic graph satisfying Theorem VIII.40 or its weaker form (Exercise VIII.ll), and if G is 2-cell embedded in an orientable or non-orientable surface 3 with Euler characteristic x ( F ) = 2 - a , (where is the number of handles in the orientable case, while cy is the number of cross-caps in the non-orientable case), we may conclude from

2 - a = p - q + f, 3p = 2q, 2q = that

42 - 21a = 21p - 219

Cifi2 7f

+ 21f 5 14q - 219 + 6q = -q

i.e., q

;

5 21a - 42 .

That is, for fixed orientable or non-orientable genus, there is only a finite number of candidates for a minimal counterexample (Corollary 111.57). Whence if one could determine an upper bound on a for a minimal counterexample to Conjecture VIII.28, this conjecture would be reduced to a finite problem (which, in principle, could be solved by computers - similar to the Four-Color Problem). On the other hand, F. Jaeger and T. Swart conjecture that no snarks with a girth of at least 7 exist. accurate references; and he does not mention the settlement of these cases in [JAEGSSa]). In fact, [MOLLSSa] contains a proof of these cases and with the aid of computers even generalizes Theorem VIII.39 to a certain extent. R. Weaver seems to have extended this result to all admissible graphs G satisfying x(G) 2 -2, [WEAVSSa]. However, Ceimins proved the foliowing important result which we quote without proof (see [JAEGSSa] for exact references).

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.57

Is there an upper bound smaller than 5 for p(G, cp) if one considers classes of graphs which are more special than just bridgeless graphs ? In fact, the following conjecture is due to Tutte (see [TUTT66a, p.221). Conjecture VIII.41. (Tutte’s Four-Flow Conjecture).If G is a bridgeless graph having no subgraph contractible to the Petersen graph, P(G,cp) 5 4 . Using the Four-Color-Theorem and matroid theory, P.N. Walton and D.J.A. Welsh proved that if G is bridgeless and has no subgraph contractible to K3 3 , it has a nowhere-zero d-fEow, [WALT8Oa]. This result points to the direction of Conjecture VI11.41. A result transforming the question on nowhere-zero 4-flows in G into a question on subgraphs covering G (as it has been done for &flows in Theorem VIII.32) can be found, e.g., in [SEYM8ld, 2.21. According to P.D. Seymour (written communication), this result is due to W.T. Tutte. Theorem VIII.42. A graph G has a nowhere-zero 4-flow if and only if G can be covered by two eulerian graphs.

Proof. Suppose G = G , UG,, where G i , i = 1,2, is an eulerian subgraph of G. Let D j be an eulerian orientation of G i , i = 1 , 2 (see Corollary IV.3). Now define cp : E ( G ) --t {1,2,3} by

E E ( G , ) - E(G,), cp(e) = 2, if e E E(G,) - E(G,), y ( e ) = 3, if a, E A ( D , ) n A(&), cp(e) = 1, if a , E A(D,) n A(DF). y ( e ) = 1, if e

To construct D,, an orientation of G corresponding to cp, define

A(D,) = ~

(

~

u2( ~1 ( ~ 1 1 -{ a e

E A ( D , ) / ~ E; ~

( ~ 2 1 ) )

D, is an orientation of G since one deletes only those arcs of D,for which the opposite arc is in 0,. Moreover, cp can be thought of as being obtained from cpl : E(G,) -+ {l}, p2 : E(G,) t (2) which are nowhere-zero 4flows. Precisely because of this interpretation of cp and the definition of A(D,) it follows that cp is a nowhere-zero 4-flow with corresponding D, as required. Conversely suppose G to have a nowhere-zero 4-flow cp : E ( G ) -, { 1,2,3}. We have for every v E V ( G )and D, corresponding to cp

I { e E E,/cp(e) E {1,3H I =

cp(4 2 c€Eu

u(e)#a

= cp(At)

+ cp(A,) = 24A;)

cp(4 = eEE,

0 (mod2) ;

VIII. Various Types of Closed Covering Walks

VIII.58

i.e., G , := ({e E E(G)/cp(e)E {1,3}}) is an eulerian subgraph of G. Let D, be an eulerian orientation of G , and define

$ ( e ) = cp(4 if e E E(G) - E ( G, ) $ ( e ) = cp(e) 1 if a, E A(D,) n A(D,) $(e) = cp(e)- 1 if a, E A(D,) - A(D,)

+

Thus $ : E ( G ) t {0,2,4} because of the definition of cp and G,. Clearly, G, := ({e E E(G)/$(e)= 2)) 2 G - G,; whence it sufEces to show that G, is eulerian. In fact, D+ := D, is an orientation of G corresponding to 5; thus $(A:) = $(A;) for every v E V ( G ) . This follows from the definition of $ and the fact that cp is a (4-)flow, whereas D, corresponds to a (2-)flow of G,. Consequently,

D, := ( { a e E A P J / $ ( e ) = 2)) is an orientation of G, corresponding to $, : E(G) -+ {0,2} defined by $,(e)

= $(e)

if $ ( e ) = 2

,$,(e) = 0 otherwise

.

Since for every v E V(G,)

it follows that $, is a (mod4)-flow We thus obtain from the definition of G, and $a 20dD2(V) = $,(A:)

&(A:) = 2idD2(v) ( m o d 4 ) ;

hence 2 d D 2 ( v ) O(mod4), i.e., dG2(v) is even for every w E V(G,). Having found a covering of G by two eulerian graphs G, and G, , we may now conclude the validity of the theorem. Thus Conjecture VIII.41 is equivalent to saying that every bridgeless graph having no subgraph contractible to P5 can be covered by two eulerian graphs (for further discussion of Conjecture VIII.41 and its relation to other conjectures, see [SEYM8ld]).

Remark VIII.43. The definition of cp in the first part of the proof of Theorem VIII.42 shows that one can even obtain cp(e) = 1for a prescribed edge e E E(G). For this definition of cp is symmetric in GI and G, (i.e.,

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.59

it is irrelevant which of the two eulerian graphs is considered f i s t and which second); and the sets Eln E ( G , ) n E ( G , ) , E3 n E ( G , ) n E ( G , ) can be interchanged because of Remark VIII.20 (simply consider OF instead of D, if necessary). By the same token we can prescribe the orientation of a , E A ( D J .

It has been mentioned in [JAEG88a] that in the 3-regular case Conjecture VIII.41 is reduced to the Trivalent Four-Flow Conjecture: every cubic bridgeless graph having no subgraph homeomorphic to P5 is ,%edgecolorable (which by Theorem VI11.31 is equivalent to saying that this graph has a nowhere-zero 4-flow); but it is not clear at this point whether these two conjectures are equivalent.

Another way of restricting G to a special class of bridgeless graphs in order to have p(G,‘p) as small as possible, is to increase connectivity. For this case we have another conjecture.

Conjecture VIII.44. (Three-Flow Conjecture). If G is a bridgeless graph without 3-cuts, p(G, ‘p) 5 3.

F. Jaeger pointed out to me that it is sufficient to prove Conjecture VIII.44 in the following weaker form (see [BOND76a, p.252, Problem 4811* Conjecture VIII.44.a. Every bridgeless 5-regular simple graph G without 3-cuts has an orientation D such that {od,(u), id,(u)} = { 1,4} for every u E V ( D )= V ( G ) . Moreover, Jaeger put forward the Weak Three-Flow Conjecture which states that p(G, ‘p) 5 3 for every k-edge-connected graph, for some k E N , [JAEG88a]. He had obtained on an earlier occasion a result which comes as close to a solution of Conjecture VI11.44 as Seymour’s Six-Flow Theorem to Conjecture VIII.28, [JAEG76b,79a,88a].

Theorem VIII.45 (Jaeger’s Four-Flow Theorem). Every bridgeless graph without 3-cuts has a nowhere-zero 4-flow. We shall make use of Kundu’s Lemma (Lemma 111.40) just as Jaeger did in his proof of Theorem VIII.45, but otherwise we shall proceed purely constructively in the main part of our proof. There we shall make use of the following lemma.

Lemma VIII.46. Let T be a spanning tree of the connected graph G. Then there exists a spanning eulerian subgraph H of G satisfying E(G)- E ( T ) E ( H ) .

VIII.60

VIII. Various Types of Closed Covering Walks

Proof. Consider H, := (E(G)- E ( T ) )tl V ( G ) . If H, is eulerian, then let H = H,; otherwise mark in T the odd vertices vl,. . . ,v 2 k of H,. Let Pi= P ( ' U ~v2J ~-~ be, a path joining v2i-l and v2i in T , i = 1,.. . ,k. Assume the subscripts of these 2k vertices to have been chosen in such a way that the total length of these k paths is as small as possible. We claim that this choice implies

E(PJ n E ( P j )= 0 for 1 5 i < j 5 k

.

(1)

Suppose to the contrary E ( P i ) n E ( P j # ) 0 for some i # j . Then (E(P,)n E ( P j ) ) , is a path because in a tree any two vertices have a unique path joining them (Theorem 111.29). Form

Fi,j consists solely of edges in Piand Pi,and it is a path-forest (i.e., every component is a path) with precisely two components P,I,Pj' such that { ~ ~ ~ - ~ , 2 1 ~ = ~ Vl(Fi,j). , v ~ ~ -Letting ~ , v PA ~ ~=} P, for i # rn # j , 1 5 m _< k, we have k

k

i= 1

i=l

a contradiction to the choice of the subscripts of vl,. . . ,v 2 k . Now define k

H:=H,UU<

.

i= 1

+

dH(vj) E d H o ( v j ) 1(rnod2), for j = 1,.. . ,2k, for v # v j , j = 1,.. . ,2k. 4 d v ) dHo (mod 21,

=

(4

That is, H is eulerian.

The next two corollaries are direct consequences of the proof of Lemma VIII .46. Corollary VIII.47. Let vl, . . . ,v2k be any 2k vertices of the connected graph G. Then k edge-disjoint paths exist in G, P l , . . .,Pk, such that

VIII.2. Value-True Walks and Integer Flows in Graphs

&

y(pi)= of G).

VIII.61

iV1,. . .,~ 2 k } . - (Consider vl, . . .,vak in a spanning tree

Corollary VIII.48. If a connected graph G has two edge-disjoint spanning trees T, T', G has two connected spanning eulerian subgraphs H , H' such that T C H , T' C H'. Proof of Theorem VIII.45. Proceeding indirectly, we choose G with minimal pG qG such that X(G) 2 2, G has no 3-cuts, but p(G, 'p) > 4.

+

Every block B of G also satisfies X(B) 2 2, and a 3-cut of B is a 3-cut of G. Moreover, generally speaking, if ' p j is a nowhere-zero k-flow of B j , where B = {Bl,. . . ,B,} is the set of blocks of G, e E E ( B j ) , j E {I,. . ., s}

cp(e) = cpj(e)

is a nowhere-zero k-flow of G since E(G)= Uj"=lE(Bj)(and thus cp is uniquely determined by { ' p j / j = 1,.. . ,s}). It thus follows from the choice of G that 23 = {G}. Whence K(G)2 2. We claim that X(G) > 2.

Note first that the choice of G implies S(G) > 2 (see also the proof of Proposition VIII.30). Supposing X(G) = 2 we find a 2-cut S, = {z1y1,z2y2} whose elements are nonadjacent since n(G) 2 2 and such that the graph (G {51527 Y1y2) is disconnected having precisely two components G,, G2; w.1.o.g. z1z2E E(Gl), y1y2 E E(G2). Observing that any 3-cut in Gi, i E {1,2}, induces a 3-cut in G, and noting pGj qGj < pG qG, i = 1,2, it follows from the hypothesis of the theorem and from the choice of G (we note n(Gi) 2 2, i = 1,2) that Gi admits a nowhere-zero 4-flow 'pi. Denote a corresponding orientation of G iby D i , i = 1,2. By Remark VIII.43 we asSume w.l.0.g. that (PI(Z152) = 'p2(Y1Y2) = 1, (51,521 € A ( D , ) , ( Y 2 , Y l ) € A(D2).Defining now

"

+

+

'p(e)=

{

D, = (Ol -

Pl(4 cp2(e>

cp1(z152) = cp2(YlY2) 52>}>

"

(D2

if e E E(G,)- ( ~ ~ if e E E(G2) - { Y l Y 2 1 if e E s,

- ((Y27 yl>))

"

5

~

)

3

{(z17 Yl), ( Y 2 7 z 2 ) }

it follows that cp is a nowhere-zero 4-flow in G with D, being a corresponding orientation of G. We conclude X(G) 2 4 because of the choice of G and due to the nonexistence of 3-cuts in G.

VIII.62

VIII. Various Types of Closed Covering Walks

By Kundu’s Lemma (Lemma 111.40), G contains two edge-disjoint spanning trees T ,T’ which, by Corollary VIII.48, yield two connected spanning eulerian subgraphs H 3 T and H‘ 3 T‘.G = H U H‘ since G - E(T’) & H , G - E ( T ) C H’ (see Lemma VIII.46) and TnT‘ = V(G). Theorem VIII.45 now follows from Theorem VIII.42.’) As a consequence of the last part of the preceding proof we obtain the following. Corollary VIII.48.a. Every 4-edge-connected graph has a connected spanning eulerian subgraph.

This corollary will serve as the basis for later considerations on the existence of connected spanning eulerian subgraphs. However, by adopting the method used to prove Theorem VIII.45 we obtain a proof of the 8-Flow Theorem: doubling the edges of a 3edge-connected 3-regular graph G yields a 6-edge-connected graph H which contains, by Kundu’s Lemma, three edge-disjoint spanning trees T ,T’, T” corresponding to respective spanning trees T I ,T,, T3 C G such that E(T,) n E(T,) n E(T,) = 8 (see [JAEG88a, Lemma 4.21). Whence G can be covered by three eulerian graphs G,, G,, G,. Defining pi : E ( G i ) -, {2i-1}, i = 1,2,3,yields a nowhere-zero 8-flow in G. Apart from Theorem VIII.45 Jaeger also proved (implicitly) that the validity of the Three-Flow Conjecture implies Seymour’s Six-Flow Theorem, [JAEG79a, Proposition 111. As for partial solutions of the Three-Flow Conjecture, we mention [STEISSa] where this conjecture has been verified for the projective plane. In [JAEG84b,88a] the following conjecture has been proposed. Conjecture VIII.49 (Circular Flow Conjecture). For every k E N and every 4k-edge-connected graph G, an orientation D of G exists such that id,(v) 5 o d D ( v )(rnod(2k 1)) for every v E V ( D )= V(G).’)

+

8, The paper [SEYM8ld], which contains Theorem VIII.42, and where Seymour refers t o Jaeger’s 8-Flow Theorem (which had preceded Seymour’s 6Flow Theorem and which contains Theorem VIII.45 as well), does not explicitly note the link between Theorem VIII.42 and Jaeger’s 4-Flow Theorem (with the former admitting an easy way t o prove the latter). In fact, Jaeger deduces his results from more general considerations; see [JAEG79a, Propositions 5,6,7,8].

F. Jaeger calls this orientation of [JAEG88a].

G a (rnod(2k

+ 1))-orientation,

VIII.2. Value-The Walks and Integer Flows in Graphs

VIII.63

Not only does this conjecture follow in the footsteps of Conjecture VIII.44.a, but it is for k = 1 equivalent indeed to the Three-Flow Conjecture; moreover for k = 2 it implies the Five-Flow Conjecture. Namely, if G is 3-edge-connected, by replacing each edge with three p a r d e l edges (i.e., with an edge of multiplicity 3) one obtains a 9-edge-connected graph H. Any orientation D of H satisfying id,(v) G od,(v) (mod5) can be transformed into a nowhere-zero 5-flow in G. We leave it as an exercise to do this (Exercise VIII.13). Yet another way of describing nowhere-zero 4-flows in terms of covering subgraphs (see Theorem VIII.42) has been found by P.A. Catlin who defines a nowhere-zero 4-flow in the equivalent form as expressed by Theorem VIII.42, [CATL86b]. Theorem VIII.50. Let G be an arbitrary graph. The following statements are equivalent.

1) G has a nowhere-zero 4-flow. 2) G = G , U G2 U G,, E(G,) n E ( G j )= 0 , l

d~(V)(mod2),21 E V ( G ) ,1

5 i < j 5 3, and dGi(v)

3

5 i 5 3.

3 ) G, C G exists such that dGl(V) dG(v)(m0d2), v E V ( G ) ,and each component of G - E ( G , ) contains an even number of odd vertices of

G. (Note that in 3 ) , G - E ( G , ) is an eulerian graph because of the vdency condition).

Proof. 1) implies 2). By Theorem VIII.42, G has a nowhere-zero 4-flow if and only if G = Hl U H2 and H I ,H2 are eulerian graphs. Define

(Of course, if G is eulerian, E(G,) =

0 may hold true for some i

E

(I, 2,3}). Since in general,

and since E ( G, ) n E ( G j ) = 0, 1 5 i < j 5 3, by construction, we also have d G i ( v ) = d H i ( v ) - I E( H l) n E(H,) n E, I, i = 1,2 (2)

VIII.64

VIII. Various Types of Closed Covering Walks

for every v E V(G)(where one counts loops in H , d H i ( v ) F O(rnod2) combined with (1) and (2) yield

nH,

twice). Now,

This congruence also holds for i = 3 since G , U G, is eulerian and G, U G, U G, = G is an edge-disjoint union. 2 ) implies 3 ) . For G,, G,, G, satisfying 2), each graph Gi, i = 1 , 2 , 3 , satisfies dGi(v) dG(v)(mod2). W.1.o.g. i = 1. Since G - E(G,)= G, U G, is eulerian, and since every component of G,, k = 2,3, has an even number of odd vertices of Gk which are also odd vertices in G , and because every component C of G,UG, is the union of certain components of G2 and G,, it follows that C contains an even number of odd vertices of G. 3 ) implies 1). Let G , C G be chosen such that G, satisfies 3). Each component of H := G - E(G,)is eulerian and every component C of H contains an even number of odd vertices of G ; denote them by 211 . * , 212kc* By Corollary VIII.47, k, edge-disjoint paths P,, . . . , p k c exist whose odd vertices are precisely these vertices v,, . . . , 2)2kc. Thus G, := U(,U~ f, Pi)U V ( G ) ,where the first union is taken over all components C of H , also satisfies 3). Consequently, both H and G, U G, are eulerian subgraphs of G , and they cover G. Theorem VIII.50 now follows from Theorem VIII.42.

We note in passing that Theorem VIII.50 will be of relevance in dealing with cycle covers of graphs.

As for k-flows cp : E ( G ) t { O , l , . . . , k - 1) in general, one can put restrictions on the actual value of rnin{cp(e)/e E E(G)}. This has been done in [JAEG76a].

Definition VIII.51. a) For a , b E N U {0}, a 5 b, an integer k-flow cp of G is called an [a,b]-flow if a 5 p(e) 5 b for every e E E(G). If G admits an [a,b]-flow cp, we call G [a,b]-orientable, and an orientation of G corresponding to cp is called an [a,b]-orientation. b) For a graph G, call a mapping w : V(G)-+ R a balanced valuation of G if

I2

I

~ ( v )5 e ( X ) for every X C_ V ( G ) .

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.65

Thus the existence of a nowhere-zero k-flow in a graph G is equivalent to saying that G is [1,k - 11-orientable. Observe the affinity between [a,b]flows and Definition 111.83. As for balanced valuations, we note that the mapping w is bounded by the condition I w ( v ) 15 d(v), v E V(G). However, there is a close link between [a,b]-orientability and balanced orientation. First we prove two lemmas. The first classifies the outdegree sequences of orientations of a given graph.

L e m m a VIII.52 ([HAKI65a]). Let G be a graph, and let d, ,. . . ,d, E NU (0}, p = pG. An orientation D of G exists such that d, = od(vi), i = 1,..., p , if and only if P

Edi= qG

and

c

d, 2 q(x)G for every X

C V(G) .

vi E X

i= 1

Proof. Suppose for some orientation di, i = 1,... , p . Then

D of G that we have od(v,)

D

D

i= 1

i= 1

=

Moreover, for every X C V ( G )one has

This proves necessity.

zEl

xviEX

di = qG and d, 2 q ( x ) , for every x Conversely, suppose V(G).We assume w.1.o.g. that G is connected (note that CvjEV(C) d. > qc for every component C implies equality because of

Cy=ld,

= qG).

Since the case p = 1 is trivial (any orientation of G will be as required in this case), we assume that vivj E E(G)for some i # j, 1 5 i < j 5 p. W.1.o.g. i = p - 1 , j = p . Identify wp-l and vp to obtain zp-l 6 V(G) and add X(v,-,vp) new loops incident with zPTl. Denote x j = wj, 1 5 i 5 p - 2, and define for the graph H thus obtamed

VIII.66

VIII. Various Types of Closed Covering Walks

By construction of H we still have d: = q f f and CZiEX, d: 2 q ( X l ) H for every XI C V ( H )regardless of zp--lE XI or zp-l $! XI. Applying induction we obtain an orientation D' of H such that 0 d D , ( z j ) = d:, 1 5 i 5 p - 1. We transform D' into an orientation D of G which is uniquely determined by D' except for the orientation of the X(up-lwp) 2 1 edges joining and up. In particular, we have d: = d j = o d D ( u j ) ,1 5 i 5 p - 2. Assume w.1.o.g that ~ d D ( u , - ~ )2 dp-l. Thus O C Z ~ ( V ~ )5 d p since CE1di= u D = qG. Whence assume bp-! := ~ d ~ ( u , --~ dp-l ) 2 0. Among all possible choices for D satisfying di = 0dD(vi),1 5 i 5 p - 2, choose one with minimal 6p-1. If O ~ ~ ( U , - ~=) dpTl then D is as required. Whence 6p-l > 0 has to be assumed. We clam that a path P = P ( V , _ ~u,p ) runs from up-l to up in D. Suppose to the contrary that such P does not exist, and define

By definition and by assumption we have

a contradiction to the hypothesis. Whence we may conclude that P exists indeed. Forming D,:=(D-P)UPR we have odDl(vi)= odD(vi) = di, 1 5 z 5 p - 2, odDl(u,-l) = ~dD(u~- ~ I,) ~ d D , ( u , ) = 0 d D ( u p ) I. Whence O 5 6L-l .._

+

~ d ~ , ( u , --~dp-l ) < 6p-l. The lemma now follows.

This contradicts the minimality of 6 p - l .

The next lemma relates out-degree sequences of orientations of a graph to balanced valuations.

Lemma VIII.53 [JAEG76a, Proposition 31. Let G be a graph on p vertices, and let d j E N U { 0 } , i = 1,.. . , p . The following statements are equivalent. 1) G has an orientation D such that od(vi) = di, i = 1,. . . ,p.

VIII.2. Value-True Walks and Integer Flows in Graphs 2) w : V(G)-+

R defined by

w(zli)

= 2d,

- dG(vi), i

VIII.67

= 1,.. .,p,is a

balanced valuation.

Proof. w is a balanced orientation if and only if

i.e., since

cUiEx= 2q(,) + e ( X ) ,if and only if dG(zli)

which is equivalent to

We claim that ( u ) is equivalent to

Note that ( u ) implies (b) simply because the inequality of ( b ) is part of ( a ) , while for X = V ( G ) ,( u ) becomes qG 5 di I QG, i.e., Cf=ld; = q G . Conversely, assuming the validity of (b) we have for every X C_ V ( G )= V the inequalities

xEl

and the equation

combined they yield

v i EX

VjEV-x

'

VIII. Various Types of Closed Covering Walks

VIII.68

Thus (b) and (a) are equivalent. However, by Lemma VIII.52, (b) is equivalent to the existence of an orientation D of G such that od(v;) = d;, i = 1,.. . , p . That is, statements 1) and 2) of the lemma are in fact equivalent. By using Lemma VIII.53 we can prove the following result, [JAEG76a, Theorem 21 (2denotes the set of integers).

Theorem VIII.54. Let G be a graph, and let n , m E N , n following statements are equivalent.

> m.

The

1) G is [m,n]-orientable.

E V ( G ) ,y, E 2 exists satisfying y, G d ( v ) (mod2), and is a balanced ) y, is such that w : V ( G ) -+ R defined by ~ ( v = valuation.

2) For every

Proof. By Hoffman’s Theorem (Theorem 111.84) G has an [rn,n]orientation D if and only if for every X V ( G )= V ( D ) n.u+(X) 2 m.a-(X), i.e., if and only if for every X (n (n

n.a-(X) 2 m.u+(X)

,

E V(G)

+ m)(a-(X) - a + ( X ) )I (n - m ) ( a - ( x )+ u + ( x ) ) + m)(a+(X) - .-(x))I (n - m ) ( a - ( ~+) a + ( x ) ) ,

which is equivalent to saying that for every X

However, in any case we have for any X viewed as an arbitrary orientation)

V ( G )= V ( D )

E V ( G ) = V ( D ) (where D

is

Thus, having an [m,n]-orientation D is equivalent to (1) which, by (2), is equivalent to

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.69

> 1). Defining w : V(G) + R by

(note that

we conclude that w is a balanced valuation of G if and only if (3) is satisfied, i.e., if and only if G is [m, n]-orientable. Assuming now G to be [m, n]-orientable, we may conclude for

that w ( v ) = yu

-

,

E V(G), defines a balanced valuation of G.

Conversely, suppose that an integer yu v E V(G) and is such that

dG(v)(mod2) exists for every

=

defines a balanced valuation of G. Since 1 w ( v ) 1s dG(v) (see the paragraph following Definition VIII.51), we have I y, 15 dG(v) < dG(v), i.e., -dG('u)

< yu < dG(v) -

Since -dG(v) 5 2du - dG(v) 5 d G ( v ) for every d, E (0,.. . ,d G ( v ) }

,

it follows that

d, E (0,.

. .,dG(v)} exists such that

7, = 2du - dG(v) .

(6)

Since w is a balanced valuation of G we have for every X & V(G)

That is, w o ( v ) := yu already defines a balanced valuation of G. By Lemma VIII.53, G has an orientation D such that 0dD(v)= d,, 21 E V(G).This combined with (6) and (5) yields (4). Since w is a balanced valuation of G it follows that G is [m,n]-orientable. This finishes the proof of the theorem.

VIII.70

VIII. Various Types of Closed Covering Walks

Corollary VIII.55 ([JAEG76a, Proposition 41). A cubic graph G is [m,n]-orientable, 0 < rn < n, if and only if G has a balanced valuation w : V(G)+

{-s, e}.

Corollary VIII.56 ([JAEG76a, Proposition 5); see also [BOND72a, Theorem]). A cubic graph has a nowhere-zero 4-flow if and only if it has a balanced valuation w : V(G)--t {-2, +2). Corollaries VIII.55 and VIII.56 are direct consequences of Theorem VIII.54 and the interpretation of 4-flows in terms of [l,31-orientability; their proofs are therefore left as an exercise. As for Corollary VIII.56, we can rephrase it equivalently by applying Theorem VIII.31: a cubic graph is 3-edge-colorable if and only if at has a balanced valuation with values in { -2, +2}. We shall discuss balanced valuation and [ a ,b]-orientability in the context of coloring problems. However, we take note of another conjecture of F. Jaeger, [JAEG76a].

Conjecture VIII.57. Every planar bridgeless graph is [2,7]-orientable.

F. Jaeger notes that this conjecture is stronger than the 5-Color Theorem but weaker than the 4-Color Theorem. He also pointed out to me that [a,b]-orientability can be viewed as a problem on closed covering walks which are not necessarily value-true walks. Proposition VIII.58. A connected graph G admits an [m,n]-flow, 0 < m 5 n , if and only if G has a closed covering walk W using each edge exactly 2n times and such that each edge is used at least rn n times in one of the two possible directions.

+

Proof. Consider an [rn,n]-flow in G. By definition, G has a nowherep(e) n for every zero flow cp and an orientation D, such that rn e 6 E(G),and cp(A$) = cp(A;) for every TJ 6 V ( G ) = V ( D ). This, p. however, is tantamount to saying that G has a value-true covering walk using every edge precisely cp(e) times in one of the two possible directions, but not in the other direction. Define

<

D,:=D,UD,R

,

and define ‘p, : A ( D , ) + (1,. ..,2n} by

+

p(e) n n-cp(e)

if a = a, E D, i f a = a , R E 0,” .

<

VIII.2. Value-True Walks and Integer Flows in Graphs We then have for every v E V(D,) (where A$,Dl,A& responding arc sets in A(D,))

+

'Pl(A:,DJ = ' P ( A 3 n IA: = 'P(A,)

VIII.71

denote the cor-

1 +n IA, I -cp(A,)

+ n 1-4, I + n ( A 9 - ' P ( A 3

= 91(A&

1 -

That is, cpl is a flow in G , , the graph obtained from G by doubling each edge of G . A value-true walk W, in G, corresponding to 'pl is, however, equivalent to a closed covering walk W in G using every e E E(G) exactly cp(e) n times in one and n - cp(e) times in the other direction. Thus, e is being used exactly 2n times and at least m n times in one of the two possible directions (note cp(e) 2 m).

+

+

Conversely, suppose G has a closed covering walk W using every edge e exactly 2n times and at least m n times in one of the two possible directions, say it uses it cpl(e) 2 m n times. Write 2n - cpl(e) = cp2(e), and for e = zy set a,,, = (z,y), ae,2 = (y,z) assuming e is being used cp,(e) times from x to y. Then we have for

+ +

D, := V ( G )u iae,l, ae,2/e E E(G)l the equation

'P(A:,DV) = ' P ( K , D v ) 7

v E V(G)

(where cp(a,+) := cpi(e), i = 1 , 2 ) since W corresponds to a closed walk Wv in D, using every arc a,,* exactly cpi(e) times, i = 1,2. Noting that cp2(e) < cpl(e) and defining

Do := D, - +%,2/e E E(G)), 1

'Po(e):= ~ ( c p ' ( " > - cpz(eN7

E

W)

(observe that cpl(e) - cp2(e)= 2(n - (p2(e))= 2((p1(e)- n ) ) we still have 'Po

= 'Po (Av,D, )

-

That is, p0 is a nowhere-zero flow in G since Do is an orientation of G corresponding to y o , and because yo satisfies O

1 1 < m 5 cpl(e) - n = po(e) = 5(pl(e) - cpz(e)) 5 5(2n - 0 ) = n .

VIII.72

VIII. Various Types of Closed Covering Walks

That is, cpo is an [m,n]-flow in G. Proposition VIII.58 follows.

A different kind of nowhere-zero flow has been studied in [BOUC83b]. Suppose for a graph G that two mappings $ : E*(G) + {+l,-1)

and

cp : E ( G ) + 2

are given. $ is called a signature or bidirection of the elements of E*(G). A graph G with signature $ is called a bidirected graph. Let

define A;,+ analogously, and call (cp, $) a bidirectional flow if

cp(Ai,+)= cp(A,+) for every w E V(G) (we set cp(e(w)) = cp(e)). Bidirectional (nowhere-zero) k-flows are defined analogously by using the definition of (nowhere-zero) k-flows. Note that a signature can be interpreted as an orientation of the half-edges e ( v ) , w € V(G), whereby one replaces e ( v ) with a,(v) E (A:)+ if $(e(v)) = +1, respectively with a,(w) E (At)- if $(e(v)) = -1. In particular, if the sign of an edge e = zy, defined by sgn(e) := -$(e(x))$(e(y)), satisfies sgn(e) = +1 for every e E E ( G ) , the above interpretation corresponds to defining an orientation of G and a bidirectional flow ( c p , $) then becomes a flow in the corresponding orientation of G. It has been noted in [BOUC83b] that in a graph G with signature $, an edge e may exist such that cp(e) = 0 for every bidirectional flow (cp, $) of G, although G itself is bridgeless (note that a graph having a bridge has no nowhere-zero flow, while a bridgeless graph with at least one edge has a nowhere-zero 6-flow). An edge of this kind is called a signed graphic isthmus; and it has to be distinguished from a bridge (= isthmus in A. Bouchet's notation) of G. In fact, the following is true, [BOUC83b, 2.4. Lemma].

Lemma VIII.59. If G is bidirected and sgn(e) = +1 for every e E E(G)- {f},while s g n ( f ) = -1 then f is a signed graphic isthmus. Proof. In view of the above interpretation of signatures having only edges with positive signs, we transform G into a graph having positively signed edges only. To this end, let G, and G2 be isomorphic copies of G - f , denote f = zy (possibly z = y), and form

H := Gl u G2 u (z1z2, Y l Y 2 )

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.73

where xi and yi are the vertices of Gi corresponding to x and y, respectively, i = l , 2 . Denote fl = z1x2,f2 = yl?/z, and let el E E*(G,) denote the half-edge corresponding to e' E E*(G- f). Define $H : E * ( H )+ {+l,-1) by $,(e:)

= (-l)i-l+(e')

,

$H(f1(4)

= (-Vl+(f(xN

7

$H(f2(Yi))

= (-1Ii-l+(f(d)

7

= 192

-

Thus, sgn(e) = +1 for every e E E ( H ) , and H has an orientation D such that a,(v) E (A:)+ if $,(e(v)) = +1, while a,(v) E (A:)- if t,bH(e(v))= -1, where e f E, E ( H ) . However, either S = {(x1,x2),(y1,y2)} or S = { ( x 2 ,xl),(y2, yl)} is an arc-cut of D. Consequently, assuming the existence of a nowhere-zero bidirectional flow ( c p , + ) in G, we can transform this flow into a bidirectional nowhere-zero flow ( p H,q H )in H by defining cpH(ei)= cp(e), e E E ( G ) , i = 1 , 2 . This is equivalent, however, to saying that H has a nowhere-zero flow, namely pH, where D is an orientation of H corresponding to (pH.'') However, we have for X = V(Gl), Y = V(G,) = V ( H )- V(G,)

i.e., pH(A+(X,Y ) )# p , ( A - ( X , Y ) ) ,a contradiction to the assumption that ' p H is a flow in H . Hence, G has no nowhere-zero bidirectional flow. The lemma now follows.

On the other hand, the existence of nowhere-zero bidirectional flows is secured if G has no signed graphic isthmus. This follows from [BOUC83b, 4.3. Theorem] which we state without proof. Theorem VIII.60. If G is a bidirected graph having no signed graphic isthmus, G admits a nowhere-zero bidirectional 216-flow.

In fact, A. Bouchet conjectures that 216 can be replaced by 6 ; by showing that P5 with the bidirection as exhibited in Figure VIII.10 lo) The fact that p(e) < 0 may hold for some e = uw E E(G), is of no essential relevance; for if we replace $ ( e ( u ) ) , +(e(w>) with -$(e(u)), -$(e(w)), and p(e) with -cp(e), the hypothesis of the lemma is still satisfied and we still have a nowhere-zero bidirectional flow.

VIII.74

VIII. Various Types of Closed Covering Walks

(= [BOUC83b, Figure 21) has no bidirectional nowhere-zero &flow, it is demonstrated that the value 6 in Bouchet’s conjecture is best possible. We note that J.L. Fouquet, [FOUQ84b], improved Theorem VIII.60 by showing that 216 can be replaced by 103; for various classes of graphs he obtained even better upper bounds. However, 0. Z$ka recently showed that 216 can be replaced by 30, [ZYKA88a], whereas Bouchet’s conjecture has been verified by A. Khelladi for a certain class of 3-connected graphs, who also showed that under certain additional structural assumptions, 216 can be replaced by 18, [KHEL87a,89a].

Figure VIII.10. A bidirection of the Petersen Graph which has a bidirectional nowhere-zero 6-flow, but no such 5-flow. The signature $ is expressed by half-arcs and by arcs, respectively, if the sign of the corresponding edge is positive.

We finish this section by considering two papers. The first, [BANG80a], deals with a special type of positive integer flows. Denote q = qG. Definition VIII.61. a) Call a flow cp : E ( G ) -, N a directed numbering if cp is injective, i.e., cp(e) # cp(f) if e # f . A directed numbering cp will be called a conservative flow if 1 5: cp(e) 5: q, e E E(G). Call a conservative flow eulerian conservative if it has a corresponding ederian

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.75

orientation of G. Correspondingly, we speak of (eulerian) conservative graphs if they admit (eulerian) conservative flows. b) A graph is said to be graceful if there exists an injective mapping 'pv : V ( G ) + (0,. . .,q } such that 'pg : E(G) + (1,. . . ,q } defined by cp,(e) =Icpv(z) - cpv(y) I for every e = xy f E(G), is a bijection. p ' , is called a graceful labeling of G. Graceful graphs have been the object of considerable research over the past twenty years or so, while conservative graphs seem to have found only little attention (see [wHIA8Oa] for a topological application of [BANG80a]). Note that a graceful graph is always simple, while this need not apply to conservative graphs. The starting point for considering conservative graphs is [BANG80a, Theorem 1.11.'1)

Theorem VIII.62. For a connected graph G,the following statements are equivalent.

1) G is 3-edge-connected. 2)

G has a directed numbering.

Outline of Proof. Since graphs with bridges admit no nowhere-zero k-flow whatsoever, X(G) 2 2 follows. However, if S = { e , f } is an edgecut of G then p(e) = p(f) for every flow of G. Thus, X(G) > 2 is necessary for the existence of a directed numbering. Consider a strongly connected orientation of G (see Lemma 111.22). Then, for every e , f E E(G), there exists a cycle C c D such that { a , , a f } A ( C ) although { a , , a f } n A(C) # 0 (this follows from X(G) > 2). A flow 'po in D can be obtained by labeling each arc a of D with the number of cycles of D containing a . If 'po(al) = ' p o ( a z ) a1,a2 , E A(D), then find a cycle C such that al E A(C),a2 $! A ( C ) . Define cpl(a) = 'po(a) if a $! A(C), and cpl(a) = po(a)+ maz{cpo(b)/ b E A ( D ) } , otherwise. 'pl is a flow in D,and 'pl(al) # 'pl(a,). Continue this procedure until for some k 2 0, 'pk(al)# 'pk(a2)for any pair of distinct arcs a1,a2of D. Just as in the general case of flows, Theorem VIII.62 leads to the problem of determining

In this paper, the authors assume a priori that G is connected - an assumption which is not really necessary, on the one hand; on the other hand, it is no loss of generality either. 11)

VIII.76

VIII. Various Types of Closed Covering Walks

where 'pd is i ~ l l arbitrary directed numbering of the 3-edge-connected graph G. It seems that this problem has not been dealt with yet in general. Looking for conservative graphs is equivalent to determining graphs G for which p(G, (Pd) = qG . Two of the main results of the cited paper are summarized in the next theorem (see [BANG80a, Theorems 1.6, 1.7, 3.21).

Theorem VIII.63. For any integers m 2 2 , n 2 2, G is a conservative graph if G = K2,+,, or G = K,,, or G = K2m,2n. In contrast to Theorem VIII.63, we have the following result on graceful graphs, [KOTZ78b].12)

Theorem VIII.63.a. Let G be a simple d-regular graph with c(G) components, a l l of which are complete graphs. G is graceful if and only if E { K 1 ? Kz7 K 3 , K 4 ) . The proof of Theorem VIII.63 rests on the following observations and special edge labelings: 1) If G is the edge-disjoint union of a conservative graph and an eulerian conservative graph, it is conservative; 2) If G has a cycle decomposition into two hamiltonian cycles H,, H,, it is conservative (starting in u0 E V ( G ) label the edges in H , in ascending order with 1,3, . .. ,2pG - 1 and in H2 in descending order with 2,4, . . . ,2pG, where uo, . . . ,wPG-, is the order of the vertices in H,); 3) If pG is odd and G has a cycle decomposition into three hamiltonian cycles H,, H,, H3, then G is conservative (starting at q, label the edges of H3 with 3 , 9 , 1 5 , .. . ,3pG, 6 , 1 2 , . . . ,3(pG - l), and label the edges in Hiin descending order with the integers E 1 (mod 3), i = 1,2, starting at uo with 3pG - 2 if i = 1, and at ZI(~~-,)/, with 3pG - 1 if = 2, and where the vertices of G appear in H3 in the order uo,u,,. . . ,vpG-,); 4) K2n+l is the edge-disjoint union of hamiltonian cycles (see Theorem 111.50);5) K4,4,K4,6 and K6,6are eulerian conservative; 6) K2m,2n, m,n 2 2, can be decomposed into subgraphs H E ( K 44 , K4,s,K6,6}; 7 ) The wheel W , is conservative. On the grounds of 1)-7) and the following Theorem VIII.64 it is not difficult to produce an accurate proof of Theorem VIII.63. We leave it as an exercise. - We note that 7) is used in proving that K,, is conservative: for K,, is the edge-disjoint union of ( K Z n - ,- H,) and W,,, where H , is an element in a decomposition of K2n-1 into n - 1 Instead of 'graceful labelings', A. Kotzig and I. Turgeon speak of pvaluations, a term ascribed to A. Rosa who initiated the study of graceful graphs.

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.77

hamiltonian cycles. The fact that W,,, is conservative follows from a more general result on planar graphs.

Theorem VIII.64. [BANG80a, Theorem 3.11. For any plane G, if G is graceful, its dual G* := D(G) is conservative. Proof. Consider G and G* embedded in the plane in such a way that are placed in the corresponding faces F of G, the elements vF E V(G*) and such that vFvFIE E(G*)intersects the corresponding edge eF,Fr E bd(F) n bd(F') G . Define cp, : E(G*)--+ { 1,.. .,q } by

where 'pg is a graceful labeling of G with underlying 'pv : V(G) (0, - * - 7 q}.

4

Now orient G* by replacing v F v F i with ( v F , v F Iif) pV(y) > cpv(z), where eF,Fl = sy, provided y lies to the right of V F V F ~as one passes from vF to vF,. Denote by D*the orientation of G* thus obtained. To finish the proof, it now suffices to show that dzff,c(v) = 0 for every v E V(G*) = V(D*), where cp, is defined on A(D*)in the obvious way. Consider a shortest closed walk W along bd(F) where F is the face of G corresponding to the arbitrary but fixed v E V(G*). Write W as a vertex sequence, 7 = w l , w2, * * * 9 w k + l where

wk+l = w l .

we then have

k

Thus, by definition

k

VIII. Various Types of Closed Covering Walks

VIII.78

The theorem now follows. It follows that W, is conservative because it is self-dual and graceful ([FRUC79a],[HOED78a]). We note that the edge labeling cp' : E(G) + 2 given by cp'(e) = (pv(y) - cpv(z) for e = z y E E(G) defmes a tension in G corresponding to the potential defined by cpv. Thus, in the planar case, the duality between tensions and flows seems to correspond to a duality between graceful and conservative graphs. However, the authors of [BANGSOa] quote an example which is conservative, self-dual, but not graceful. Furthermore, they also determine classes of graphs which are not conservative. However, if G is a graph consisting of a cycle plus a chord then it is graceful, [DEL080a]. Finally, a connected eulerian graph G is graceful only if qG = 0 (mod4) or qG 3 (rnod4), [GOL072a]. It has been shown in [BODE76a, BODE77al that this condition is also sufEcient in the case where G consists of two cycles having just one vertex in common. Additional references on graceful graphs and related topics can be found in the bibliography of this volume. They are marked with an asterisk in order to distinguish them from quoted references.

A correspondence between l-factors in a graph G without even vertices and certain closed covering walks in G has been established in [KALU77a]. Call a graph H a double-star if it can be written in the form E ( H ) = {zy} U E , U Ev,z # y, and call zcy the axis of H . Note that a double-star may have loops, multiple edges and even triangles. Call a double-star H C G bisecting, if d H ( z ) = $(dG(z) l),d H ( y ) = +(d,(y) l), where z y is the axis of H . The following is true (cf.[KALU77a, Satz I], which allows multiple edges and is, therefore, false; this can be seen by doubling the edges of a l-factor in a hexagon).

+

+

Theorem VIII.65. Let G be a simple graph without even vertices. G has a l-factor if and only if it admits a decomposition into bisecting double-stars. The axes of these double-stars form a l-factor and the edges of a l-factor are the axes of the double-stars in such a decomposition. 13) The proof of Theorem VIII.65 rests on the following facts: 1) G - F is eulerian for any l-factor F c G. Considering an eulerian 13) A variation of this theme of decomposing a graph into double-stars has been treated in [DEAN88a], a special case of Theorem VIII.65 has been proven in [JAEG83d].

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.79

orientation D of G - F and defining for zy E F, Hzy := {zy} UE,+UE:, where EZ = {uz E E ( G ) / ( z , u )E A$ c A ( D ) } ,z E {z,y}, one obtains a decomposition as required. 2) In a decomposition into bisecting double-stars, no two axes are adjacent. However, looking at 1) we can deduce the following result (see [KALU77a, Satz 2’1 and the discussion following Proposition VIII.66). Its proof follows easily by applying the Splitting Lemma and is therefore omitted.

Proposition VIII.66. Let G be a connected graph with 6(G)> 1, and let E, E(G) be such that d G ( z ) = d G ( y ) for every zy E E,. Then E, is a 1-factor if and only if G has a closed covering walk W (of even length) such that edges of E, and El := E(G) - E, alternate in W and such that W uses every e E E, just once, while it uses every f E E, exactly ; d G ( f ) - 1 times (note that d G ( f ) G O(mod2) for f E E, by assumption). However, even if d G ( z ) = d G ( y ) 1(mod 2) for every zy E E,, it may very well happen that W uses some f E E, more often in one direction than in the other (just consider the 3-regular loopless graph on 2 vertices). That is, W may not be a value-true walk whose associated edge labeling cp satisfies cp : E(G) + {0,1}.

An inaccuracy in defining a 2sp-edge of E,, i.e., an edge used precisely 2sp times by W allows an interpretation such that [KALU77a, Satz 2’:(1)] is false. It claims that a loopless graph having a closed covering walk W (of even length) which is alternating with respect to some E, C E(G) and El := E(G) - E,, and which uses every edge in El just once and every f E E, 2sf times, sf E N , has E, as a l-factor (sf may vary). First of all, it becomes apparent from Kaluza’s arguments that G is meant to have no even vertices. But the discussion of the 9-regular graph of Figure VIII.ll shows that even in this case Satz 2’:(1) remains false. However, Satz 2’:( 1) can be saved in any case if one assumes that for f = zy E E, ,the number 2sf satisfies the equation 2sf = dG(z) - 1 = d G ( y ) - 1 = l2 d G ( f 1 - l (see Proposition VIII.66).14) In fact, this seems to be meant (cf. [KALU77a, 14) In this case (which is suggested by the proof of Satz 2’:(1)) it follows immediately that E, is an independent set of edges covering V ( G ) ;i.e., E, is a l-factor.

VIII. Various Types of Closed Covering Walks

vIII.80

B

H

Figure VIII.ll. The 9-regular graph H written in a formal way as H = K4 12B where one attaches three copies of B at each z E { a , b , c , d } = V(K,).E, C E ( H ) consists of E ( K 4 ) and 36 edges of the form z y , three taken from each copy of B. H has a closed covering walk W using every e E El := E ( H ) - Eo just once and each f E E, precisely twice, and such that edges of E, and Elalternate in W .

+

Satz 2

I).

We leave it as an exercise to show that W exists with the properties described in the text of Figure VIII.ll. Noting that one even can obtain W such that all elements of Eo are used in one direction only except for a l-factor L of K4, we conclude that H contains a 6-regular subgraph H , := (K4 - L) 8.(B - 3.(zy}), where exactly two edges of the form z y belong to E ( H ) - E,, and the third to E,. Thus E; := E ( H , ) n Eo consists of 4 2.8 = 20 edges. H , has a nowhere-zero %flow ‘pl with cpl(e) = 2 if e E EA , cp,(e) = 1 otherwise. That is, H , has a value-true walk W , using every e E EA precisely twice in one direction and not at all in the other direction, and using every e E E ( H , ) - Ei just once (that is, W , has the ‘one-way property’ as Kaluza calls it). However, H , is not bipartite and is thus a counterexample to [KALU77a, Satz 31, unless it is meant again that G must not have even vertices.

+

+

However, if we consider a nowhere-zero %flow cp in a simple connected cubic bipartite graph G (see Theorem VIII.21), and if p G =

VIII.2. Value-True Walks and Integer Flows in Graphs

VIII.81

0 (rnod6),pG = 2k > 14, one can construct a non-bipartite 9-regular graph H having a nowhere-zero %flow. We give a rough description of the construction. First of all, the bipartite complement B of G has a hamiltonian cycle C since d,(z) = k - 3 and k 2 8 (see Corollary 111.76). Denote the vertices according to their cyclic ordering in C by a,, b,, u2,b,, . . . ,a,, b,, and identify in G the three pairwise in G nonadjacent vertices ~ 3 ; + 1 b3;+1, , do the same for the pairwise in G nonadjacent vertices b , ; + , , ~ , ( ~ + ~ ) , b , ( ~0 +5~ )i, 5 $k - 1 (:k E N since 2k = pG 0 (rnod6)).H thus obtained is 9-regular, loopless, and 'p induces in H a nowhere-zero %flow 'pH (perform in an orientation D of G corresponding to 'p the same identification procedure to obtain DH corresponding to pH).Correspondingly, a value-true walk W in G induced by 'p corresponds to a value-true walk W , in H induced by 'pH. However, H is not bipartite. This follows directly from the following facts (see also Theorem 111.47): G{a, ,bl r a z is not bipartite since G is 2-connected;15)thus G has a cycle C containing both a, and bl

*

a l , b l , a 2 being independent vertices implies that the path in C joining al and b, but not containing a,, corresponds to an odd in G{a, ,bl ,az} identifying independent vertices in a non-bipartite graph renders a non-bipartite graph. The construction of H from G c a n be interpeted as performing a sequence of contractions; in each step the corresponding three vertices are independent; thus H is loopless but may have multiple edges. It follows that H is 9-regular, non-bipartite, loopless, and H admits a nowhere-zero Sflow. H is thus a counterexample to [KALU77a, Satz 31.''I Nonetheless, this Satz 3 remains valid if one interprets the number 2sf as for the case of Satz 2':(1).

15) G being cubic and bipartite implies that it is 1-factorable, so it cannot have bridges. '1 We note, however, that Satz 2 of that paper remains valid although its validity can no longer be deduced from Satz 2', unless... (see above).

VIII.82

VIII. Various Types of Closed Covering Walks

VIII.3. The Chinese Postman Problem So far, we have studied closed covering walks W in graphs satisfying certain conditions such as a) W uses each edge precisely once in each direction (bidirectional double tracings), b) W uses each edge always in the same direction (nowhere-zero flows, value-true walks respectively). But what if our aim is simply to get through the whole graph (digraph, mixed graph, respectively) as fast as possible ? That is, given a connected graph G, find a closed covering walk W in G of minimal length. This is, basically, how the Chinese Postman Problem (CPP for short) was originally formulated (e.g., see [GUAN62a]). Such a walk W will be called a postman’s tour (PT for short). The idea of dealing with this problem arose at the end of the 1950’s when G u m Meigu (= Kwan MeiKO, according to the old transcription) studied the following question. “ A mailman has to cover his assigned segment before returning to the post ofice. The problem is to find the shortest walking distance for the mailman” (quoted from [GUAN62a, $11). Whence the name Chinese Postman Problem; and it is quite common among Guan Meigu’s colleagues worldwide to call him “The Chinese Postman”. In fact, the C P P has become one of the famous topics in operations research today. Accordingly, we shall discuss the C P P in terms of graphs, digraphs and mixed graphs separately.

VIII.3.1. The Chinese Postman Problem for Graphs Clearly, since every connected graph G has a closed covering walk W using every edge exactly twice (Corollaries V.8 and V.9), the C P P is not an existence problem but a characterization problem concerning those closed covering walks which are postman’s tours, and an algorithmic problem inasmuch as one seeks to determine a PT by a method which is as efficient as possible. Although the CPP was originally formulated only for unlabeled graphs, we shall now consider graphs G together with a cost function c : E(G)-+ R+ U (0); c ( e ) can be viewed as the “length” of the edge e = xy, or the time it takes to travel along e from vertex 2 to vertex y, or the costs

VIII.3.1. The Chinese Postman Problem for Graphs

VIII.83

connected with driving a vehicle from x to y along e, a.s.0. For H an edge sequence W = el, e2,. . .,e, in G respectively, define

E G,

respectively.

T h u s the (general) CPP asks for a closed covering walk W i n G such that c ( W ) is minimal, and, subject t o this condition, Aw(e) 5 2 i f c(e) = 0 and every cycle K satisfying c ( K ) = 0 contains at least one edge appearing just once in W (CPP) (the second condition takes care of the instances where c(e) = 0). In this case we shall also speak of a postman's tour PT, PT(G) respectively. The following result has already been proved, in principle, in [GUAN62a, §3-§4] and [GOOM73a, Theorem 21 (there c(e) = 1, e E E(G),is assumed throughout).

Theorem VIII.67. A closed covering walk W in the (connected) graph G with cost function c : E ( G ) -+ R+ U (0) is a postman's tour if and only if W satisfies the following conditions: 1) 1 5 A,(e) 2)

5 2 for every e E E ( G ) ; c(H n C ) _< f c ( C ) for every cycle C E G , where

H = ( { e E E ( G )/ Aw(e) = 2 ) ) ; 3) if C E G is a cycle satisfying e(C) = 0, thenCnH#C.

Proof. Necessity. Suppose W is a PT. Transform G into a supergraph G+ of G by adding, for every e = zy E E ( G ) , precisely Aw(e) - 1 edges joining x and y. Consequently, W can be transformed into an eulerian trail T+ of G+; i.e., G+ is eulerian. Extend the cost function c onto E(G+) by assigning the same value c(e) to each of the additional Aw(e) - 1 edges, e E E(G). Thus, c(G+) = c(T+)= c(W) . Now, if Aw(e) > 2 for any e = x y E E(G), then delete in G+ two of these additional Aw(e) - 1 edges to obtain a connected eulerian graph H+. We have G C H+ c G+. For every eulerian trail T of H+ we have

c(H+)= c ( T ) 5 c(T+)= c(W) .

(1)

VIII.84

VIII. Various Types of Closed Covering Walks

Since T corresponds to a closed covering walk W’ in G satisfying c(W’)= c(T), and since W is a PT, equality must hold in (1). Consequently, c(e) = 0. This and Aw(e) > 2 contradict, however, the second part of the statement (CPP). Whence A,(e) 5 2 for every e E E ( G ) ;i.e., W satisfies condition 1). Also, since W is a PT it follows by definition that C C H is impossible if C E G is a cycle satisfying c(C) = 0. That is, W satisfies condition 3). Now suppose for H = ({e E E(G)/Aw(e) = 2)) and for some cycle C C G that c(H n C) > c(C - H n C ) . (2) Construct G+ as above and define H , by

E(H,) := ( E ( H )- E ( H n c))u (E(C) - E ( H nC ) )

(3)

(note that e E E(C)- E ( H n C) implies Aw(e) = 1). Construct GT 2 G from G by introducing an additional edge for every element of E(H,). G: is connected and, by (3), eulerian. Extend the cost function c onto E(G:) in the above manner. Gt has an eulerian trail T, which corresponds to a closed covering walk W, of G, and

c(GT) = c(T,) = c(W,) = c ( W )- c(H n C )

+ c(C - H n c)

.

Whence we conclude from (2) that C(Wl>

< c(W>

>

a contradiction to W being a PT. Consequently, we must have

2 c ( H n c )Ic(C-HnC)+c(HnC)=c(C)

,

i.e., 1 2 C G. That is, W satisfies condition 2) as well.

~ ( H n c5) -c(C)

for every cycle C

Suficiency. Let W be any fixed PT, while W’ is an arbitrary closed covering walk of G satisfying the three conditions stated in the theorem. We have to show that c(W) = c(W’). Consider H and H‘ defined correspondingly. It follows from the definition of these graphs, and since W and W‘ are closed covering walks that d&)

= d H , ( V ) = d,(v)

(mod2),

21

f

V(G) .

(4)

VIII.3.1. The Chinese Postman Problem for Graphs

VIII.85

Thus, Ho := ( H U H ’ ) - ( H n H’) is eulerian. By Theorem IV.l, H, has a cycle decomposition So (possibly So = 0). Note that

We conclude from the fact that W is a PT

Assuming now c(W) # c(W’)it follows that (5) is a strict inequality; i.e.,

c(HnC,)

< c(H’nC,)

for some Co E So

.

(6)

However, by definition of H ,

+

c(C,) = c(H n C,) c(H’ n C,) .

(7)

Thus we obtain from (6) and (7) 1

-@,) 2

;

i.e., W’ violates condition 2) for C, E So. This contradiction proves c(W) = c(W’). Moreover, condition 3) secures that a cycle K satisfying c ( K ) = 0 contains some e with A,,(e) = 1. That is, W‘ is a PT. This finishes the proof of the theorem. Although Theorem VIII.67 characterizes postman tours in a simple way, it is impractical from an algorithmic point of view since it requires, in principle, the examination of all cycles of G in the context of conditions 2) and 3). Guan Meigu himself, being aware of this problem, showed that it suffices to examine chordless cycles when checking conditions 2) and 3) for a closed covering walk W of G, [GUAN62b] (of course, first one has to check whether W satisfies condition 1) of Theorem VIII.67). But even this restriction to chordless cycles does not seem to be feasible for constructing a good algorithm to solve the CPP.

VIII.86

VIII. Various Types of Closed Covering Walks

However, despite the shortcomings of Theorem VIII.67 just mentioned, this theorem already contains the starting point for what is needed to produce such a good algorithm. To this end, let us consider, for a postman’s tour W in G, the graph

H = ( { e E E(G)/Xw(e)= 2)) as defined in condition 2) of that theorem, together with the graph G+ obtained from G by duplicating the elements of E ( H ) . Since G+ is eulerian we have d H ( v )E dG(u) (mod2), 2, E V ( G ) (4’) (see (4) in the proof of Theorem VIII.67, and define d H ( v ) = 0 for

V ( H ) ) . Moreover, H is acyclic because of conditions 2) and 3) v of Theorem VIII.67, and the odd vertices of H are the odd vertices of G by the above congruence; let these vertices be denoted by q ,. . . ,‘upk. By applying Corollary VIII.47 to each component of H , we obtain k-edgedisjoint paths Pl, . . . ,Pk in H such that

On the other hand, the application of Corollary VIII.47 as such does not require knowledge of H to find such k paths Pl,. . . , p k ; and if one has such k paths, then G t which is obtained from G by duplicating precisely the edges of these paths is a connected eulerian graph satisfying condition 1) (but possibly not conditions 2) and 3)) of Theorem VI11.67. However, if one knows H , it follows of necessity that H can be written k as H = Ui=l Pisince G+ is eulerian and W is a PT. Generalizing this idea of finding Pl,. . . ,P k such that k

Vodd(G):= {U E V ( G ) / d ( v ) 1( m o d 2 ) ) =

UV,(Pi) i= 1

we consider n,(Vo,,), an arbitrary partition of V,,,(G) into k classes of size 2, and denote by P2(Vodd) the set of all these partitions. For fixed n, E P2(Vo,,)and arbitrary {vil, v i a }f I&, let Pi1,i2 be a ‘shortest’ path (in terms of the cost function c ) joining vil and vi2 in G, and denote Cil,i2

=C(Pil,i2)

.

VIII.3.1. The Chinese Postman Problem for Graphs Denote 4 3 2 )

c

= {.;I

Ci1,iZ Iviz}En~

VIII.87

*

With this notation we are led to an alternate characterization of the solutions of the CPP (see also [GOOM73a, Theorem 11, which considers the case c(e) = 1, e E E(G)).

Theorem VIII.68. The problem of finding a solution of the CPP for the connected graph G with cost function c : E(G)-+ R+ is equivalent to finding nq E p 2 ( v o d d ) such that

That is, if W is a PT in G, c(W) = c(G)

+ c(l$)

.

The proof of Theorem VIII.68 can be deduced easily from the preceding discussion and is therefore left as an exercise. We note, however, that the k paths corresponding to the elements of II! are edge-disjoint; this follows from the minimality of c(II;) and because c(e) > 0, e E E(G).We also note that [GOOM73a] contains solutions of the CPP for various classes of graphs for the special cost function c : E ( G ) + (1). If, however, the cost function in Theorem VIII.68 is defined by c : E (G) + R+ U {0}, II: may not correspond to a postman’s tour. However, by deleting certain edges e with c(e) = 0, these k paths corresponding to rI; can be transformed into k edge-disjoint paths corresponding to nk E p 2 ( V o d d ) such that C(IIt) = C ( r I S ) .

At a first glance, Theorem VIII.68 does not seem to be an essentially better characterization of postman tours than Theorem VIII.67. For, a simple combina.torial argument shows that for k 2 4

Ip2(vodd) I= (2k - 1)(2k - 3). . .3.1 2 3k (Exercise VIII.19). That is, if G has no even vertices, and p :=p , 2 8,

VIII.88

VIII. Various Types of Closed Covering Walks

i.e., the number of partitions to be considered in determining c(IIi) grows, in principle, e ~ p o n e n t i a l l y . ~ ~ Nevertheless, ) Theorems VIII.67 and VIII.68 yield the following lower and upper bound for a PT (Exercise VIII.20; see also [HEDE68a, Propositions 5,6] and [GOOM73a, Corollaries la,6b]).

Corollary VIII.69. For any postman's tour W in the connected graph G and any cost function c : E ( G ) -+ R+,

the lower bound being achieved if and only if G is eulerian, and the upper bound being achieved if and only if G is a tree. For the case c s 1, upper bounds can be found in [WATT79a,KESE87a].

However, Theorem VIII.68 points in the right direction. Using the notation introduced in the discussion preceding Theorem VIII.68, we may obtain the following criterion.

Theorem VIII.70. Let G be a connected non-eulerian graph with cost function c : E ( G ) R+, and let C* be the cost function of K2k defined by C * ( U 1. U3. ) = C 2. J ' Ui,'Uj E V,dd(G) = V(K2,) . (4 --$

'

For E, C_ E(K,,) define c*(Eo):= CeEEo c*(e).

A closed covering walk W of G is a PT if and only if c ( W ) = c(G)

+ m i n { c * ( L ) / Lis a l-factor of K Z k } .

The proof of Theorem VIII.70 follows from the observation that there is a l-l-correspondence between the partitions IT2 E P2(v0,d) and the l-factors L of K2k and vice versa, and that c(II,) = c*(L) by the very definitions of c(II,) and c*. Thus, the above discussion shows that the Chinese Postman Problem can be transformed into the following equivalent algorithmic form (CPP') which is more explicit than the original statement (CPP), and the idea 17) In a different approach using dynamic programming, Bellman runs into similar problems concerning the number of steps needed to determine a postman's tour W and c(W),[BELL69a].

VIII.3.1. The Chinese Postman Problem for Graphs

VIII.89

of which is contained in [EDMO73a] as one of the possible approaches to solving (CPP) (see also [CHRI75a, p.2051).

1) Input: Given the connected graph G with cost function c : E ( G ) -+ R+ U (01, let V0dd(G)= {q,. . . , v ~ denote ~ } the set of odd vertices of G,k 2 1 (see Corollary VIII.69). 2) S P P (Shortest Path Problem): Determine = r n i n ~ ( P ~1, ~5> , i < j 5 2k, where Pi,j is an arbitrary path joining vi, vj E V,,,(G) in G . Store precisely one P<j satisfying c ; , = ~ C ( P < ~1>5, i < j 5 2k. 3) MMP (Minimum Weight Perfect Matching Problem): For K z k , V ( K 2 , )= Vodd(G), and for c* : E ( K z k )+ R+ U{O) satisfying c ’ ( v j v j )= ci,j7determine a l-factor L E(K,,) such that c*(L) is minimal. 4) EUL (Euler Tour Problem): For every vivj E L duplicate in G the 18) to obtain the connected eulerian graph G+. Construct edges of

an arbitrary eulerian trail T+ of G+ and describe W , the closed covering walk of G corresponding to T+.W is a PT. This idea of transforming (CPP) into (CPP’) is already contained basically in the note [EDM065b] and has been restated in [GOOM73a, p.271 (but only for the case c : E(G)-+ (1)). As for the complexity of an algorithm based on (CPP’), we take note of the following facts:

SPP. The Shortest Path Problem stated in 2 ) can be solved in polynomial time. For, if we apply Dijkstra’s algorithm, say, for a fixed ZI; E Vodd(G), we obtain in O(p2) time at the most not only all values ci,j but also corresponding paths P&, 1 5 j 5 2k,j # i (see, e.g., [BOND76a, p. 17-19]). Thus, SPP can be solved in 2k0(p2) 5 O ( p 3 )time at the most. Thus estab lishing K,, with cost function c* and a list of the above paths P [ j can be done in polynomial time S(p). MMP. The Minimum Weight Perfect Matching Problem (i.e., finding a l-factor L in K 2 k with minimum cost c * ( L ) )can be solved in O ( p 3 )time at the most (Theorem 111.88.~)).The fist person to show that MMP Is) As we have noted above, these k paths P;:j are pairwise edge-disjoint, or at least can be transformed into such paths by deleting certain e E E ( G ) with c(e) = 0. By the same token, we may assume w.1.o.g. that this edge E(P<j)/c(e) = 0)) is acyclic. deletion procedure is such that ( { e E UvivjEL

VIII.90

VIII. Various Types of Closed Covering Walks

can be solved in polynomial time was J. Edmonds [EDM065a],19)while [PAPA82a, Theorem 11.31 contains an explicit and concise “ Weighted Matching Ahpm’thm” yielding directly a 1-factor L with minimal c* ( L ) . We note, however, that the development of this algorithm is based on [EDM065a]. In any case, MMP can be solved in polynomial time M ( p ) (see also [LOVA86a, Theorem 9.21, and p. 376]).20)

EUL. G+ can be constructed from G in polynomial time E,(P) since the paths corresponding to the elements of L can be found in polynomial time in the above list of all <:j ’s. As we shall see in the chapter on algorithms, an eulerian trail T+ in G+ can be found in polynomial time E 2 ( p ) ,where 2’ can be transformed into W , a PT of G, in polynomial time E3(p). Thus, the Euler TOUTProblem can be solved in polynomial time E(p) := E,(P) E,(p) E&).

+

+

Summarizing these considerations, we can say that the Chinese Postman Problem can be solved in polynomial time C ( p ) ,where

In fact, in their famous paper [EDM073a], J. Edmonds and E.L. Johnson formulated the Chinese Postman Problem as an integer linear programming problem (ILP for short). Their starting point is the loopless connected graph G with cost function c : E ( G ) -+R+ U (0). Associating a variable z, with every e E E(G), the original (CPP) can be expressed 19) Actually, J. Edmonds does not provide an explicit complexity study in his paper. Moreover, Edmonds’ algorithm finds a 1-factor L of maximum cost. This, however, is no real obstacle; for, if we define a new cost function i. by ?(vi,vj) = c(G)- c ~2 , 0,~we have e(L) = k.c(G) - c*(L)for any 1-factor L E(K2,).In short, finding L with maximum ;(L)is tantamount t o finding L with minimum c*(L). ’O) The same formulation (CPP’) can be found, basically, in [SERD74a]. This paper also contains the discussion of SPP as outlined above. However, observing that a 1-factor is a bipartite graph and making use of an O ( k 3 )time algorithm for finding a minimum weight 1-factor in Kk (ascribing this algorithm t o E.A. Diniz and N.A. Kronrod), A.I. Serdjukov’reduces the MMP t o 2k-1 solving the problem for each of the ( k-l ) subgraphs Kk,, of Obviously, this algorithm is inadequate. However, neither M. Guan’s nor J. Edmonds’ work is mentioned in this article.

VIII.3.1. The Chinese Postman Problem for Graphs

VIII.91

in the following equivalent form: Determine

5,

f (0, l}, e E E ( G )

c(e)x, is minimized

such that

(CPP”)

eEE(G)

+

(1 xe) = 0 (mod 2 ) , 21 E V ( G ).

subject to eEE,

The equivalence of (CPP”) and (CPP) follows from the fact that any postman’s tour W of G corresponds to a solution of (CPP”) satisfying

H = ({e E E ( G ) / x , = 1)) and

c(H)=

c(e)z,= efE(G)

c(e)x,

,

eEE(H)

(where H is the subgraph of G defined by W in condition 2) of Theorem VIII.6?), and vice versa (also note that c ( W ) = c(G) c ( H ) ) . Moreover, the p congruences in (CPP”) are equivalent to saying that G+ obtained from G by duplicating the edges of H is eulerian.21)

+

In order to reduce (CPP”) to a special case of the general matching problem as treated in [EDMO?Oa], Edmonds and Johnson transform (CPP”) into another equivalent form: starting from the incidence matrix B = [bJ of G, let di E {0,1} be defined by d ( v i ) = CjZ1 P bij = di ( m o d 2 ) , and attach variables wi and x j to the vertex ‘ui, the edge e j respectively, 1 5 i 5 p , 1 5 j 5 q, such that (1) w; E 2, xj E 2 (2) wi 2 0, xj

20

a

(3)

1b i j x j - 2wi

(CPP”’)

j=l

4

(4) z = x C ( e j ) x j j=1 21) In (EDM073a1, the x, are only required t o be non-negative integers, for, as the authors later note, if 2, > 1 then c(e) = 0; and replacing x, with x, - 2 will also yield a solution of the CPP as stated in that article. Also, the acyclity of H as defined in Theorem VIII.67, may not hold for H derived from a solution of (CPP”),but it can easily be obtained by deleting whole cycles K if necessary; for such cycles satisfy c ( K ) = 0.

VIII.92

VIII. Various Types of Closed Covering Walks

In order to see that the solutions of (CPP”) correspond to solutions zj E ( 0 , l ) of (CPP”’), and vice versa, it suffices to observe that

c

(1

c

+ Xj) = d(2J;)+

ejEEvi

ej

xj

+

= d(2Jj)

EEv;

c P

bij”j

j=1

i.e., P

b;jzj

= d(2Jj) 2 d; (mod 2)

j=1

if and only if

(1 ej

+ zj)

0 (rnod2).

EEvi

On the other hand, any solution of (CPP”’) can be reduced to a solution of (CPP”) satisfying xj E (0,l) (see the preceding footnote). We note that [EDMO7Oa] gives an explicit polynomial upper bound on the number of steps needed to solve (CPP”’).

In (CPP”’), the restriction (1) (the solutions w i , z j , 1 5 i 5 p , 1 5 5 q must be integers) can be replaced by what the authors call Blossom Inequalities. This modified version of (CPP”’) is the starting point for describing the BZossom AZgom’thm which is a matching algorithm and which allows one to produce the eulerian graph G+ by duplicating the ‘matching edges’.22) As before, any eulerian trail of G+ corresponds to a PT of G. For more details, see [EDM073a], and the chapter on algorithms, where we shall also briefly discuss the End-Pairing Atgorithm and Nezt-Node Algorithm as described in that paper. We remark that a modified version of the Blossom Algorithm can be found in [HECK76a]. Unfortunately, neither of these two papers contains an explicit complexity

j

22) The first version of the Blossom Algorithm appears in [EDM065c] which is also the starting point for [EDM065a]. However, [EDM065c] covers the case c : E(G)---t (1) only. According t o [LOVA86a, Table 9.1.11, this Blossom Algorithm basically has a running time of O(p4). This fact has also been

noted in [EVEN75a], where the authors develop an O@295) algorithm t o find a maximum matching in a graph; their algorithm’s starting point is [EDM065c] and subsequent improvements of the Blossom Algorithm by various authors.

VIII.3.1.1.1. Applications

VIII.93

study of the respective Blossom Algorithm.23) For a thorough discussion of matching algorithms, see [LOVASGa, Chapter 91, but also [LAWL76a] which contains a modified version of the Blossom Algorithm with a running time of 0(p3). We also quote [BARA88d] as a recent example of improving the ILP ‘side’ of the CPP.

VIII.3.1.l. Some Applications and Generalizations of the CPP The CPP and the way to solve it (discussed above) has found numerous applications - in day-to-day life as well as for other mathematical problems. In fact, Guan Meigu himself did not foresee the significance and wide range of applications of the CPP when he first formulated it.24)

VIII.3.1.1.1. Applications Although the formulation of the CPP rests on the idealizing assumption that the postman has to deliver at least one piece of mail in each streetsection (which corresponds to an edge in the associated graph), there are practical problems which - by their very nature - automatically require that every edge is passed at least once in the graph associated with the street system under consideration. Some of the following examples relating to such practical problems are quoted in [CHRI75a]. a) Garbage collection, milk delivery. Just as in the case of a PT, one is also faced in these cases with restrictions such as vehicle capacity (and/or the weight of letters and parcels a postman can carry) and the length of a working day, which are not explicitly formulated in (CPP) - (CPP”’). However, these restrictions axe implicitly taken into account when considering graphs which are ‘not too large’. That is, these restrictions are S.A. Viches, [VICH83a], starting from a problem in computer graphics (minimization of drawing time for a plotter) and reducing it to (CPP’) improves the Blossom Algorithm so as to obtain a way of finding a solution of (CPP’) in (basically) O(p3) time. Another feature of Viches’ algorithm is that it drastically reduces the size of the memory needed during the running of the Blossom Algorithm. 24) Mentioned during a short conversation at his home in Jinan, China, in July 1987. 23)

VIII.94

VIII. Various Types of Closed Covering Walks

reflected by the number of postmen serving a certain district of a city, or by the number of garbage trucks (milk vans) used to collect garbage (deliver milk) in a certain area.25).

b) Street cleaning, gritting roads in winter, cleaning ofices and corridors. In the first two cases, we are faced with the same type of restrictions as in a) (maximum capacity of tank used to sprinkle water on the streets, or maximum load of the truck carrying grit), while such technical restrictions usually do not occur in the case of cleaning office buildings, say (here, the essential factor is the work load allotted to the cleaning staff). c ) Reading of (gas, water, electricity) meters. In some of the above applications (e.g., garbage collection, street cleaning, milk delivery), considering the CPP for graphs may not be an adequate reflection of reality. For, if a truck (van) serves houses on a rather broad street, it would seem more appropriate for the truck to serve the two sides of the road separately, i.e., in each of the two directions. This observation points towards the directed or mixed postman problem. We note that T.M. Liebling has undertaken an exhaustive study of the street-cleaning problem for the city of Zurich, Switzerland, [LIEB7Oa]. Unfortunately, at the time he did not use a polynomial time matching algorithm. Generally speaking, the CPP can be applied whenever one has to distribute or collect certain ‘goods’ in every section of a network of streets, and if one has just one means of transport to hand: a bicycle, a truck, or just the feet of the postman. However, P. Jiirvinen has studied the question of distributing newspapers by two m e a s of transport; e.g., by private van or by post. In this case, we have two types of unit cost associated with two means of transport: c1 > 0 for transportation by the van, c2 > 0 for mail delivery. Of course, the unit cost c1 is related to mileage, while the unit cost c2 is related to sending a newspaper by mail. Denoting by d j j the ‘length’ of edge xixiE E(G)and by f i j the number of copies to be sent by mail to sixiE E(G), P. J k i n e n arrives at the following formulation of the problem: find a closed walk W in a simple 25) The fact that garbage collection usually takes place once a week only, can be reflected by solving the CPP for five graphs associated with one truck. Also, in the course of time there may be an increase in the amount of mail to be delivered to a house, or in the amount of garbage to be collected from a house. These changes may require redefining the graphs on which one seeks a solution of the CPP, or introducing larger garbage trucks.

VIII.3.1.1.1. Applications

VIII.95

connected graph G such that

becomes a minimum ([JARV73a]).

d) The maximum weight cycle packing problem (MCPP for short) asks for a set S = {el,. . . ,C,) of edge-disjoint cycles Ci, i = 1,.. .,T , in a graph G with non-negative cost function c such that

[GUAN84b]. (CPP) and the MCPP are in fact equivalent.

Theorem VIII.71 ([GUAN84b, Theorem 11). Let S = {Cl, -..,C,} be a set of edge-disjoint cycles of the connected graph G with cost function c : E(G)+ R+ U (0). The following statements are equivalent. (1)S is a solution of the MCPP. (2) c(S) + 2c(H) = c ( W ) , where H := (E(G) any PT of G .

UI=l E(Ci))and W is

Proof. In fact, if G+ is an eulerian graph obtained from any graph G by doubling the edges of some subgraph H _C G, then G - H is eulerian since G - H = G+ - UeEE(H) C 2 ( e ) ,where C2(e)is the digon of G+ obtained by duplicating e E E ( H ) . Thus, for any cycle decomposition S of G - H we have c(G) = c(S) c ( H ) = c(G - H ) c ( H )

+

and

+

+

+

c(G+) = c(G) c ( H ) = c( S) 2c(H) .

That is, c (H ) is minimum if and only if c(S) is maximum (I). Moreover, for any closed covering walk W using edges once or twice only we have 1 c(W) = c(G) C ( H , )

+

where H, VIII.67).

= ( ( e E E(G)/X,(e)

= 2)) (see the proof of Theorem

VIII.96

VIII. Various Types of Closed Covering Walks

That is, W is a PT (or can be transformed into a PT by deleting cycles K c H , with c ( K ) = 0) if and only if c(H,) is minimal (2). Combining (1) and (2) for H = H , with the equations concerning c(G+) and c(W), we now conclude the validity of the theorem. We note that the proof of Theorem VIII.71 also shows that a solution of the MCPP for a graph G can be found in polynomial time. For, since we can find a PT W in G in polynomial time, G - H , can also be found in polynomial time. As we shall see later, a cycle decomposition S of G - H , can also be found in polynomial time; and S is a maximum weight cycle packing of G by Theorem VIII.71. Finally, we note that Theorem VIII.71 implies Corollary VIII.69.

On the other hand, we can already note that if we consider cycle coverings S of G (thus requiring G to be bridgeless) and seek such an S where c(S) is minimal, there is no such simple equivalence between the CPP and this minimum weight cycle covering problem (MCCP for short), as expressed by Theorem VIII.71 for the CPP and the MCPP. However, in the context of compatible cycle decompositions we shall prove that the CPP and the MCCP are equivalent in the planar case, while this does not generally hold true for the non-planar case. For applications of the CPP to finding ground states of Ising spin glasses and to the design of VLSI circuits, see [BARA88b,88c].

VIII.3.1.1.2. t-Joins, t-Cuts and Multicommodity Flows Following A. Sebo, [SEBO84a, SEBO85a, SEBO86a1, we introduce the following concepts. Throughout this subsection, we assume G to be connected.

Definition VIII.72. For a graph G, define t : V ( G ) --t 2 and t ( X ) := C I E X t (for z ) X C V ( G ) . Suppose t ( V ( G ) ) = O ( m o d 2 ) . F E(G) is called a t-join of G if d(,)(v) E t ( v ) (mod2) for every v E V ( G ) . For X S V ( G ) ,the coboundary E ( X , V ( G )- X ) is called a t-cut of G if t ( X ) f 1 (rnod2) (Note that t(Y) t ( V ( G )- Y )(mod2) for every Y C V ( G )because of t ( V ( G ) ) O ( m o d 2 ) ) . We observe that these concepts are equivalent to T-joins and 2'-cuts as defined, for example, in [SEYMglb, KORA82a, LOVASSa]. There, T &

VIII.3.1.1.2. &Joins, t-Cuts and Multicommodity Flows

MII.97

V(G) is chosen such that ITI= 0 (rnod2), which means in the context of Definition VIII.72,

T = ( v E V ( G ) / t ( v ) 1(mod2))

.

vodd((F))= T for any t-join F

,

This implies

t ( X ) =lX n TI+ (V(G) - X)n TI (mod2) for any X

c V(G) .

It is for practical purposes, however, that we have chosen A. Sebb’s definition of t-joins and t-cuts.

In particular, if t satisfies t ( v )-= d ( v ) (mod2), v E V ( G )

,

then the above T satisfies

Moreover, for such t we have dealt already with t-joins: namely the subgraphs H C G satisfying condition 2) of Theorem VIII.67 and used in producing the eulerian graph G+ (see (4’) in the discussion following the proof of Theorem VI11.67). Finally, for these t the tcuts are precisely the odd (edge) cuts. This follows from the fact that V ( G ) . On the I fl V,dd(G) E ( X , V ( G )- I (mod2) for any other hand, the crucial steps in (CPP’), the algorithmic form of (CPP), are SPP and MMP, and there it was only for practical purposes that we considered V,,,(G) rather than an arbitrary even subset T of V(G). In the language of Definition VIII.72 and its subsequent equivalent form, for T = V,,,(G) we were looking for a T-join H with minimal c ( H ) : this was equivalent to finding a 1-factor L in K Z k V(K,,) , = T , such that c*(L)is minimal. Thus, even in the general case of a non-negative cost function c associated with G , the problem of determining a t-join F with minimal c(F) can be solved in polynomial time.

x

x)

x

However, for the subsequent considerations we assume G to have cost 1. We define for fixed t : V ( G ) 4 2 with t ( V ( G ) ) G function c 0(mod 2)

T ( G t, ) = min{lFI / P i s a t-join of G } , v(G,t ) = maz{ IEs I /&, is a set of disjoint t-cuts of G } .

VIII. Various Types of Closed Covering Walks

VIII.98

Furthermore, for z,y E V(G),z

# y, define

Using this notation, we summarize some properties of minimum t-joins in the next results, [SEBO85a]. Lemma VIII.73. Let F be a t-join and let F' be a t'-join of G. The following statements are true. a) I F

C

I=

T(G,t ) if and only if IF

n E ( C )151E(C)- F I for every cycle

C G. 1 F I= T ( G ,t ) and 1 F'

b) If for every cycle

C

)=T ( G ,t'),then IF C_ ( F U F' - F n F') .

n E(C)I=) F' n E ( C )I

The proof of Lemma VIII.73 is left as an exercise since it can be derived from the proof of Theorem VIII.67. Lemma VIII.74. Let F be a minimum t-join of G , p , > 1, and let x,y E V(G),z # y, be arbitrarily chosen. There exists a minimum tx,Y -join F1 of G arid a path P ( z , y ) joining x and y such that F = Fl u W(? Y)) - Fl n E(P(x, Y)).

Proof. By Definition VI11.72 and the definition of t,,, we have for any t-join F and any t,,,-join F'

+ d(F,)(w)f 0 (mod2) and

whenever

z

#w#y ,

+

.

d ( F ) ( ~ ) d ( F O ( z )5 1 (mod2) for z E {z, y} Thus we have for H := (F U F' - F

n F')

d H ( v ) 3 1 (mod2) if and only ifw E Thus we can write

(2,y}

.

VIII.3.1.1.2. &Joins, t-Cuts and Multicommodity Flows

VIII.99

where P ( z ,y) is an arbitrary path from z to y in H and S = {Cl,. . . ,C,} is an arbitrary cycle decomposition of the eulerian graph H - E ( P ( 2 ,y)) (see (1)).Of course, S = 8 is possible. Assume, in addition, that F, F' have been chosen such that

Consider the above S = {C,, . . . ,C,} and let

i = 1,.. .,m. By definition, F ( i ) is a t,,,-join for i = l , . ..,m. The definition of F ( i ) and the first part of (2) yield

F ( ~=) ( F ( ~ - --~(F' )

n E(c;))) u ( F n E(c;)).

(4)

(4)and Lemma VI11.73.b) now yield

Moreover, the definitions of H and

P(')and the second part

of (2) yield

F' n E ( P ( z ,9)) = F (4 n E ( P ( z , y ) ) ,i = 1,. . . , m , F(") n E ( S ) = F n ~ ( s ) . Since and

F = ( F n F ' ) u ( F n E ( S ) ) u ( F n E ( P ( 2 ,y)))

(5) (6)

,

F(") = ( F n F ' ) u ( ~ ( " 1 n E ( s ) )u (F(") n E ( P ( ~y))) ,

we obtain from (5) and (6)

That is, F, := F(") is a t,,,-join as required. Lemma VIII.75. Consider a minimum t-join F of G, where t ( v ) f 0 (mod2) for some 21 E V(G), and choose z, y € V(G),x # y, such that T(G, is as small as possible. Then d ( F )(2) = d ( F )(y) = 1.

VIII.100

VIII. Various Types of Closed Covering Walks

Proof. By Lemma VIII.74, there exists a minimum t,,,-join Fl of G and a path P ( z ,y), x # y, such that F = Fl U E ( P ( z ,y)) - Fl n E ( P ( z ,Y)). Since t(v) $ 0 (mod 2) there exists a path Q(v, w)C ( F ) such that t ( v ) t ( w ) 1(mod2). Whence F -E(Q(v, w)) is a t,,,-join, i.e., T(G,t,,,) < 7 ( G , t ) . Consequently, the choice of z , y in the statement of the lemma implies 7.(G,G,y)< G t ) * (4

=

We claim that d ( q ) ( z ) = d(Fl)(y) = 0 must hold (**). Otherwise, assuming w.1.o.g. d ( q ) ( z ) > 0 we have some e E Fl n E, joining z and z , for example. But then the definition oft,,, implies that Fl - { e } is a L Y -join provided z # y. In this case we have T ( G , ~ , ,<~T) ( G , ~ , , ~a ) , contradiction to the choice of z and y. Assuming z = y for all possible choices of e E Fl nE, and, symmetrically, z’ = z for all possible choices of e’ E Fl n E, (e’ joins y and z ’ ) , we conclude from the minimality of Fl that (FlnE,) is a path PI N K, and thus a component of ( F l ) (see Lemma VIII.73.a)). If Pl = P ( z ,y), we obtain a contradiction to (*) or else t(v) 0 (mod2) for every v E V ( G ) which contradicts the hypothesis. Whence C = PI U P ( z ,y) is a cycle. We have F n Fl 2 E ( P l ) and

On the other hand, this equation and Lemma VIII.73.a) yield

Combining these inequalities we obtain

an obvious contradiction which implies the validity of (**). It now follows from the relation between F and Fl that for P = P ( x ,y)

This finishes the proof of the lemma.

VIII.3.1.1.2. t-Joins, t-Cuts and Multicommodity Flows

VIII.101

With the help of these lemmas we are able to present A. Sebii's short proof of an important result of P.D. Seymour, [SEYM8lb, SEBO85al.

Theorem VIII.76. If G is a bipartite graph, T ( G ,t ) = v(G,t ) . Proof. If t ( v ) O(mod2) for every z1 E V ( G ) then F = 0 is the only minimal t-join of G, and E, = 8 is the only set of disjoint t-cuts. Thus 7 ( G , t ) = v ( G , t ) = 0 in this case. Whence suppose t ( v ) f O(mod2) for some v E V ( G ) ;let F # 0 be a minimum t-join and choose x , y E V ( G ) ,z # y, such that T ( G , ~ , , is ~ )as small as possible. By Lemma

VIII.75,

d(,)(4

= d(&)

=1

(1)

Let G be the graph obtained from G - x by contracting N ( z ) C V ( G ) onto a vertex. G is connected and bipartite by construction. Let

t^ : V ( & )+ 2 be defined by

t^(G) 0 (mod 2) follows from the definition of t^. Define 3 = F - Ex and note that (1) and the choice of F imply

Also, the definition of ?i and F imply that t-join of G. Thus 1 312 .(G,t^>.

F

is a t^-join of G since F is a

If I F I> .(G,t^), then Lemma VIII.73.a) and G being bipartite imply that G contains a cycle K such that p(k)nPpIE(2)-%1+2

.

(3)

Since F is a minimum t-join in G, and because of (3), z E V ( k )must hold, where z E V ( G )- V ( G ) . Then E ( k ) = E ( P ( w l , w 2 ) ) , where P(w,, w2)is a path in G joining wl, w 2 E N ( z ) ,and wl # w2. Whence there exist wlz,w2x E E, such that K := @(I?) U {wlz, w2z}) is a cycle in G. By Lemma VIII.73.a) we have

vIII. Various Types of Closed Covering Walks

vIII.102

However,

( E ( K )n F p l E ( k ) nP1,

( E ( K )- F I I ( E ( ~-)PI +2

(5)

by definition of K , K , F, #. Thus we have in (3), (4),( 5 ) equality throughout, implying wlx,w2x @ F . But 1 E ( K ) n F !=I E ( K ) - F I classifies Fl := ( F U E ( K ) )- ( F n E ( K ) ) as a minimum t-join of G. It follows that {w1z,w2z} U F n E, C F,, i.e., d F,)(x) = 3 since d(F)(z)= 1. On the other hand, the definitions o f t an Fl and the choice of x,y permit the application of Lemma VIII.75 which implies d ( F (z) = 1, an obvious ?) contradiction to d(F,)(z)= 3. This contradiction implies the falsity of

d

IF\>T(G,i). Now, having proved I F I= .(G,i) and observing that E, is a t-cut satisfying E, n E(G) = 8, we consider is, a set of disjoint ;-cuts of G, satisfying I I= v(G,I!) .

E,

It follows from the very definition of G and t^ that every E , E 2, is a t-cut. Thus E, := 2, U (E,} is a set of disjoint t-cuts of G which implies

Now we proceed by induction on T(G,t). Having shown at the beginning that the theorem holds if T(G,t ) = 0, and applying induction to I k )= ~ ( d we , obtain from (2), (6) and the definition of v(G, i) = T ( G ,I!) =IF I -1 = T(G,t ) - 1 5 v(G,t ) - 1

,

On the other hand, the very definition of a t-join F and a t-cut E ( X ,V ( G )- X ) implies I F n E(X,V ( G ) - X ) 1- 1(mod 2). Whence G cannot have more than IF I disjoint t-cuts; i.e., v(G, t ) 5 T ( G ,t ) . This and (7) finish the proof of Theorem VIII.76.

As P.D. Seymour notes if G is not bipartite, the conclusion of Theorem VIII.76 is no longer generally valid. Just take G = K4 and t(v) = 1, 2, E V(K,). A characterization of the graphs satisfying the conclusion of this theorem has not been found yet. However, A.M.S. Gerards very recently generalized Theorem VIII.76 by establishing a sdlicient

VIII.3.1.1.2. &Joins, &Cuts and Multicommodity Flows VIII.103

condition under which T(G,t) = v(G,t). It is formulated in terms of forbidden homeomorphic subgraphs of two types (see [GERA88b,Theorem 1.11).

In fact, P.D. Seymour obtained his result as a generalization of a result . which was extracted from (but not explicitly stated in) [EDM070a, 731 first proven in [LOVA75a] (see Corollary VIII.78 below). In order to state this result in a short form we introduce the following concepts.

Definition VIII.77. For a graph G consider t as in Definition VIII.72. A k-packing of t-cuts of G is a collection Es k of t-cuts of G (repetitions allowed) such that every edge of G belongs t; at most k (not necessarily different) elements of

Denote by

and observe that vl(G,t) = v(G,t ) .

Corollary VIII.78. v2(G,t ) = 27(G,t) for any graph G.

Proof. Consider S(G) which is a bipartite graph and extend t to t‘ : V ( S ( G ) )-+ 2 by defining t’(v)

1(mod 2 ) if and only if

t ( v ) f 1(mod 2)

.

By definition oft’, there is a 1-1-correspondence between t-joins F of G and t’-joins F‘ of S(G), defined by F‘ = E ( S ( ( F ) ) ) ;therefore, I F I= 7 ( G , t )if and only if I E ( S ( ( F ) ) )I= 7 ( S ( G ) , t ’ ) On . the other hand, for every t-cut E, of G there are 2po1 t’-cuts Ek of S(G) corresponding to E,: for if s, E V ( S ( G ) )- V ( G )is the subdivision vertex corresponding to e E E ( G ) , such an E; can be obtained by taking precisely one f E Ese C E ( S ( G ) )for every e E E,. However, for each of these EL there is another one, E{ say, such that

Conversely, the definition of t‘ implies for every t’-cut of S(G), EL = E ( X ,V ( S ( G ) )- X ) , that

J?$ $ E8, , or else N ( s , ) C X or N ( s , ) c V ( S ( G ) )- X

.

(2)

VIII.104

VIII. Various Types of Closed Covering Walks

Whence we conclude that Ek defines, in a unique way, a t-cut Eo of G. Summarizing these considerations and applying Theorem VIII.76 we obtain for a minimum t-join F of G

I=I

27(G, t ) = 2 IF E ( S ( ( F ) ) I= ) =T(S(G),t’) = v ( S ( G ) t’) , 5 v2(G,t )

(3)

(the inequality follows from (1) and (2)). Moreover, if €s,z is a 2-packing of G, the above construction of EA from Eo and (2) imply that can be transformed into a 1-packing of S(G), i.e.,

4 G t t ) I W G ) ,t’)

(4)

*

The corollary now follows by combining (3) and (4). Note that if we define t(v) = d(v), it follows from Corollary VIII.78 that the solution of the ‘unweighted’ CPP (i.e., c = 1) is given by qG 7 , where rn is the maximum size of a 2-packing of odd cuts of G, [LOVA86a, Corollary 6.5.111.

+

Corollary VIII.78 can be generalized as follows. Corollary VIII.79 ([SEYMSlc, (2.5)]). Let G and t be given as before, and let w : E ( G ) t N U (0) be such that w(C) := CeEE(,-,) w(e) = 0 (rnod2) for every cycle of G. Let F be a t-join with minimum w ( F ) := C f E F w ( f ) . Then there are w ( F )t-cuts such that every e E E(G) belongs to at most w(e) of them. By subdividing every e E E(G) with w(e) vertices one can prove Corollary VIII.79 in a way similar to the proof of Corollary VIII.78 (Exercise VIII.22). Note, however, that Corollary VIII.79 is reduced to Theorem VIII.76 if w 1. Also, if one deletes in Corollary VIII.79 the parity condition concerning w(C),one obtains a ‘weighted’ result which is related to Corollary VIII.79 in a way analogous to the relation between Corollary VIII.78 and Theorem VIII.76 (see [LOVA86a, Theorem 6.5.201).

We note in passing that the treatment of t-joins and t-cuts as developed in [SEBO84a, SEBO86a, FRAN84al yields an algorithm for solving the CPP which no longer requires the type of matching theory developed in [EDM073a]; see also [LOVA86a, p.245-2471. Furthermore, some of the following topics of this subsection have also been treated in [GROT85a, 8.5,8.6].

VIII.3.1.1.2. &Joins, &Cuts and Multicommodity Flows

VIII.105

However, if we combine Lemma VIII.73.a) and Theorem VIII.76, we arrive at the following result which opens the door to the consideration of the Multicommodity Flow Problem (MFP for short) in planar graphs (for the formulation of the MFP, see below).

Corollary VIII.80 ([SEYMSlb, (5.1)]). Consider F C E ( G ) , where G is a bipartite graph. The following statements are equivalent: 1) For every f E F, there exists an edge cut Es,p 3 E,,, n Eslg= 8 if f # g , where f , g E F . 2)

(E(C) n FI
{f}of G such that

G.26)

Proof. By defining t : V ( G )4 2 by t(v) = d(F)(v), we classify F as a t-j oin.

If 2) holds, IF I= T ( G t, ) by Lemma VIII.73.a). Since G is bipartite we have I F I= T ( G , ~=) v ( G , t ) by Theorem VIII.76; i.e., there are I F I pairwise disjoint t-cuts of G, each of which intersects F (see the last paragraph of the proof of Theorem VIII.76). This implies the validity of 1)If 1) holds, we have 1 F n E,,, I= 1(mod 2), f E F , which classifies Esl, as a t-cut of G for every f E F , so E, := { E , , f / f E F } is a set of pairwise disjoint t-cuts of G. Consequently,

However, since G is bipartite the application of Theorem VIII.76 yields equality in (*), so F is a minimal t-join of G which, by Lemma VIII.73.a), implies the validity of 2). The corollary now follows. Now, if we assume G to be planar, then there is a 1-1-correspondence between cycles (or minimal edge cuts) of G and minimal edge cuts (or cycles) of D(G),where D(G) is the dual of G in an arbitrary embedding of G in the plane (note that the existence of such a 1-1-correspondence is independent of any actual embedding). Moreover, G is bipartite if and only if D ( G ) is eulerian. Thus, we can express Corollary VIII.80 in the planar case in the following equivalent form [SEYMSlb, (5.2)]. 26) In fact, if one replaces in 2) the term ‘cycle’ with ‘eulerian subgraph’ the equivalence with 1) remains valid (since one is considering a set of edgedisjoint cycles simultaneously); and P.D. Seymour stated his result originally in that manner (which leads more directly to Corollary VIII.81 below).

VIII.106

VIII. Various Types of Closed Covering Walks

Corollary VIII.81. Consider F

E E(G),where G is a planar

eulerian

graph. The following statements are equivalent.

1) G contains pairwise edge-disjoint cycles Cf3

{f},f E F .

2) IE, n F 151E, - F I for every (minimal) edge cut E, C_ AT(G).’~) P.D. Seymour shows by example, though, that neither of the hypotheses ‘planar’ and ‘eulerian’ can be dropped. Following [KORA82a7SEBO???] we formulate the MFP in the following way2’) For a graph G, F E ( G ) , and p : E(G) 4 N U { 0 } , does there exist a set of cycles S = {Cl,. . . , C,} and a function ~!,t S + R+ such that

a) IE(C)n F

b)

I=

1 for every C E S;

Ce€C;€S $(Ci)

5

cp(e) for arbitrary e E E ( G ) , with e p a l i t y

holdang zf e E F ?

A necessary condition for the existence of a solution of the MFP is expressed in the next lemma [SEYM8lb, (5.3)]. Lemma VIII.82. Let G, F , and p be as in the formulation of the MFP. A solution of the MFP (i-e., a system of cycles satisfying a) and b)) exists, only if cp(E, n F ) 5 p(E, - F ) for every edge cut E, of G. Proof. For every cycle C E S containing f E F n E,, there is at least one edge in E, - F that is also contained in C (see a) in the formulation of the MFP). This and the second part of b) ensures that for each term of the sum in b), Ci is also accounted for in p(E, - F ) , if e E F . Hence,

This proves the lemma. 27) If the inequality holds for minimal edge cuts, it also holds for arbitrary edge cuts, and vice versa (See also the preceding footnote).

’‘1 We note that the MFP as defined here is equivalent t o the MFP as expressed in [SEYM8lb]. In fact, P.D. Seymour defines the MFP such that cp is just a non-negative real valued function, but he notes that it suffices t o consider the case when cp is integral.

VIII.107

VIII.3.1.1.2. t-Joins, t-Cuts and Multicommodity Flows

Suppose now that G is also planar. Assuming F C E(G) and 'p : E(G) 4 N U (0) be given, we construct G$ by first deleting the edges e for which p(e) = 0, and then replacing every other edge g = zgyg by 2'p(g) edges joining zg and yg. By construction, G$ is eulerian and F C E(G) corresponds to a set F, E E(G$) containing all edges joining zf and yf and corresponding to any f = zfyf E F . Moreover, I Fv I= 2 4 F ) and, similarly, every edge cut E, of G corresponds to an edge cut E, of G$ having precisely I E, I= 2'p(E,) elements. On the other hand, every edge cut E, of G$ corresponds to an edge cut E, of G . If G,F, 'p satisfy the inequality of Lemma VIII.82, we have for every edge cut E, E(G$)

IE, n F,

I=

2 4 E , n F ) 5 2'p(E, - F ) =IE, - q,I

.

By Corollary VIII.81, for every f E F,, G$ contains a cycle C, 3 {j} such that S, := (Cf/j E F,} is a set of pairwise edge-disjoint cycles. S, corresponds to a set S of cycles in G . However, it follows from the construction of G t that different cycles of S, may correspond to the same cycle of S. Let ?r : S, + S denote this correspondence. We define for C E S,

(we call such a

t+!J

i-integer-vahed).

Nevertheless, it follows from the definition of G $ , S,, and S that S satisfies property a) of the MFP. Property b) holds by definition of S, and S, and by definition of ?r and (+). In particular, if f = zfyf E F , the 2 4 f ) cycles of S, containing an edge joining zf and yf and each edge corresponding to f , satisfy

Summarizing the above considerations we obtain the following, [SEYM78a, SEYMSlb].

Theorem VIII.83. For a planar graph G let there be given F E ( G ) and 'p : E(G) + NU (0). A solution S of the MFP exists if and only if

'p(E, n F ) 5 p ( E , - F ) for every edge cut E, of G .

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VIII. Various Types of Closed Covering Walks

Moreover, S can be chosen such that $ is i-integer-valued. Of course, cannot be chosen integer-valued in general because one has to construct an eulerian G;. However, if - in addition to the hypotheses p(e) 0 (rnod2), a solution S of Theorem VIII.83 - ‘p satisfies CeEE, of the MFP exists for which is integer-valued (in this case it suffices to replace every e E E(G ) with p(e) edges to obtain an eulerian graph G:). On the other hand, P.D. Seymour conjectured, that in general, if there is a solution of the MFP, a solution S can be found in which t+h is $-integer-valued. This conjecture was supported by Theorem VIII.83 and its extension to such graphs which have no subgraph contractible to K5 (see [SEYM8lb, 81c] where, in the latter paper, the MFP is being studied for various types of matroids). However, a counterexample to this conjecture has been presented in [HASS84a, Figure 6(a)]. $J

+

Although the inequality of Lemma VIII.82 is, in general, not sufficient for the existence of a solution of the MFP, it is a sufficient condition in some special cases other then those already quoted, such as (see also [SEYM8lb, KORA82al): 1) whenever IF

I=

1 (in this case we obtain the mas-flow-min-cut theorem; see Theorem 111.85 or [FORD62a]);

2) whenever IF )=2 (this yields Hu’s 2-commodity flow theorem, a short

proof of which can be found in [SEYM79a]);29)

3) whenever (E(G)- F ) has a plane embedding in such a way that the open edges of F lie in the unbounded face of (E(G) - F ) , [OKAM8la] (also here S can be found such that $ is i-integer-valued). In fact, the general MFP can be solved in polynomial time since it can be reformulated as an LP having polynomially many constraints and variables associated with the edges of G, provided F E E(G)is of given size k (this was pointed out to me by P.D. Seymour). In the case of the planar graphs described in 3), good algorithms for solving the MFP have been developed in [HASS84a, MATSSSa]. Shortly afterwards, the authors of the latter paper found a good algorithm for solving the MFP for another class of planar graphs G, [MATS86a]; its running time is O(n5I2log n). By reducing the MFP for G to a CPP in the dual D ( G ) ,one can obtain an 29) The dual result has been proven in [SEYM78a]; a strengthening of Hu’s result can be found in [ROTH66b].

VIII.3.1.1.3. Hamiltonian Walks, the TSP, and the CPP VIII.109

algorithm with O(n3I2log n) running time, [BARA87a]. Very recently, A. Sebii gave a polynomial algorithm for the plane M F P (i.e., G is planar), provided IF1 is bounded by a fixed integer k, [SEBO89b]. On the other hand, the integral MFP is an NP-complete problem [EVEN76a] (integral refers to the range of t,b).30) For other problems related to the CPP, t-joins respectively, we refer the interested reader to [JOHN87a].

VIII.3.1.1.3. Hamiltonian Walks, the Traveling Salesman and Their Relation to the Chinese Postman Our starting point is again a connected graph G with cost function t R+ U (0). Recall that a V(G)-covering walk is a closed walk W in G containing every vertex of G, and that hamiltonian cycles and closed covering walks in G are special cases of V(G)-covering walks. Just as one asks, in the case of the CPP, for a closed covering walk W of minimum c ( W ) ,the Hamiltonian Walk Problem (HWP for short) asks for a closed V(G)-covering walk W’ of minimum c(W’); such W’ will be called a hamiltonian walk. As can be seen from Figure VIII.12, it may very well be that no hamiltonian cycle of G is a solution of the HWP in G. On the other hand, if G is hamiltonian and c is a constant, any hamiltonian cycle of G is a solution of the HWP in G, and vice versa. However, as we shall see below, in general the HWP in G is nothing but a special case of the Traveling Salesman Problem (TSP for short). For the TSP we are given p “cities” vl, . . . ,v p , p 2 3, and “distances” d,,j E R+ U (0) between vi and vj, 1 5 z,j 5 p , i # j , d i , j = dj,i. The problem is to c : E(G)

+

(putting p 1 = l), where T is an arbitrary permutation of the integers 1, . . . ,p (see e.g., [PAPA82a, p.4; GARE79a, p.2111). But this is 30) In the case of 2-commodity flows, the integral MFP has been solved for some special classes of graphs in [ROTH66a]. Moreover, it has been shown in

[FRAN85a] that the integral MFP can be solved in polynomial time for plane graphs in which every odd vertex belongs to the outer face boundary. For a solution of the integral three commodity flow problem, see [OKAM83a].

VIII. 110

VIII. Various Types of Closed Covering Walks

X

1

Y

.

F,gure TIII.1 K 3 with cost function c : E(G) t F ’ satisfying c(zy) = c(zz) = l,c(yz) = 3. One has c(H) = 5 for the unique hamiltonian cycle H , whereas c(W) = 4 for the unique solution W of the HWP in K 3 . nothing more than considering the complete graph K p with cost function c* : E(K,) + R defined by c*(wjwuj) = d i , j , and to find a hamiltonian cycle H C K p such that c*(H) is minimum. (TSP’)

However, Figure VIII.12 shows that solving (TSP’)is not equivalent to solving the HWP in K p . The theoretical reason for this difference between HWP and (TSP’)lies in the definition of c*, c respectively.

Definition VIII.84. We say that a cost function c : E(G) t R+ U (0) satisfies the triangle inequality if c(z, y)

5 c(z, 2) + c(z, y) for every

2,

y, 2 E V(G)

,

where c(u,w) := rnin(c(P(u,v))}, the minimum taken over a l l paths P(u,w) C G, where G is (w.1.o.g.) connected. The cost function of K3 as described in Figure VIII.12 does not satisfy the triangle inequality; and this is the reason why (TSP’) and HWP are not equivalent problems in this case. On the other hand, since every hamiltonian cycle of a graph G is a V(G)-covering walk in G, we conclude that every solution H of (TSP’) and every solution W of the HWP (in K p ) satisfy c(H) 2 c(W) for any non-negative cost function c

(*>

(i.e., regardless of whether c satisfies the triangle inequality).

Proposition VIII.85. If the cost function c : E ( K ) 4 R+ U (0) satisfies the triangle inequality, every solution of (TSP’Pis a solution of the HWP in K p with respect to c.

VIII.3.1.1.3.

Hamiltonian Walks, the TSP, and the CPP vIII.111

Proof. Consider W , a solution of the HWP, and suppose that W is not a hamiltonian cycle of K p . Then we can write W as an edge sequence in the form

w = . . . ,uv,vw,. . . ,xv,vy,. . . , where u,v, w,2,y are not necessarily a l l distinct. Because of the triangle inequality and by the choice of W , we conclude that the V(G)-coveringwalk W‘ obtained from W by replacing the section x v , v y with xy, i.e.,

W’ = . . . ,uv,vw,. . .,xy, . . .

,

satisfies c(W‘) = c(W). That is, W’ is also a solution of the HWP. W ’contains fewer edges than W . So, by repeating this reduction step, if necessary, we arrive finally at a V(G)-covering walk W(’) satisfying c(W(’))= c(W) and containing no vertex of K p more than once. That is, W ( i )is a hamiltonian cycle which is a solution of the HWP as well as of (TSP’)(see (*) above). Whence we conclude that every solution of (TSP’)is a solution of the HWP. This finishes the proof.

So, let us consider the HWP for arbitrary connected G and arbitrary cost function c : E(G) + R+ U (0). Similar to what we did in transforming (CPP)into (CPP’)(see the discussion following Theorem V111.70), we define for K p , V(KJ = V(G) := { v i / i = 1,. . . , p } the cost function c* bY c*(v.v.) = c;,j , 3 % where c , , ~= min c(P(vi,vj)). P( ~i ,uj)

It follows from the very definition of c* that this cost function satisfies the triangle inequality. By Proposition VIII.85 and its proof, every solution is a solution of the HWP in K p with this cost function c * , and of (TSP’) every solution of the HWP in K p can be transformed into a solution of (TSP’) in K p with the same c* as cost function. Note that if G = K p and if c does not satisfy the triangle inequality, the definition of c* amounts to “lowering the cost” of certain edges of K p (Note that the very nature of the TSP implies that w.l.o.g., G can be assumed to be simple).

As for any solution W of the HWP in G and the solutions of (TSP’) in K p with the respective cost functions c and c*, we note that because

VIII. Various Types of Closed Covering Walks

VIII.112

of G & K p and because of the choice of W we can view W as a V ( K p ) covering walk in K p ;c IE(W)= c* IE(W) follows of necessity. Otherwise, by definition of c* we would have c(zeye) > c*(zeye) for some e = zeye E E ( W ) , which amounts to the existence of a path P ( z , , y e ) G such that c(P(ze,ye)) = c*(zeye); and replacing in W the edge e with the path P ( z e ,ye) would render a V(G)-covering walk W' satisfying c(W') < c( W ) ,a contradiction. We claim that W is a solution of the HWP in K p as well: for, any solution W* of the HWP in K p with cost function c* corresponds to a V(G)-covering walk W, in G, m

j=1

where m = Z(W*), and for j = 1,.. . ,rn

+

(putting m 1 = 1). By the above, by the choice of W * and by the construction of W,, it follows that

c ( W ) 2 c(W,) = c*(W*)L c * ( W ) . This and c I ~ ( ~ =) c * I ~ ( ~imply ) the validity of the above claim. Moreover, it follows that any solution of the HWP in K p with cost function C* corresponds (via the above definition of W,) to a solution of the HWP in G with cost function c. Summarizing the considerations including and following Proposition VIII.85 we obtain our next result.

Theorem VIII.86. Let G be a connected graph with cost function c : E(G) --t R+ U (0}, and let K p ,p = p G , with cost function c* be defined as above. Then every solution W of the HWP in G corresponds to a solution H of (TSP') in K p and vice versa, with respect to the corresponding cost functions c and c*, and c ( W ) = c * ( H ) .

In other words, solving the Hamiltonian Walk Problem in G with cost function c is polynomially equivalent to solving the Traveling Salesman Problem in K p 3 G,p = p G , with the corresponding cost function c*.

VIII.3.1.1.3. Hamiltonian Walks, the TSP, and the CPP

VIII.113

The details of this equivalence can be extracted from the discussion of (CPP’) (see SPP and EUL) and from the construction of W, above; it is therefore left as an exercise to do this. Whence we conclude that the crucial difference between solving the Chinese Postman Problem on the one hand and the Traveling Salesman Problem (respectively, the Hamiltonian Walk Problem) on the other hand, lies in the difference between finding a l-factor L in KZksuch that c*(L) is minimum, and finding a hamiltonian cycle H in K p such that c*(H) is minimum, where 2k =I V,,,(G) 1 and p = p G . However, (TSP) is an NP-complete problem even if c : E ( K p )-+ NU (0) (see [GARE79a, ~ . 2 1 1 ] ) . ~ ~ ) However, instead of transforming the HWP into (TSP’) one can seek a characterization of hamiltonian walks similar to Theorem VIII.67 regarding postman tours, if one is more interested in finding structural differences between the CPP and the HWP. In fact, Theorem VIII.67 itself expresses the difference between these two problems. Proposition VIII.87 ([GOOM74a, Lemmas 5a, 6al). If W is a hamiltonian walk in G then W satisfies conditions 1)- 3) of Theorem VI11.67, while a V(G)-covering walk satisfying these conditions need not be a hamiltonian walk. The proof of the first part of Proposition VIII.87 can be extracted from the proof of Theorem VIII.67 by observing that W induces an eulerian trail in a supergraph of the connected spanning subgraph ( E ( W ) )of G (it is left as an exercise to work out the details). The second part follows from the example of Figure VIII.12, where one takes W := z, xy,y, yz, z , z z , z. In other words, solving the HWP for G amounts to finding a connected spanning subgraph G‘ of G and doubling certain edges of G‘ to obtain an eulerian graph (GI)+ such that .((GI)+) is minimal. This and Proposition VIII.87 yield the following bounds for the length of a hamiltonian walk (see also [GOOM74a] which considers only the case c G 1). Corollary VIII.88. Let G be a connected graph with cost function c : E(G) + R+ U (0). If W is a hamiltonian walk and W’ a postman’s tour in G, then c ( W ) 5 c(W’) 5 2c(G). Usually the TSP is formulated in such a way that one asks whether exists such that c*(H) 5 U , where u is a given (positive) integer. This, however, has no impact on the above considerations. 31)

H

VIII.114

VIII. Various Types of Closed Covering Walks

Moreover, since a double tracing in a spanning tree B of G is a V ( G ) covering walk, and since the determination of such B with minimal c(B) can be achieved in polynomial time (see Theorem III.88.b) or [PAPA82a, Theorem 12.2]), we obtain c ( B ) 5 c(W) 5 2c(B). These are relatively good bounds for a solution of the HWP which can be determined in polynomial time. We note that for the case c = 1, better upper bounds have been determined for special classes of graphs such as triangulations of the plane [ASAN8Oa], graphs satisfying certain degree conditions, [BER075c], and n-connected graphs having diameter d, [GOOM74a].32) Moreover, in the latter paper a procedure is described by which G+,the eulerian supergraph of G corresponding to a given PT, can be transformed into (G’)+, the eulerian graph corresponding to a given hamiltonian walk of G.

As for a formulation of the TSP as an ILP, we refer to [PAPA82a, p.3081. Moreover, the reader interested in a thorough treatment of the TSP is referred to [LAWL85a]; an account of research on the TSP from the late sixties to the early eighties is given in [PARK83a]. As recent example of solving the TSP for graphs with up to 1000 vertices we quote [GROT88a].

VIII.3.2. The Directed Postman Problem Here, the starting point is a strongly connected digraph D and a cost function c : A ( D ) + R+ U ( 0 ) . The Directed Postman Problem (DPP for short) is (as in the case of the CPP) to f i n d a closed covering walk W an D such t h a t c(W) is minimum, and every cycle K satisfying c ( K ) = 0 contains a t least one arc appearing j u s t once in W (DPP). Call such W a directed postman’s tour (DPT for short). Analogously to the case of graphs, the construction of such W in D amounts to duplicating arcs in D so its to obtain an eulerian digraph D+ such that c(D+ - 0 ) is minimum and D+ - D is acyclic; and an eulerian trail in D+ corresponds t o a DPT in D. Contrary to the case of graphs, a result strictly analogous to Theorem VIII.67 cannot hold in generai. For, if I od(v) - i d ( v ) [> 2rnin(od(v),i d ( v ) } for some v E V ( D ) , the construction of D+ yields Aw(a) > 2 for at least one a E A,. To 32)

max,

The diameter of G , symbolically diam(G),is defined by diarn(G) := where d ( z , y) is the distance between 2 and y in G.

,yEV(G) d ( z , y),

VIII.3.2. The Directed Postman Problem

VIII.115

establish a characterization of a DPT analogous to Theorem VIII.67 we need an auxiliary result on digraphs. For this purpose consider a digraph D and K c A ( D ) and denote by G , the graph underlying ( K ) .

L e m m a VIII.89. Let a digraph D be written as the arc-disjoint union of two subdigraphs D, and Db; w.l.o.g., v(D,)= v ( D b ) = v(D).Suppose

Then either A ( D ) = 0 or A ( D ) can be decomposed into non-empty classes K j , 1 5 i 5 rn, such that

1) GKi is a cycle for i = 1,.. .,m. 2 ) D , ; := (A(D,)fl Kj) U v(D)and Db,;:= ( A ( D b )n Ki) U v(D) satisfy (DIFF) for i = 1,.. . ,rn (with D,,; in place of D, and

Db,jin place of Db).

=

Proof. First of all, d ( v ) O(mod2) for every w E V ( D ) ;this follows from A(D,) f l A(Db) = 8 and (DIFF). Whence we conclude that if A (D) # 8, then A ( D ) has a decomposition S = { K l , .. .,K,; m 2 1) whose elements satisfy 1). In order to see that such an S exists whose elements also satisfy 2 ) , consider an arbitrary 2) E V ( D )for which d(v) = A(m If A(D) = 2, then K i = A(CJ is as required for every C;, where Cj is a weakly connected component of D. Hence S = { K i / C iis a weakly connected component ofD} is a partition of A ( D ) as required. Note that if G,; is given an orientation, and if one considers in ( K i )a walk W following this orientation, then (x,y) E A(Dr,j)is passed by W from x to y (from y to x) i f and only if every (u,v) E A(Db,i)is passed by W f r o m v to u (from u to v ) (D). This follows from (DIFF) which is satisfied by the elements of S because they satisfy 2 ) . Consider the case A ( D )

>

2.

In order to apply (D) and (DIFF)

we replace every vertex v for which d ( v ) = 2k, > 2 with kV2-valent vertices as follows: first split away pairs of arcs u - , u + E A,, where a- E A;, a+ E A:, and { u - , u + } C A(D,) or { u - , u + } C A(Db),un-

til such arc pairing is no longer possible; then split away pairs of arcs U r , a b E A,, where U, E A(D,) and a b E A(Db). It fOllOWS from (DIFF) that either { u , , a b } c A: or { u , , a b } c A;. In any case, we end up

VIII.116

VIII. Various Types of Closed Covering Walks

with a digraph D’such that A(D’)= 2 and, by construction, D’ satisfies (DIFF) as well. By the first part of the proof, A(D’) has a decomposition S’ whose elements are the arc sets of the weakly connected components of D’ and which satisfy l), 2) and, therefore, (D). Consequently, either S’ corresponds to a decomposition S of A ( D ) such that each of the elements of S satisfies 1) and 2), or else 1 K’ n A, I> 2 for some K‘ E S’ and some v E V ( D ) . In the latter case we can write

K’ = K: u K 3! , K: n K’.3 = 0 such that {vi,w3} = V ( ( K { ) Dn , )V((K;),,) where v i , v j are two of the k, 2-valent vertices corresponding to v E V ( D ) . Whence by identifying v i and vj we obtain D” from D’. D” also satisfies (DIFF), and by (D), S” := (S’ - { K’}) u { K:, K ;} is a decomposition of A(D”) satisfying 1)and 2) (as for D,, D, C D , we assume that they are transformed together with D,D’, a.s.0.). Now we argue as above with D“ and S” in place of D‘ and S’, a.s.0. Finally we arrive after j 5 CvEv(D) k, steps at S ( j ) , a decomposition of A ( D ( j ) ) satisfying 1) and 2), which corresponds to a decomposition S of A ( D ) as required; possibly D ( j ) = D. The lemma now follows.

We make the following observations the proof of which is left as an exercise. a)If S is a decomposition of A ( D ) satisfying 1) and 2) of Lemma VIII.89 then D, and D, satisfy (DIFF). b)If D is weakly connected, (DIFF) is a necessary and sufficient condition for the existence of a closed covering arc sequence To of D which corresponds to an eulerian trail T of the underlying graph G,, such that every (x,y) E A(D,) is passed by T, from x to y, while every ( u , v ) E A(D,) is passed by To from v t o u. Thus To can be viewed as lying inbetween the concept of an eulerian trail and the concept of an antidirected eulerian trail (see Definition VI.20). c)Lemma VIII.89 and a), b) generalize Theorem IV.8. For, if A(D,) = 8, then (DIFF) reduces to id,(v) = od,(v).

We are now able to characterize a DPT in a way similar to Theorem VIII.67. Recall that Aw(a) denotes the number of times a E A ( D ) appears in the walk W .

VIII.3.2. The Directed Postman Problem

VIII.117

Theorem VIII.90. For a strongly connected digraph D with cost function c : A(D ) R+ U {0}, let W be a closed covering walk in D ,and let Do := ( { a E A(D)/Xw(a)> 1)).W is a DPT if and only if it has the --f

following properties:

1) Do is acyclic; 2) For every K

A ( D ) such that G, is a cycle, it follows that

c(Dr,K)5 + c ( K ) whenever condition (DIFF) of Lemma VIII.89 is fulfilled by a spanning Dr,,

c ( K n A(D,)) u v(D) and

Db,,

:= ( K - A(Dr,,))

u v(D)

*

Proof. Suppose W is a DPT. Similar to the proof of Theorem VIII.67, construct the eulerian digraph D+ 2 D by duplicating arcs of D such that W corresponds to an eulerian trail T+ of D+, and extend c onto A(D+). Suppose Do contains a cycle C.0: := D+ - A(C) contains an isomorphic image of D and is eulerian; an eulerian trail 2': of 0: corresponds to a closed covering walk W, of D. Since W is a DPT we must have

c(W) = C(T+)= C ( D + )= c(D;) = c ( T . )= c(W,) and thus c ( C ) = 0. This contradicts the requirement of (DPP) and implies the validity of 1). To see that 2) holds as well we consider a corresponding arc set K A ( D ) , and let D+ be as above. Form 0: by deleting in D+ for every a E A(Dr,,) one of the arcs corresponding to a and add one extra parallel arc to every a' E A(Db,,). It follows that D$ is eulerian if (DIFF) is satisfied by K , D,,,D,,, respectively. For this condition means that in the transition from D+ to D:, at any vertex both in- and out-degree are increased by 1 or decreased by 1, or they remain unchanged; and D+ is eulerian. Transforming an eulerian trail 2'; of 0: into a closed covering walk W2in D it follows from the choice of W and the construction of W2 that

VIII. Various Types of Closed Covering Walks

VIII.118

i.e., 2) holds as well. Similar to the proof of Theorem VIII.67 we now consider a DPT W together with a closed covering walk W, satisfying 1) and 2); let D+ and D: be the eulerian superdigraphs of D corresponding to W,W, respectively. We have for every 2) E V ( D )= V ( D +- A ( D ) )= V(DF - A ( D ) ) , WD+-A(D)(")

= M D : - A ( D , ( 4 = -d.1,9,(4

(1)

*

Define

D,

:= D+ - A ( D ) ,D, := D

t - A(D) ,

and viewing D, and D, as arc-disjoint digraphs, define

.

D:=D,uDb

D , D,, D,

satisfy the hypothesis of Lemma VIII.89 by definition and because of (1). So, if A ( D ) = 0, then c ( W ) = c(Wl), and W , is a DPT as well. Assuming A ( D ) # 0, let S be a decomposition of A ( D ) as described in Lemma VIII.89, 1) and 2), and consider an arbitrary Ki E S, and Dr,i,Db,ias defined in that lemma. Moreover, let Do = ( { a E A(D)I&V(a) > I)), Di = ( { a E A(D)IXw,(a) > 1)). Consider in D the arc set K corresponding to K;. If I K (= 1, then K C A ( D o )n A(DA),and 1 c(Dr,K)

where D,,,

= C(Db,K) =

corresponds to Dr,i and

Db,,

Zc(Ki)

(2)

7

corresponds to Db,;.

If I K (> 1, then G, is a cycle, and by definition

c where D,,,

(Db,$-)

A(Db,K)

corresponds to Dr,i

c

A(D,')

i

(3)

(Db,i).

Note that some a E Do n DA n K may be accounted for in A ( Dr l j )but not in A(D, .) for some K i E S, and vice versa; this follows from the satisfy (DIFF) it definition of D and S. However, since D,,i and D,,; follows that D , , and D,,, also satisfy (DIFF). By the same token, we may just as well let D,,, (Db,,) correspond to D,,;(D,,i) thus obtaining 7'-

c

A(Db,K)

&

*

(4)

VIII.3.2. The Directed Postman Problem

VIII.119

Having proved already that the DPT W satisfies 1) and Z), and because W, has these properties by assumption, we obtain from (3) and (4)

That is,

D,,, and D,,,

also satisfy (2) in the case where G, is a cycle.

Having proved (2) for every element of S we conclude from

(where these unions are arc-disjoint) that

hence

+

+

c(W) = c(D) c(Dr)= c(D) c(Db) = C(W1) . That is, W, is a DPT (Observe that the acyclicity of Dt automatically secures the validity of the second part of (DPP)). This finishes the proof of the theorem. We note in passing that Theorem VIII.90 has been published in [FLEISOb]; moreover, certain types of A(D)-covering walks have been studied in [BOGN87a]. However, (DPP) can be formulated as an ILP much the same way as (CPP) has been formulated in (CPP”): namely (see [EDM073a, KOHK74a]), determine xa E N U {0}, a E A ( D ) such that

c ( a ) z , is minimal aEA(D)

subject to

z, = id(v)

xu aEAt

and

2,

- o&(v)

(DPP’)

aEA,

is minimal.

aEA(D)

Note that the second constraint makes sure that x, does not become unnecessarily large whenever c(u) = 0. Therefore, it also secures that H = ( { u E A ( D ) / z , > 0)) is acyclic. This constraint is of no decisive

VIII. Various Types of Closed Covering Walks

VIII.120

impact, though. It does not appear in [EDMO73a] since the authors do not demand additionally that a DPT should contain as few arcs as possible (see also [KOHK74a, Lemma]). So, while H as defined above may contain cycles if this constraint is omitted, the successive deletion of the arc sets of cycles to obtain an appropriate acyclic subdigraph of D is a polynomial problem. With this observation in mind we recognize (DPP’) as a special case of the transportation problem (see [BERG62a, p.2071 and p.III.72), in which the arcs have infinite capacity, and where source and sink are deleted. However, the DPP can be treated as a matching problem in a way similar to (CPP’). The idea behind this approach can be extracted from the proof of Proposition VIII.26 and is analogous to the transition from (CPP) to Theorem VIII.68 and Theorem VIII.70. We know from the proof of Proposition VIII.26 that if d i f l ( v ) > 0 for some v E V ( D )then w E V ( D ) exists for which diff(w) < 0 and a path P ( w ,v) C D (since D is strongly connected), such that D, obtained from D by doubling the arcs of P ( w ,v) satisfies

vEV(D)

VEV(Di)

That is, we can construct the eulerian digraph Dc by successively doubling the arcs of certain paths; this guarantees a solution of (DPP’). However, since these paths cannot be chosen arc-disjoint in general (see the discussion preceding Lemma VIII.89), the determination of these paths as corresponding edges of a matching in an auxiliary graph requires a modification in the definition of this graph when compared with the definition of in Theorem VIII.70, and/or (CPP’).For this purpose, associate with every v E V ( D ) for which dZ;trD(v) # 0, 6 , :=I d Z f f D ( v ) I vertices vl, . . . , v6,, thus obtaining two sets of vertices,

V + :=

U u€V(D)

diff

(Y)>O

{ v i/l

5 i 5 b V ) , V - :=

U u€V(D)

di#Df*)
{v i / l

5 i 5 bv}.

VIII.3.2. The Directed Postman Problem

it follows that I V + I=I V -

1;

denote rn

:=I

vIII.121

V+1.

,with V(K,,,)

= V +U V - and with cost function c* : E(K,,,) -+ R+ U (0) which is derived from c in a way analogous to the case of graphs, namely

Now consider K ,

C*(ViWj)

= min(c(P(w,v))}

,

where

w j E V + ,w j E V - , 1 5 i 5 6,, 1 5 j 5 6,

,

and the minimum is taken over all (directed) paths from w to v, where d i . ( w ) < 0, difl(w) > 0.

Our next result is basically the same as [KOHK74a, Lemma and Theorem]. Theorem VIII.91. Let D be a strongly connected digraph with cost function c : A ( D ) + R+ U {0}, and let K,,, with cost function c* be defined as above. Let D+ be an eulerian digraph obtained from D by duplicating certain arcs of D in such a way that H := D+ - A ( D ) is acyclic. Then the following statements are equivalent (with c extended onto A(D+). See also Theorem VIII.70 and the paragraph thereafter). 1) c ( H ) is minimal. 2 ) A ( H ) has a decomposition into rn paths P I ,. . . ,P, which correspond to a 1-factor L of K,,m such that c*(L)is minimal.

Proof. Since D+ is eulerian, we have in any case

and H is acyclic. Construct a digraph H* from H by splitting the vertices of H in the following way. a) Whenever d i f l H ( w ) = 0, split v into i d H ( v )= odH(v)2-valent vertices vi each having id(vi) = od(vi) = 1, i = 1,.. . ,odH(v); b) if diffH(v) > 0, then split v into 6,l-valent vertices vi which are sources in H * , i = 1 , . . . ,6, and into (odH(v)- 6), 2-valent vertices as described in a), 6, =I digH(.)I; c ) if di#,(v) < 0, then produce analogously 6, sinks of H* and (idH(w)6,) 2-valent vertices as described in a).

vIII.122

VIII. Various Types of Closed Covering Walks

By construction and since H is acyclic, each (weakly connected) component of H* is a nontrivial path or an isolated vertex. That is, H* has a path decomposition S, := {Pl,.. . ,P,} which in turn is a path decomposition of H precisely because H is acyclic. Moreover, by construction of H * , each Pj, j = 1,.. . ,n, joins some x satisfying d z f l , ( z ) > 0 with some y satisfying dzfl,(y) < 0; and every such x (y) is the initial (end-) vertex of precisely 6, (6,) paths in S ., From this and since

we obtain, by utilizing (1) and the definition of rn, the equation m=n

.

(2)

Moreover, the structure of the elements of S, permits the association of S, with a whole variety of 1-factors L of K , ., For, if Pi,, . . . ,Pis, E S, all have v as their initial (end-) vertex, we can arbitrarily associate with them certain edges incident with vl, . . . , v6, E V - ( V + ) ,the only restriction being that if Pi,has w as its end- (initial) vertex and has { T , k} C (1,. . . ,b v } , then e E Uf:l E,. been associated with e E EVp, must hold as well, where wl,. . . ,wg, E V +( V - ) . We note, however, that c(Pi,) > c * ( e ) may hold if Pi,E S, has been associated with e E L. In any case, c(Pik)< c*(e) is impossible by definition of c*; whence we conclude

c(H)1 c*(L)

(3)

for any 1-factor L of Km,, associated with any path decomposition S, of H . Conversely, if L is any 1-factor of K,,,, and if we add A, - 1 parallel arcs to every a E A ( D ) , where A, is the number of paths containing a and corresponding to edges in L , the resulting digraph D+ is eulerian. This follows precisely from the construction of Km,m,the definition of c * , and because L is a 1-factor of K,,,. Moreover, the construction of Dt guarantees a path-decomposition S, of H := D+ - A ( D ) such that the elements of S, correspond bijectively to those of L , so we also have

c ( H )= c * ( L ) .

(4)

VIII.3.2. The Directed Postman Problem

VIII.123

Now, (3) and (4) immediately imply the equivalence of statements 1) and 2) which finishes the proof of the theorem. We note that Theorem VIII.91 and its proof directly yield a criterion for a closed covering walk to be a DPT in the strongly connected digraph D,which is analogous to Theorem VIII.70. But the above proof also yields an algorithmic formulation of the DPP analogous to (CPP’),and it reveals (see the discussion following (CPP’))that solving the DPP is polynomidy equivalent to finding a l-factor L C Km such that c*(L) is minimum, [KOHK74a, PAPA76a1, and that for a DPT W one has

+

+

c ( W )= c(D) c*(L)= c(D) c ( H )

(*I

( H := D+ - A ( D ) and c ( H ) is minimum). One way of finding an L of this kind is to formulate this problem as a linear programming problem known as assignment problem (AP for short). It says the following. Assign m persons v i t o m jobs w iwith each person doing precisely one job and each job being performed by precisely one person, the cost of person ui doing the job w j is c*(viwj) 2 0 , 1 1. i , j 5 m; and do this an such a way that the total cost of getting all m jobs done, is minimal. The equivalence with the above l-factor problem in Km,mbeing evident, the AP can be phrased as a linear programming problem as follows. Given c ~2, 0 ~for 1 5 i, j

5 m, find xi,i 2 0 in order to m

minimize

C ci,jxi,j i,j=l m

m

= 1 and

subject to i= 1

=1 j=1

(see, e.g., [KOHK74a, Remark; FORD62a, p.111; PAPA82a, p.248; (AP) can be solved in O(rn3)arithmetic SYSL83a, ~ . 7 4 ] ) . ~In~ fact, ) operations, [PAPA82a, Theorem 11.11 (for an earlier account of polynomial time solvability of (AP), see [KOHK74a, PAPA76al). Interestingly, the above equation (*), i.e., finding an eulerian D+ by duplicating certain arcs of D,such that c ( H ) is minimum for H = D+ ~~

33)

In the main part of [KOHK74a],the authors reduce (AP) to a min-cost

max-flow problem.

VIII.124

VIII. Various Types of Closed Covering Walks

A(D) has already been dealt with in [VIZI69a, Theorem 21, but only for the case c G 1. In the same paper it is proven that if od = m for every E V(D), then a D P T contains at most p D ( m - l)(””,”)arcs,34) while in an arbitrary digraph a DPT contains at most arcs (Theorems 3 and 4). It is also shown that the corresponding minimum cycle covering problem has the same ‘length’ as a D P T (Theorem 1); this will be shown to be true for digraphs with arbitrary cost function c (note the difference to the case of graphs). Most of these results have been extended to minimal open covering walks joining distinct vertices a , b E V(D), [GRUN73a]. In each case, D is strongly connected.

+

[(w)21

VIII.3.3. The Mixed Postman Problem Similar to the cases of the CPP and DPP, for the Mixed Postman Problem (MPP for short) one considers a mixed graph H with cost function c : E ( H )U A ( H ) 4 R+ U (O}, and tries to find a closed covering walk W in H such that c ( W ) is minimum and W contains as few edges and arcs as possible (MPP). Call such W a mixed postman’s tour (MPT for short).

As in the case of the DPP, the MPP for H may not have a solution if H is just weakly connected. However, it follows from the next result that it is algorithmically easy to check whether the MPP for H has a solution (see also [EDM073a]). Proposition VIII.92. The following statements are equivalent for a mixed graph H . 1) The M P P for H has a solution. 2) The digraph D H obtained from H by replacing every zy E E ( H ) with the arcs (z, y) and(y, z), is strongly connected.

3) For every non-empty X C V(H), if e ( X ) = 0, then U + ( X ) # 0 # a-(X).

Proof. Let W be a solution of the MPP for H. W induces a closed V(DH)-covering walk W , in DH. This suffices for D , to be strongly 34) Note that a D of this kind can be interpreted as the next state function of an automaton on p D states with an input alphabet on m letters.

VIII.3.3. The Mixed Postman Problem

VIII.125

connected since V ( D , ) = V ( H ) by definition. Hence 1) implies 2). Note that W, may not cover all arcs of D,. Suppose e ( X ) = 0 and w.1.o.g. a + ( X ) = 0 for some X c V ( H ) ,X # 8. Thus u+(X)= 0 with respect to DH since V (DH ) = V ( H ) and e ( X ) = 0. It follows that there is no path in DH from any z E X to any y E V ( D , ) - X . Since X # 8 # V( D H)- X it follows that DH is not strongly connected. Whence we conclude that 2) implies 3). If H satisfies 3), then DH as defined in 2) is strongly connected: otherwise, there are x , y € V ( H ) = V( D H ) such that there is no path from z to y in DH, and X - the set of vertices accessible from x by a path - satisfies 8 # X # V ( H ) ,and u + ( X ) = 0 in DH. By construction of DH it follows that already in H we have e ( X ) = 0 = a + ( X ) , a contradiction. So, DH being a strongly connected digraph it follows that D, has a closed covering walk which in turn induces a closed covering walk in H.Since c is non-negative it follows that H has a closed covering walk W with minimum c(W); i.e., a solution of the MPP for H exists. So, 3) implies 1). Proposition VIII.92 now follows. In the following discussion let DH be defmed as in statement 2) of Proposition VIII.92. Moreover, assume that the cost function C, in DH is obtained from the cost function c in H by c D ( a )= .(a)

if

a E A(H)

c,((x, Y)) = CD((Y7 x)) = c ( 4

if ZY E E ( H ) .

Thus cD is non-negative with domain A(DH).

Corollary VIII.93. Let H be a mixed graph such that DH is strongly connected and let c and cD be the respective cost functions. Let W H and W , be solutions of the MPP for H , DPP for D, respectively. Then c(WH>

5 cD(WD)-

Proof. It follows from the hypothesis and Proposition VIII.92 that W H and W , exist. W , is a closed covering walk in DH and induces a closed By definition of an covering walk W, in H such that c(W,) = c,(W,). MPT, c( WH)5 c(W,) = c D ( W D ) , which proves the corollary. We observe that a solution W of the MPP for a mixed graph H with cost function c may not only yield Aw(u) > 2 for some a E A ( H ) but also AW(e) > 2 for some e E E ( H ) . However, the following is true, [MINI79a, Lemma 11.

VIII. 126

VIII. Various Types of Closed Covering Walks

Lemma VIII.94. If an MPT W uses

e

E E ( H ) in both directions, it

does so precisely once.

Proof. H can be transformed into a digraph D+ by duplicating arcs and replacing edges z y with arcs (z,y) and/or (y,z) and duplicating these new arcs as well, such that W corresponds to an eulerian trail T+ of D+,and c+(T+)= c(W),where the definition of c+ is similar to the definition of cD (see the paragraph preceding Corollary VIII.93). Now, if X,(zy) > 2 and arcs of the type (z, y), (y, z) corresponding to z y belong to A ( D + ) ,the deletion of one arc (z,y)and one arc ( y,z ) from A(D+) results in a weakly connected eulerian digraph 0: c D+. An eulerian trail 2': of Df in turn corresponds to a closed covering walk W , of H ; whence we conclude c(W,) = c(W) and therefore, c ( q ) = 0 because W is a solution of the MPP for H . By the same token, however, we obtain a contradiction, since W , contains fewer edges and arcs than W does. Thus, an edge used more than twice by W is being used in one direction only. Lemma VIII.94 now follows.

As can be seen from the proof of the lemma, the reason for generally not being able to conclude A,(). 5 2, lies in the fact that an orientation of an edge e may be implicitly imposed by the structure of H or at least by the fact that W is an MPT. It follows from Corollary VIII.93 and Lemma VIII.94 that if in H the cost of traversing edges is small as against that of traversing arcs, and if I E ( H ) I is small, producing a DPT W D in D H yields a fairly good approximation of a solution of the MPP. For, in general, if W , is an MPT in H and W Dis a DPT in the corresponding D,, and if we denote by Wh (Wh)the closed walk in H (D,)corresponding to W D( W H )then , we obtain the following inequalities:

where the sum is taken over all e E E ( H ) used by W , in just one direction, and the right hand side of these inequalities denotes the length of a certain closed covering walk in DH. Note that W k , in general, is not a closed covering w& in D H . However, it can be extended to such a covering walk by adjoining digons corresponding to the edges considered in the sum in (0). Moreover, W Dcan be found in polynomial time. So,

VIII.3.3. The Mixed Postman Problem

VIII.127

this is of all the more interest as it has been shown that the MPP is NP-complete even if H is planar, d(v) 5 3 for all v E V(H) and c = 1, [PAPA76a, Theorems 1 and 21. On the other hand, the same paper contains a polynomial time algorithm which, for a given mixed graph H, constructs a smallest possible eulerian mixed graph H+ 2 H satisfying V(H+) = V(H), E ( H + )= E(H). In any case, an arbitrary closed covering walk W in the mixed graph H corresponds to an eulerian trail T+ in an eulen'an digraph D+ satisfying

}

V(D+) = V(H), A ( D + ) 2 A ( H ) , (u7

sc A(%)

+

(u7 0)

6w+>

I(A(D+) - A(H)) n A ( D + ) , n A(D+), 121E(H) n E,

n EyI

(D+)

*

That is, every xy E E ( H ) is represented in D+ by (x,y) and/or (y,x); also, x = y is possible (A(D+), denotes the set of arcs incident with z in D+,while E, denotes the set of edges incident with z in H ) . Moreover, any eulerian trail in D+ satisfying (D+) corresponds to a closed covering walk in H . However, to find an eulerian digraph D+ satisfying (D+) does not require knowledge of a closed covering walk in H , provided D , is strongly connected (Proposition VIII.92). On the other hand, if W is an MPT, it may use some e E E ( H ) in one direction only; hence for D+ corresponding to W (and satisfying (D+)) we may have D , D+ (see Corollary VIII.93, Lemma VIII.94). Call an eulerian digraph D+ satisfying (D+) an H-absorbing eulerian digraph, 35) and suppose c : E ( H ) U A ( H )-+ R+U(O} has been extended to A(D+) in the obvious way. It follows from the very definition of such D+ that finding a n MPT in H i s equivalent t o determining a n H-absorbing eulerian digraph D+ with minimum a(D+) subject to minimum c(D+) (MPP').

An eulerian trail in such D+ then corresponds to an MPT in H . Lemma VIII.95. If D+ is an H-absorbing digraph corresponding t o a solution of the MPP in H then D+ contains no proper eulerian subgraph which is H-absorbing. Proof. Suppose an H-absorbing eulerian digraph 0: C D+ exists. D: corresponds to a closed covering walk W , in H . Since D+ corresponds 35)

In [KAPP79a] such D+ is called an assigned euler network.

VIII.128

VIII. Various Types of Closed Covering Walks

to an MPT W in H , and since Df c

D,we have

C(W1)= .(Ill+) 5 C(D+)= c(W) 5 C(W1) ; i.e.,

C(W1)= c(W)

.

However, u(D+ - A(D:)) > 0 shows that u(D+) is not minimal which violates (MPP’). The lemma now follows.

On the other hand, an MPT W in H determines a digraph D ( H ) := D, C DH which satisfies (D+) (with D ( H ) in place of D+)and which contains as few arcs as possible. W corresponds to a closed covering walk W D in D,. Hence, apart from A ( H ) , D , contains only one arc or two oppositely directed arcs for every e E E ( H ) (see Lemma VIII.94). Whence W and W Dcorrespond to the same eulerian trail in D+,and W , is a solution of the DPP for D ( H ) with the corresponding H-absorbing eulerian digraph D+ _> D ( H ) . Conversely, for a given H-absorbing eulerian digraph D+ one can always determine at least one subdigraph D ( H ) & D+ as above. This follows from (D+) and the structure of D+. However, for a DPT W , in D ( H ) we may have c(W>< C D ( H ) ( W D ) < c ( D + ) 7 where c D ( H = ) cD IA(D(H)) and W is an MPT in H . In any case, though, these considerations reduce the MPP to determining

where W is a closed covering walk in D ( H ) . However, solving (MPP”) requires the solution of the DPP for 3 p ( H 1 digraphs D ( H ) at the most. A somewhat different approach to the MPP lies in narrowing the scope of H-absorbing eulerian digraphs in order to solve (MPP’); this has been done in [KAPP79a]. For this purpose consider an arbitrary H-absorbing eulerian digraph D+;define AD+ (f) as the number of arcs in D+ corresponding to f E E ( H ) UA(H), and let

D+ := DH n D+ - {af E A ( D H ) / A D + ( f=) 1) , where u f = f if f E A ( H ) , and af corresponds to f E E ( H ) , otherwise.

VIII.3.3. The Mixed Postman Problem

VIII.129

Comparing b+with the definition of Doin Theorem VIII.90 we observe that b+= D oifandonlyif I A ( b + ) - A ( H ) ( = ( ( eE E(H)/A,+(e) > 1}1 (where an eulerian trail of D+ corresponds to a closed covering walk W in H inducing Do).

c

Suppose D+ has a cycle C such that either is a cycle in some orientation of H, or else AD+(f) > 2, where f E E ( H ) is represented in D, by two oppositely directed arcs forming Let D: := D+ - A ( C ) , where C c D+ - Dz-corresponds to 6 and DH is a certain orientation of H, and define 0: correspondingly (replacing D+ with 0: in (1)). It follows from the choice of that 0: is also an H-absorbing eulerian digraph, and c(D:) 5 c(D+).Repeating this procedure we finally arrive at an H-absorbing eulerian digraph D+ i 2. 1, such that I): defined correspondingly, has the property that n ( A ( H ) )is acyclic and every cycle of D: - A ( H ) is a digon C, corresponding to e E E ( H ) for which A,+(e) = 2. That is, D: 'almost' satisfies property 1) of Theorem * ~111.90.In any case, c (D + ) 5 c ( D + ) .

c.

c

bf

For the second reduction step suppose that the above 0: contains two arc-disjoint paths pl,p, with end-vertices u , v E V ( H ) = V(D:) = V ( D f )and such that

Q i is the mixed subgraph of H underlying pi,! = 1,2. Since D+ n A ( H ) is acyclic and because of (2) it follows that Pi = pi(u,v), i = 1,2; i.e., w.1.o.g. both start at u and end at v: otherwise, plUp, D:

where

is eulerian and has, therefore, a cycle decomposition S1,2, which in turn implies for every C E S,,, that C = C, and A,t(e) = 2 . That is, PI and p2 violate (2). Suppose next w.1.o.g. that c(Q,) 5 c(Q2). Then for a directed path P3 = P3(u,v) satisfying A(P3)n A(DT) = 8 but having Q1as its underlying mixed subgraph, it follows that 8

D z l := (0: - A(P,))U A(P3) is an H-absorbing eulerian digraph satisfying c(DZl) 5 c(D:) (P,c Df - DH corresponds to p . ) . Define Dh, correspondingly and repeat this procedure until D t is obtained such that D t contains no arc-disjoint pl(u,v),p2(u,v) as described above, for any pair u,v E V(bk+). This is possible since the transition from D: to DLl does not create cycles in

VIII.130

VIII. Various Types of Closed Covering Walks

DG1 which allow the application of the first reduction step. Observe that (f) = A,+ (f) for f @ Q i U Q 2 - Qi n Q27 AD^+^ (f) = AD:(^) + 1 for f E Q1 - b 2 , A,+ (f) = AD+(f) - 1 for f E Q2 - & I , and that *+1 c(Dk+)5 C(D+). On the grounds of these two reduction steps we call an H-absorbing eulerian digraph D+ irreducible if and only if every cycle of d+ is a digon C, such that A,+(e) = 2 (e E E ( H ) ) , and D+ contains no pair of arc-disjoint paths Pi(”,u ) , P2(u,v) for any u, u E V ( @ ) . 3 6 )Partly as a summary we obtain the following result.

Theorem VIII.96. The following statements are true for a mixed graph H with cost function c : E ( H ) U A ( H ) + R+ U (0). 1) An H-absorbing eulerian digraph D+ is irreducible if and only if has the following properties:

Df

a) No eulerian proper subdigraph of D+ is H-absorbing.

b) For every pair of arc-disjoint paths PI ,P2 C D+ having the same end-vertices and with underlying respective Q1, Q2 H, some f E E(Q,UQ,)uA(&, UQ,) exists such that AD+(f) = 1, unless Qi

= Q2-

2) Among all irreducible H-absorbing eulerian digraphs there is one

which is a solution of (MPP’) .

Proof. 1) Let D+ be an irreducible H-absorbing eulerian digraph. Consider a proper eulerian subdigraph 0: c D+. Then 0: := D+ - 0;’ is a non-empty eulerian digraph. By Theorem IV.8 it has a cycle decomposition S2 # 0. Every cycle of 0: corresponds to a cycle of D+ or else X D + ( a )= 1 for some a E A(D;). By assumption, the only cycles of D+ are (7,’s such that A,+(e) = 2. It follows that either some a E A ( H ) or some e E E ( H ) is not represented by any arc in 0:. That is, D: is not H-absorbing which implies the validity of a). Consider two arc-disjoint paths PI,Pz C D+ having u and u as their common end-vertices. Suppose Q1 # Q2 and assume AD+(f) > 1 for every f E E ( Q ) U A(Q), where Q := Q1 U Q 2 . We may further assume w.1.o.g. that A ( P , ) n A, and A(P,) n A , do not correspond to the sitme element of A(D+);otherwise, we may consider Pi- u instead, i = 1,2. It 36)

In [KAPP79a], the term ‘prime’ is used instead of ‘irreducible’.

VIII.3.3. The Mixed Postman Problem

VIII.131

follows that pi corresponds to a path pi D+ with u and v as its endvertices, i = 1,2; and pl # p2 since Q1 # Q2. W.1.o.g. Pl = Pl(u,v). Starting in u let z # u be the first vertex of PI which also belongs to P2; possibly z = v. Depending on whether P2 = P2(v,u ) or P2 = Pz(u,v) it follows that the corresponding subpaths of Pl and P2 having u and x as their common end-vertices, define in D+ either a cycle C 3 { u ,z} or two internally-disjoint paths starting in u and ending in 2. The second case is impossible since D+ is irreducible. By the same token C = C,follows. That is, e E E(Ql nQ,), and Q-u c H is the underlying mixed subgraph of (Pl - u) U (P2- u). However, Pl - u = P l ( z , v ) , P2 - u = P 2 ( v , x ) . Repeating the above argument we conclude that Q1 = Q2 2 (E(H)),a contradiction to the assumption. Hence Q1 # Q2 implies the existence of some f E E(Q) U A(Q) for which AD+(f) = 1. Conversely, suppose now that D+ is not irreducible and that it has no proper eulerian subdigraph which is H-absorbing. It follows that if e E E ( H ) is represented by oppositely directed arcs in D+ which define a cycle C , then AD+(e) = 2 must hold. Otherwise, e would still be represented in 0: := D+ - A(C,); i.e., 0: is a proper H-absorbing eulerian subdigraph, a contradiction to the assumption. By the same token, D+ cannot contain a cycle other than the above Ce’s since, for C c D+ - A ( H ) corresponding to D: := D+ - A ( C ) would be a proper H-absorbing eulerian subdigraph. Thus, for some u,v E V(o+) we have arc-disjoint pl( u ,v), p2(u,v) C D+. They are arc-disjoint in D+ and satisfy Q1 # Q2 as well as XD+(f) > 1 for every f E E(Q) U A(Q) because of the choice of p l ( u ,v), p2(u,v) and the definition of D+. The validity of 1) follows.

c

c,

2) Let D+ be a solution of (MPP’) and suppose D+ is not irreducible. Supposing that there is an H-absorbing eulerian D: C D+ we conclude for 0; := D+ - 0: that ~ ( 0 :=) 0. This contradicts the minimality of u(D+). It now follows from the second part of the proof of 1) that D+ contains arc-disjoint paths Pl = Pl(u,21) and P2= P2(u,v) for some u , v E V ( D + )such that Q1 # Q2 and AD+(f) > 1 for every f E E(Q) U A(&). Applying the second reduction step to D+ (see the transition from 0: to DLl) we obtain D: = (D+- A(P,)) U A(P3)(where P3 is defined as above, and c(Pl) 5 c(P2)is assumed) satisfying ~ ( 0 :5)c(D+). Note that the choice of Pl and P2 ensures that D f is H-absorbing and eulerian. By the choice of D+ we have ~ ( 0 :=) c(D+) and hence .(PI) = c(P2). Now we may assume, additionally, that IA(Pl)J = lA(P2)l:otherwise,

VIII.132

VIII. Various Types of Closed Covering Walks

since D+ is a solution of (MPP’) and c(Pl) = c(P2),the application of the second reduction step (with Pl and Pz possibly exchanging their roles) yields an H-absorbing eulerian DLl such that c(DL1)= c(D+)but u ( D L l ) < u(D+). As we noted in the discussion of this reduction step it follows that if D: is not irreducible, Dfhas no cycles other than the C,’s for which A,,+(e) = 2. Consequently, a sequence of m applications of the second reduction step finally yields the irreducible H-absorbing eulerian digraph D L which must satisfy the equation c ( D k ) = c(D+) because of the choice of D+. This proves 2) and finishes the proof of the theorem. We note that Theorem VIII.96 and the preceding discussion of the two reduction steps are basically the content of [KAPP79a, Definition 3.1, Theorem 3.11.37) Observe, however, that an irreducible H-absorbing eulerian digraph D+ containing a certain D ( H ) (see the paragraphs following the proof of Lemma VIII.95) does not necessarily correspond t o a solution of the DPP for D ( H ) . However, comparison with Theorem VIII.90 shows that if D+ corresponds to a solution W of the D P P for D ( H )then Do D ( H ) as defined in Theorem VIII.90 and are related by the equation

o+

u

Do = D+ -

- w e )

(3)

7

C,CD(H)nDH

where C, is the digon corresponding to e E E ( H ) . That is, Theorem VIII.90 and (3) constitute a theoretical tool for determining whether an irreducible H-absorbing eulerian digraph D+ is a candidate for a solution of the MPP for H (see (MPP’), (MPP”) and the inequality preceding it, as well as Theorem VIII.96.2)). However, in [KAPP79a] the M P P is reduced t o an ILP in the following

way. For a mixed graph H with strongly connected D,,

E ( H ) = {el,. * .

7

write

eq),

= { u l ,u 2 7 . .

., %?q7

u2q+l,.

. . > u2q+q,)7

~

3 7 ) However, condition (ii) of Definition 3.1 is redundant because of condition (i) which is the same as property 1)a) in the above Theorem VIII.96. Also, Case 2 of the proof of Theorem 3.1 contains a minor flaw: there, if Pl = Pl(u,v),P2 = P2(21,21)for pl,P, C D+ (A(Pl) f l A(P2)= 0) then 0; := 0’ - A(Pl UP.) is being formed. However, while 0; is eulerian, it is not H-absorbing if and only if C, f Pl UP2 for some e E E ( H ) , A,+ (e) = 2. That is, ‘too much’ may be deleted. This becomes explicitly clear if, for the above e is a bridge of H.

c,,

VIII.3.3. The Mixed Postman Problem

VIII. 133

where @ Z q + l , * * > a,,+,a } = A ( H ) (and therefore, IA(H)I = q 4 ) , and where a2i-1 and aZi are the two oppositely directed arcs correspondi q. Associate a variable x j with every ing to ei € E ( H ) , 1 a j E A(D,), 1 j 5 n := 2q q4; xj represents the number of arcs corresponding to a j in an H-absorbing eulerian digraph D+. Consequently, for j = 2q 1,.. . ,n we have xj = XD+(uj). Finally, write V(D,) = V(H)= (q, . . . ,u p } . Thus, we arrive at the ILP representing the MPP for H (we set c = cD for the cost functions of H,D, respectively; see the paragraph preceding Corollary VIII.93):

< <

<

+

+

n

minimize

c(aj)xj j=1

subject to 4j€Atk

xj20,

4j€ALk

(MPP”’)

j = l ,..., n

x j integer,

j = 1,.. .,n.

We take note of the following facts concerning (MPP”’): 1) It sufIices to consider the degree condition (2) for k = 1, . . ., p - 1. For, if O d ( v k ) = id(uk) for p - 1 vertices, this equation must hold for the pth vertex as well. 2) Since we do not know which of the arcs a2i-l,a2i representing ei E E ( W ) is used in D+ corresponding to some solution of the MPP we need restriction (3) which, together with (5), says that at least one of these two arcs is used. Hence every e E E ( H ) is represented in D+ by at least one arc; and A(D+) 3 A ( H ) follows from (4). Whence we conclude that every H-absorbing eulerian digraph D+ satisfies (2) - (6). 3) In particular, Theorem VIII.96.2) asserts that some such irreducible D+ can be found among the solutions of (MPP”’).

In fact, [KAPP79a, Theorem 4.51 which is the main result of that paper asserts that an H-absorbing eulerian digraph D+ is irreducible if and only if the values of x i , 1 5 i 5 n , correspond to the components yi of some

VIII.134

VIII. Various Types of Closed Covering Walks

estremal point y = (yl,. . . ,y,) of ( 2 ) - ( 5 ) such that x i = yi if y; E N , while x i = 2y, = 1 if yi is not an integer. However, already in [EDMO’73a]an ILP formulation of the MPP has been given which differs from the ILP (MPP”’) as presented in [KAPP79a]. Namely, in [EDMO73a] every e; E E ( H ) is associated with two oriented digons Cdi and Cti. That is, the starting point is not some D ( H ) but rather D, together with an extra copy of C, for every e E E ( H ) . For more details, see the paper cited. Another approach to the MPP has been developed in [BRUC8la]. The starting point is the fact, that if ( E ( H ) ) , and ( A ( H ) ) , are both eulerian and H is a weakly connected mixed graph, H has an eulerian trail; orienting the edges of H according to a fixed eulerian trail in each component of ( E ( H ) ) , yields an H-absorbing eulerian digraph D+, and an eulerian trail in D+ corresponds in a natural way to an eulerian trail in H . Consequently, instead of seeking H-absorbing eulerian digraphs, P. Brucker’s approach is to find a (minimum cost) eulerian H-augmentation: that is, to orient some elements of E ( H ) and/or duplicating edges and (old or new) arcs such that the resulting mixed graph HI is eulerian. Once this has been achieved, H‘ can be transformed into a mixed graph H“ such that I E: 1- 0 ( m o d 2 ) and a+({v}) = a - ( ( u } ) for every v E V ( H ) = V ( H ” ) ;and c ( H ’ ) = c ( H ” ) , where c : E ( H ) U A ( H ) 4 R+ U (0) is extended onto E(H’) U A(H’) and E(H”)UA(H”)in the natural way. Thus, if the eulerian H-augmentation H‘ is of minimum cost, then so is the even more special eulerian H augmentation H“. Thus, a solution of the MPP for H can be approximated by solving the following problems: 1)Assuming that D , is strongly connected, construct a mixed graph H ,

such that A(H,) A ( D , ) , E ( H , ) C E ( H ) , and each weakly connected component of ( A ( H , ) ) is strongly connected; 2) Solve the CPP for each component of ( E ( H l ) ) H l ; 3) Solve the DPP for each weakly connected component of ( A ( H l ) ) H.38) l Having solved these three problems, one arrives at the mixed graph H” such that each of its vertices v satisfies I E,* 1- O ( m o d 2 ) , a+({u}) = a-({4>38) It has been noted in [BRUCSla] that this idea has already led to approximation algorithms in [FRED79a].

VIII.3.4. The Windy Postman Problem and Find Remarks VIII.135

VIII.3.4. The Windy Postman Problem and Final Remarks In dealing with the MPP and also phrasing it (indirectly) as an ILP, E. Minieka introduced the Windy Postman Problem (WPP for short), [MIN179a], which has also been treated in [GUAN84a]. The starting point of the WPP is the observation that in many practical problems the traversal of a route from A to B may have a different cost when compared with the cost of traversing the same route &om 33 to A. For example, it costs less energy and takes less time to transport by boat a certain freight from Vienna (Austria) to Budapest (Hungary) than vice versa (both cities being located on the not so blue Danube river); or, modern electric locomotives produce electric energy (which is fed back into the power system) while going downhill (this corresponds to negative costs). Thus the WPP says the following:

Let a connected loopless graph G with cost functions c l , $ E(G) R+ U (0) be given such that the cost of traversing e = x,y, E E(G) f r o m x, t o ye as cl(e), whereas the cost of traversing e f r o m ye t o x, is --f

c2(e). Determine a closed covering walk W = e l , e 2 , .. . ,en, n 2 qG, in G such that c(W) := Cj”=, c(ej) is minimum, where c(ej) E {cl(ej), $(ej)} depending o n the direction in which ej is being traversed in W. ( W P P )

If we replace every e = z,y, E E ( G ) with a digon C, := ( { ( x , , ~ , ) , ( y e ,x,)}) (thus creating a symmetric digraph - that is, a digraph which also contains uR if it contains a), and define

we arrive at the following equivalent reformulation of (WPP): Let G be a connected loopless graph and let 0; be the symmetric digraph c o v e sponding t o G. For every e € E(G),let C, c 06 denote the digon corresponding t o e, and let a cost function c : A(D&)3 R+ U (0) be given. Determine a closed walk W * in 0;such that 1) A ( W * )n A(C,) # 0 for every C, C 0; and 2) c(W*) is minimum.39) (WPP’) 39) The restriction to loopless graphs is in fact of no practical relevance. If a loop in G is assigned different costs depending on the direction of traversal (in a topological sense), one can always take the ‘cheaper’ direction; and one always gets away with traversing a loop just once.

VIII.136

VIII. Various Types of Closed Covering Walks

We call the walk W (W’) in (WPP) ((WPP’))a windy postman’s tour (WPT for short) in G (D;). Thus, the transformation of (WPP) into (WPP’) shows that the WPP is not a special case of the DPP, and one might suspect that the complexity of an algorithmic solution of the WPP is somewhat more intricate than the complexity concerning the DPP. This suspicion is in fact justified. It has been shown in [MINI79a, GUAN84al that the WPP is NP-complete, where E. Minieka reduces the WPP to his way of formulating the MPP, and M. Guan transforms the MPP to the WPP in a way similar to the transformation of (WPP) into (WPP’).40)However, M. G u m gives a polynomial time algorithm for a special case which is also of practical relevance.

Definition VIII.97. For a connected loopless graph G and the correC- c G b the two cycles sponding symmetric digraph G&, denote by C+, corresponding to the cycle C C G. Let a cost function c : A (G> ) + R+ U (0) be given. We say that c is cycle-balancing if c(C+)= c ( C - ) for every C G.41) Before stating and discussing M.Guan’s algorithm, we first study some implications of cycle-balancing cost functions c. For this purpose assume a function p : V ( G ) + R be given and define a new cost function cp : A(D&)+ R by c,(u) = c ( a ) +p(vj) - p ( v i ) if a E A: n A;

,

(CB)

where c is not necessarily cycle-balancing. Note that cp(u) < 0 may hold for some u E A(Df;).In view of (WPP) and its transformation into (WPP’) we also consider implicitly the respective cost functions cl, $, c ~ , ~ ,in G.

Lemma VIII.98. Let G,D;,c , cp be given as in Definition VIII.97 and (CB). Let W be any closed (not necessarily covering) walk in G, and let W* be the corresponding walk in Df;. Then c(W) = c p ( W * ) .

Proof. By definition of c(W) (see (WPP)) we have c(W) = c ( W * )in any case. By duplicating arcs in 0; , if necessary, we transform W* into 40) Incidentally, the formulation of the WPP given in [BRUCSla] is erroneous. *l) In M. Guan’s notation this means that “G satisfies condition Q”.

VIII.3.4. The Windy Postman Problem and Find Remarks VIII.137 a closed trail T+ in some D+ _> D&. (A(T+)),+ has a cycle decomposition S+ = (Ct,. . .,C,'} by Theorem IV.8, and every Ct corresponds to a cycle Ci DL,1 5 i 5 k. Observe that p can be reinterpreted as a potential in 0;.Thus, by definition of cp and Definition 111.80 we can write cp=c+$, where $ is a tension in Dz. As a consequence of Lemma 111.81 we therefore have

0;:.

c(C) = cp(C) for every cycle C

Extending c onto A ( D + )in the natural way and observing that k

c(W*) = C ( T + ) = CC(Ci',

,

i=l

we therefore obtain the validity of the lemma by k

k

c(W) = c(W*) = CC(Ci) = CCP(Ci)= CP(W*)

.

Next we choose a special type of function p . Let G be a connected graph, let B be a spanning tree in G, and let D, c D& be an out-arborescence with v1 E V(G) = V(D,) as its root and B as its underlying graph. Define inductively: for a E A(D,) nA: fl A;, if p B ( v u ihas ) been defined already but ps(vj) is not yet defined, set

where a- E (A(D&)- A(D,)) n A:. e E E(G).

n AZi and a correspond to the same

L e m m a VIII.99. Suppose a cycle-balancing cost function c : A(D&)-+ R+ U (0) is given, where G is a connected loopless graph. Suppose p , and cpBare given according to (PB) and (CB), respectively. Then cpE(a:) = cpE(a:), where { a ~ , a E ~ }A(D&) corresponds to the arbitrarily chosen e E E(G).

VIII.138

VIII. Various Types of Closed Covering Walks

Proof. Consider first a: = a , a: = a- for a E A(D,) and a- E A ( D 5 )A(D,), where B is a spanning tree of G. Using the same indices as in (PB) and (CB) we have for p B ( v j >- p,(vi) = ~ ( c ( u -) .(a)) c p e ( a )= .(a>

+ z1( . ( u - )

= .(a-)

1 - -(.(a-) 2

1 - .(a)) = # a )

+ .(a-))

- .(a)) = c,,(a-).

Thus the lemma is true for every e E E ( B ) . Now let e E E ( G ) - E ( B ) and consider the corresponding a ~ , aE ~ A(D&) - A(D,).B U { e } contains a unique cycle C. Denote by C+ the unique cycle of D, U Di U {a:} DL containing a:. W.1.o.g.

A ( @ ) = {a: E A ( D & ) / fE E ( C ) }u {a:}

;

and therefore, the cycle C- := ( C + ) Ris given by

Since c p B ( a ; )= cpB(u;)for f E E ( B ) ,and since c is cycle-balancing, we conclude from Definition VIII.97 and (CB) (see also the last part of the proof of Lemma VIII.98) that

for every e E E(G - B ) . This proves the lemma.

We now prove the following result, [GUAN84a, Theorem 21.

Theorem VIII.100. For the connected loopless graph G let c : A(D&) +. R+ U (0) be a cycle-balancing cost function. Define a cost function co : E(G) + R+ U (0) by

VIII.3.4. The Windy Postman Problem and Final Remarks VIII.139

where ( a ~ , a C~ A(D,%) ) corresponds to e E E(G).Then every solution W of the CPP for G with respect to co corresponds to a solution W* of (WPP') for 0 5 with respect to c, and vice versa; moreover, c o ( W ) = c(W*).

Proof. Consider a spanning tree B in G, and let D B be an outarborescence with root v1 and having B as its underlying graph. Starting from c, define p B : V ( G )t R as in (PB), and cpB : A(D&)t R as in (CB). Since c is cycle-balancing we have by the first part of the proof of Lemma VIII.99 and by definition of co

whenever e E E(B). To see that (1) holds for every e E E(G) consider now f E EUjn E,,. - E(B). Assuming w.1.o.g. a$ E A$ n A; we have by (CB)

Since cpB( a $ ) = cpB(a;) in any case by Lemma VIII.99, we now obtain

implying (1) for every e E

E(G).

Now let W be a PT in G with respect to co, and let W* be the closed walk in D& corresponding to W . Since W is a closed covering walk in G, W * satisfies A(W*)n A(CJ # 0 for every C, = ( { a ~ , a ~ } )0 2 . Since cpB(a:) = cpB(a:) = c,, ( e ) it follows from the definition of cPEand from Lemma VIII.98 that c o ( W ) = c ( W * ) . Thus c ( W * ) 2 c(W,) for a WPT W , in 0; (see (WPP')). On the other hand, W , induces a closed covering walb Woin G such that by Lemma VIII.98

co(Wo)= c(W,) 2 c o ( W )= c ( W * ) That is,

c o ( W )= c ( W * )= c(W,)

.

.

Hence W* is a WPT in D& with respect to c. However, the analogous argument also implies that if W* is a WPT in 0;with respect to c, then

VIII. Various Types of Closed Covering Walks

VIII.140

the corresponding closed covering walk W in G must be a PT in G with respect to c,,. This finishes the proof of the theorem. Theorem VIII.100 immediately gives rise to the following algorithm (see [GUAN84a, Algorithm A]) whose correctness follows from this theorem.

WPT-Algorithm for cycle-balancing cost function. Step 0. Let G be a connected loopless graph together with cyclebalancing cost function c : A(D$) + R+ U (0) inducing cost functions cl,~2 : E(G)-+ R+ U (0). Step 1. Define co = $(c1

+ c2>.

Step 2. Solve the CPP for G with cost function co. Step 3. Any PT W in G with respect to co is a WPT in G with respect to c1,

cz.

However, while in respect to practical problems the very nature of the problem may imply that c : A(D&)--f R+ U(0) is cycle-balancing, checking in theory whether c(C+)= c ( C - ) for every cycle C C G is a rather tedious procedure. In fact, it suflices to check this equation for every (uniquely determined) cycle C ( e ) C B U { e } where B is a spanning tree of G and e E E ( G ) - E ( B ) , [GUAN84a, Theorem 31.

In further studying the WPP, M. Guan considers the case where the cost function comes close to being cycle-balancing, and derives from the above algorithm another one which produces a good approximation of a WPT (for details see [GUAN84a, Theorems 4,5,6, Algorithm .?I). There are still other variations of the CPP than those discussed above. We briefly mention two of them. The first of these problems asks for a minimum cost closed covering walk W in a connected graph G such that W traverses the elements of Ei E ( G ) for the first time before it traverses those of Ej+l E E(G),where { E j / i = 1,. . .,k} is a partition of E(G). This problem has been studied in [DROR87a]; it is, in this general form, NP-complete. This is also the case for the Rura2 Postman Problem (RPP for short) which asks for a minimum cost closed walk W in G , where W need not be a covering walk but includes a specified E ( G ) . The RPP remains NP-complete if reformulated for subset El digraphs (see e.g. [GARE79a, p. 213, [ND27]] and [CHRI86a]).

c

c

VIII.3.4. The Windy Postman Problem and Final Remarks VIII.141

In closing our consideration of the Chinese Postman Problem for graphs, digraphs and mixed graphs we may remark that some of the applications such as street cleaning or milk rounds, have not been discussed in terms of digraphs and mixed graphs. This is partly due to the fact that the relevant literature deals - to my knowledge - mainly with graphs, but it is also due to the theme of the book. As for the latter point, I purposely dedicated considerable space to the theme of t-joins and t-cuts not only because it represents (historically as well as with respect to its content) a generalization of the CPP, but also because it represents an important and fairly new development in graph-theoretical research (I frankly admit that subjective aesthetical reasons also governed my decision). However, it is not difficult to realize that a street system of two-way streets represented by graph G can also be represented by a digraph D, by replacing every e E E(G)with a digon C,;or, for that matter, it is not difficult to reduce a mixed graph presented as a system of one-way and two-way streets to a digraph by forming D,. Of course, one has to decide in each individual case whether one describes the problem in terms of graphs or digraphs (and thus solve the problem in polynomial time), or one has to deal with mixed graphs (and relies, for the sake of saving computer time, on approximation algorithms for the MPP). The Windy Postman Problem illustrates, to some extent at least, this interaction between practical problems and dealing with them in terms of graphs or digraphs.

As for the maximum weight cycle packing problem in digraphs (DMCPP for short) one cannot expect to have as close a relation to the DPP by analogy to the case of graphs; one can thus observe that D - A(Do)need not be eulerian (see Theorem VIII.90 for the definition of Do). However, in his thesis Liu Jiannong had studied the DMCPP [LIUJ87a] and reduced it to a bipartite matching problem in a graph G D associated with the digraph D (but in a way different to the reduction of (DPP) in Theorem VIII.91). He then shows that there is a bijection between minimum weight perfect matchings in G D and maximum weight cycle packings in D. Consequently, he develops an assignment algorithm (based on this bijection) whose complexity is of the order of O(&). Fhthermore, in the same paper the DMCPP is also formulated as an LP problem. Finally, Liu Jiannong also studies the MCPP for mixed graphs and ‘windy’graphs (i.e., graphs where every edge is associated with two costs according to the two possible orientations). Interestingly, the MCPP’s complexity is basically the same as that of the respective Postman Problem (In P or

VIII.142

VIII. Various Types of Closed Covering Walks

not in P ? - that is the question; see [LIUJ87a, DULIXXa]).42) Finally, the minimum weight cycle covering problem (MCCP) becomes a relative triviality in the case of digraphs. For if D+ arises from D by duplicating arcs, and if D+ is eulerian, then any cycle decomposition of D+ corresponds to a cycle cover of D (contrary to the case of graphs). Since cycle decompositions in D+ can be found in polynomial time (see the last chapter of this volume), it follows that the MCCP for digraphs is in P and the ‘cost’ of a minimum weight cycle cover equals that of a DPT in D. We shall deal with the MCCP for mixed graphs and ‘windy graphs’ later.

VIII.4. Exercises Exercise VIII.l. Prove Corollary VIII.3 by applying Corollary VIII.2. Exercise VIII.2. Prove Lemma VIII.5. Exercise VIII.3. Let G and W be given as in Lemma VIII.5. Prove that ( E ( W ) )is a component of G, if W uses every edge of G twice or not at all.

Exercise VIII.4. Construct an infinite family of cubic graphs G with p G 2 (mod4) such that G has no strong double tracing. Exercise VIII.5. Prove Corollary VIII.14. Exercise VIII.6.Prove Corollary VIII.16. Exercise VIII.7. Prove Corollary VIII.22 (hint: construct D := D , and consider D T ) .

Exercise VIII.8. Construct a value-true cycle cover for the graph K4,cp of Figure VIII.2. Exercise VIII.9. Rephrase Lemma VIII.16, Lemma VIII.17, Lemma VIII.18, Theorem VIII.21 and Corollary VIII.22 into true statements for mixed graphs. 42) My remarks on Mr. J. Liu’s paper are based on an English translation of his thesis which he provided upon my request (following our conversations at Shandong Teacher’s University, Jinan, China, in summer 1987), and for which I wish to express my thanks.

VIII.4. Exercises

VIII.143

Exercise VIII.10. Prove Lemma VIII.38. Exercise VIII.ll. Prove: if G is a minimal counterexample to Tutte's Five-Flow Conjecture, it is a weak snark with AJG) 2 3, g(G) 2 7. Exercise VIII.12. Prove: a graph G has a nowhere-zero k-flow if and only if every block of G has a nowhere-zero k-flow. Exercise VIII.13. Prove: for k = 2, Conjecture VIII.49 implies Conjecture VIII.28 (hint: consider the 9-edge-connected graph H obtained from a 3-edge-connected graph G by replacing every edge of G with three parallel edges). Exercise VIII.14. Prove Corollary VIII.55 and Corollary VIII.56 by applying Theorem VIII.54 (see also the paragraph following Definition VIII. 5 1). Exercise VIII.15. Use statements 1) - 7) preceding Theorem VIII.64 and this theorem itself to prove Theorem VIII.63. Exercise VIII.16. Show that the graph H of Figure VIII.ll has a closed covering walk with the properties described in the text of that figure and thereafter. Show that the 6-regular graph Hl C H described in the discussion centering around Figure VIII.ll admits a nowhere-zero %flow as described. Exercise VIII.17. Prove Theorem VIII.65 and Proposition VIII.66. Exercise VIII.18. Prove Theorem VIII.68. Exercise VIII.19. Prove: if M is a set of 2k integers, P 2 ( M ) ,the set of partitions of M into k classes of size 2, satisfies ( P 2 ( M I= ) (2k - 1)(2k 3). . .3.1 2 3"l. Show that if k 2 4 then 3k-1 can be replaced by 3k. Exercise VIII.20. Prove Corollary VIII.69 (observe that in the process of constructing G + , every bridge of G has to be duplicated since G+ is eulerian). Exercise VIII.21. Prove Lemma VIII.73 by modifying parts of the proof of Theorem VIII.67 (hint: set c = 1 for the cost function c ) . Exercise VIII.22. Prove Corollary VIII.79. Exercise VIII.23. Let G be a connected graph with cost function E(G) + R+ U (0) and let c* be the corresponding cost function for K p ,p = pG. Prove: The HWP for G is polynomially equivalent to (TSP')for K p . c :

VIII.144

VIII. Various Types of Closed Covering Walks

Exercise VIII.24. Prove the first part of Proposition VIII.87. Exercise VIII.25. Prove observations a) and b) following the proof of Lemma VIII.89.

Ix.1

Chapter IX

EULERIAN TRAILS - HOW MANY ?

In this chapter we consider eulerian trails both in digraphs and graphs, but also in mixed graphs. In the first section we shall present a parity result in digraphs which enables us to give a simple proof of the Theorem of Amitsur-Levitzki; we shall also present variations on that theme. In the second section we first prove the Matrix-Tree-Theorem for digraphs; combined with Theorem VI.33 it yields the famous BEST-Theorem a closed formula for the number of eulerian trails in eulerian digraphs. The BEST-Theorem then serves as the basis for developing an analogous formula for eulerian graphs. For the sequel we recall Definition VII.l and the discussion following it which concerned the various ways of distinguishing eulerian trails of the same (mixed) (di)graph as different objects.

IX.l. ...As Many As ...- Parity Results for Digraphs and (Mixed) Graphs In this section we first consider digraphs D whose arcs are labeled a,, . . . , a 4 ,q =I A ( D ) 1, and such that D has an open or closed covering trail. That is, either D or D U { (w, t~)} is a weakly connected eulerian digraph for certain vertices w , w E V(D).In general, denote by 7 v ( D ) the set of covering trails T in D starting at TI E V ( D ) . I , ( D )= 0 if D is not of the type described above; w is arbitrary but fixed if D is eulerian, while t~ is the above mentioned vertex if DU{(w, w)} is eulerian. Contrary to Definition VII.l

we now consider two ederian trails T ,TI of D ,D U {(w,v)} Tespectiuely, to be different if they have different initial arcs or if X , # X,,. (*I

1x2

IX.Eulerian Trails - How Many ?

We say that the covering trail T is rooted at v (or, T ) . Writing T as an arc sequence,

T = a .11 ’ a i 2 ,...,aiq

is the root of

,

we have

{i,,.. , ,i,} = {I,.

.., q } .

That is, T defines uniquely a permutation

of the integers 1,.. . ,q. In other words, if we write { 1,. . . ,q } onto itself, we have

~ T ( j ) = i j , 1 Ij I g

7rT

as a bijection of

*

Calling T E Sq an eulerian permutation if T = rT for some T E I , ( D ) (where S, denotes the symmetric group on (1,. . . ,q } ) , it follows that

I I,(D) I=/ {T E S, / T is an eulerian permutation} I

.

This, (*) and the definition of rT permits us to say equivalently that two eulerian trails of D rooted at v are different i f and only i f their respective eulerian permutations are. (**> However, there are other ways of describing an arbitrary permutation T, the most common of which is t o write T as a product of disjoint (permutation) cycles. Starting from this representation of T , one can write T as a product of (not necessarily disjoint) transpositions (i.e. cycles of length two). Important for our purposes is the fact that regardless of how 7r is represented as a product of transpositions, the number of transpositions used in that representation is either always even or always odd. Correspondingly one calls T an even or odd permutation (for the preceding concepts and elementary facts, see, e.g., [HERS64a, p.671). One defines sgn(a) = +1 if 7r is an even permutation, and sgn(7r) = -1 otherwise. Thus we are led to define for the above covering trail T ,

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.3

and we call T even (odd) if rT is an even (odd) permutation. We denote correspondingly the set of all even (odd) elements in I,(D) by

';rv"(D)~'7;"(W Remark 1X.l.a) In fact, if D is eulerian, the question whether T E

7(D)is even or odd is not only determined by the choice of the root of T , but also by the given labeling of the arcs of D. That is, for given labeling of the arcs of D and for T,T'f 7 ( D ) satisfying X, = X T r , where T is rooted at v and T' is rooted at v', v # v', Sgn(rT) # S g n ( r T / ) may hold. For example if D is a directed hexagon with the vertices denoted in their cyclic order by vl, . . . ,v6 and ai E A ( D ) incident from vi, 1 5 i 5 6, and if T is rooted at v1 and T' is rooted at v4, then xT = E while rTr= (14)(2 5)(36) ( E denotes the identical permutation). That is, rT is an even but rTris an odd permutation. Similarly, if we let the directed hexagon D and T be as above and distinguish D' from D by simply rotating the arc labeling such that c ~ is~ incident from vi, 1 5 i 5 6 (taking the subscripts (mod6) with 6 in place of 0), then T' E I ( D ' ) starting in vl is an odd eulerian trail. However, if we consider, in general, I,(D) for fixed v and just permute the arc labels to obtain D' N D,then either

T ( D )= C ( D ' ) ,

I,"@)

= I,"(D')

or

I,"(D) = C ( D ' ) , T ( D ) = T ( D ' ) . (Note that I,(D) = 7 , ( D ' ) ) . For, if T E I,(D) and T' f I,(D') are arbitrarily chosen subject to having the same initial arc and satisfying X , = X,,, the corresponding permutations R, and r,, are related by rTr= AT,, where X is the permutation relating the arc labeling of D to that of D'. This consideration also applies to the case where D U {(w, v)} is eulerian. b) However, even if the eulerian trails T and T' of D,D U {(w, v)} respectively, start at the same vertex v and if the arc labeling is fixed, we still may obtain T, # rT,and even sgn(T) # sgn(T') may hold although X , = X,, . In this case this follows generally from statement (*) concerning distinguishing eulerian trails and is, in particular, exhibited by the consideration of Figure IX.l. However, if the root v is 2-valent, the initial arc is the same for every eulerian trail of D ;this implies that rT = x,, if and only if X , = X,,. Hence, in this case we have 7,(D) = I(D).

+

~

Ix.4

IX. Eulerian Trails - How Many ?

H Figure IX.l. A digraph with eulerian trails T = a l , a2 and T' = a 2 , a 1satisfying X , = X T , , whereas sgn(T) = 1 and sgn(T') = -1. Having discussed in detail the difference between Definition VII.l and the statements (*) and (w) we now turn to one of the main results of this section which is due to R.G. Swan, [SWAN63a, SWAN69al. The proof presented here can be viewed as a modified version of Swan's proof. However, there is one case missing in Swan's papers; hence Swan's proof is not complete. This was discovered by my former student Erich Wenger who closed this (minor) gap in his thesis [WENG82a].

Theorem 1x2. Let D be a digraph satisfying qD 2 2pD. Denote A ( D ) = {al,. . . , apD}, where the indices are assigned arbitrarily. Then Il,"(U)1=17,"(D)I for every vertex ZI E V ( D ) .

Proof. Suppose the theorem is false. Among all the counterexamples D choose one which satisfies the following conditions: 1) I 7 , ( D ) I is as small as possible, where fixed.

ZI

E V ( D ) is arbitrary but

2) Subject to l), rnaz{X(z, y)/(x, y) E A ( D ) } is as small as possible.

3) Subject to 1) and 2 ) , a ( D ) - p D is as small as possible.

4) Subject to l),Z), 3), C(D)is as small as possible. If 7 , ( D ) = 8, then the theorem is vacuously true. Whence / 7 , ( D )I> 0 which in turn implies that either D or D U {aw,,} is weakly connected and eulerian for some w E V ( D ) ,where w # 21, and aW,, # A ( D ) . Suppose next that ail a . E A ( D ) exist such that { a j ,u j } c A: nA;, 1 5 i < j 5 q D , z, y E V ( D j . Let T E q ( D ) be chosen arbitrarily,

IX.1. As Many As

- Parity

Results for (Mixed) (Di) Graphs

IX.5

where { I c , I } = {i, j}. That is, T' arises from T by exchanging ai and a j . By the choice of T and by the assumption concerning a;, a j , it follows that T' E 7 , ( D ) . By definition of TI, nT and rT1, it follows that KTI

;

= (i,j)KT

hence sgn(T') = -sgn(T). That is,

T E I,"(D) if and only if T'

E I,"(D)

.

(1)

Since 7 , ( D ) = 7 , ( D ) U 7 z ( D ) and since the above relation between T and T' defines a permutation of the elements of 7 , ( D ) , it follows from (1) that ll,"(D) I=ll,"(D) I 7 (2) as claimed by the theorem. Whence X(x, y) = 1 for every (x,y) E A ( D ) ; i.e., D is a digraph without multiple arcs. Suppose k := a ( D )- p D = q D - 2 p D in D and let

Dl := D U {

~ i ,( ~ i - , ,

> 0.

Let w,,. . . , wk be vertices not

w i ) / i = 1,.. . ,k ; w 0 = Z}

,

where z = 2) if D is eulerian, x = w otherwise. Regardless whether D is eulerian or not, we have 7,(Dl) # 8, and a bijection exists between the elements T, E 7,(Dl) and the elements T E 7 , ( D ) which is given by = T ,p ( w O ,

wk)

(3)

f

where P(w,, w k ) is the path defined by A(P(w,, w k ) ) = A ( D , ) - A ( D ) . Set := (wi-,, wi), i = 1,. . . ,k , q := Q D . We have g(D1) -PD1

= 40, - ~ P D= , q D -k k

- 2 p -~ 2k

=0

(4)

By the choice of D and since I T,(D,) I=/ T,(D) 1 and because also D,has no multiple arcs, we may conclude that D, satisfies the theorem. Hence

However, by construction of D, we have

IX.6

IX. Eulerian Trails - How Many ?

where TIand T are the objects related to each other by (3), and where we disregard (permutation) cycles of length 1, i.e., fixed points.') Thus we have sgn(rTl) = sgn(rT) (7) which, together with (5), also implies the validity of (2) in this case. Whence we assume q D = 2PD

Finally suppose C ( D ) > 0. Since I,(D)

.

(8)

> 0, it follows that

+

od(v) = id@) 1, od(w) = i d ( w ) - 1, id(z) = o d ( s ) for s E V ( D )- (21, w}.

(9)

Consequently, introducing a new vertex z and defining

we may conclude from (9) that D , is a weakly connected eulerian digraph. Moreover, we have 11;(WI=l%P) I (19) since a bijection exists between the elements ments T E I , (D ) which is given by

Tl E I z ( D l ) and the ele-

up+, := (w, z ) and viewing rTl as a permutation of Setting a. := ( z , ZI), the integers {0,1, . . . q l}, we may conclude that (6) and (7) also hold true in this case (see the preceding footnote). By the choice of D and since D, satisfies C(D,) = 0, a ( D l ) - p D l = 0 and X(s,y) = 1 for every (z, y) E A(Dl), we may conclude the validity of (5) with z in place of ZI, which together with (7) and (10) again implies the validity of (2). ~

+

Partly summarizing what we have shown so f a r we can conclude from the choice of a counterexample D to the theorem that Viewing rTland nT as mappings, they are different mappings in any case because their respective domains are different. However, for our purposes, we view permutations as products of disjoint cycles of length > 1 which justifies writing equation (6).

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

D is weakly connected, eulerian and satisfies

IX.7

(12) In order to restrict further the structure of D we proceed by induction on p , and show next that = 2pD.

D contains no 4-valent vertex incident with a loop.

(13)

Suppose to the contrary that D does contain such a vertex; call it x. Denote A, = {(y, z), (2,z),( x , ~ ) }possibly ; y = t. In any case, though, we assume p D > 1; otherwise, the validity of (12) implies that D is the digraph shown in Figure IX.1 for which the theorem holds anyway. Whence y # x # z. We distinguish between two cases. a) z = w. Denoting by x,l(D)( T , q ( D ) the ) set of eulerian trails of D starting (ending) with (w,w), we can write

and there is a bijection 7,,1(D)t+ 7,,,(D) given by

where

To E %(Do) and Do := D - {(w,~)} . Suppose (v,v) = ail. Now, if

then we obtain

which is an odd permutation since q is even (see also Figure IX.2). Thus,

IX.Eulerian Trails - How Many ?

IX.8

a.

‘1

/ /

\

/-

\

/

\

/

\ \

D Figure IX.2. Every T E I,(D) is of the form T = a i l , To or T = To,a i l , where To E 7,(D0).

which, together with the bijection (13b) and equation (13u) implies the validity of (2), again a contradiction to the choice of D.

b)

5

# ZJ. In this case we form

where ay,z joins y to z , ay,z $? A ( D ) . Observe first that D , satisfies I and P D , < P D . By induction, the theorem (12), I %(Ol) /=I holds for D,. Given the last part of Remark IX.l.a), we assume w.1.o.g. that the arc labeling in D has been chosen in such a way that uq-2 = ( Y , 4 aq-1 = ( X c , X ) , a,, = (x,2 ) . Denote bi := ai, 1 5 i < Q - 2, bq-2 = u ~ , Observe ~ . the bijection between the elements T E I , ( D ) and the elements TI E 7,(Dl) which is given by

where TI and TI’ are subtrails of T (see Figure IX.3). Expressing in case a ) we obtain

7rT~=

(

1, 2 ) . ” , j - 2 , j - l , i , , t.2 , . . . , q - 2 , Zj+,

j ,”.) i.3 + ” “ ”

2,

7rT

as

IX.l. As Many As - Parity Results for (Mixed) (Di) Graphs

Y

IX.9

Y

D,

D

Figure IX.3. If z # 21, then every T E I,(D) corresponds T1E 7,(Dl), and vice versa.

to precisely one

for some j E (3, . . . ,q } . We have for

and 7r;pF1 = ( j - l , j + l , j + 3

).")

q - l ) ( j , j + 2 ,.", q )

+

+

= ( j - l , j + l>(j - 1 , j 3 ) . . . ( j - 1 , q - l)(j,j 2 ) . . . ( j , q )

if j is even, whereas we obtain for odd j 7r;7rT1

= ( j - l , j + l ,...,q , j , j + 2 ,..., q - 1 )

+

= ( j - 1 , j l)(j- 1 , j . . . ( j - 1 , q - 1)

+ 3 ) . .. ( j - 1,q ) ( j - 1,j ) ( j - 1 , j + 2 )

(note that (kl,k,, . . . ,km) = (kl,k2)(k1,kg). . . (k ,km)). Thus the number of transpositions used in expressing 7rt,"' is q - j 6, where 6 E ( 0 , l ) satisfies 6 j (mod 2). That is, q - 3 5 0 (mod 2); hence

=

+

+

IX.10

IX.Eulerian Trails - How

Many ?

which implies, together with (13d), that

where T, and T are related to each other by (13c). Thus, we conclude from (13c), (13e) and the validity of the theorem for D,,that

This contradiction to the choice of D implies the validity of (13). Next we show that D contains no 2-valent vertices. (14) Suppose it does. Then it contains one, for example x, such that (y,x) E A(D) and d ( y ) > 2; this follows from qD > p D . Let z' denote the vertex for which A, = {(y, x),(2, 2')); possibly y = z'. Let z denote the first vertex reachable from z' such that d ( z ) > 2; possibly z = z' or z = y. We consider the following two main cases.

a) I {x,y, 2) - {v} 12 2. Then either y # v or z # v; let w E {y, z } be such that w # v. For the path (cycle, respectively, if y = z ) P = P ( y , z ) containing x let a j be an element of A,nA(P); a j is uniquely determined if y # z . (al) ZI # z. Form Di,j for any ai E A , in such a way that Dilj is eulerian; i.e., ai E A; if w = y and ai E A;f; if w = z . Suppress the (unique 2-valent) vertex w ; , ~E V ( D i , j )- V ( D ) which stems from the x*), where x* denotes the splitting operation, and attach a loop (z*, neighbor of w ~ in, V~( P )n V ( D i , j )(thus x = z * if w = y). Denote by D i the eulerian digraph thus obtained (see Figure IX.4). We have

by construction. Relabel the arcs al E A ( D )n A ( D i )by defining b, := a l , denote the unique arc of A ( D i ) - ( A ( D )U {(x*,x*)}) by bi, and define bj := ( x * , x * ) . By the hypothesis of this case a) and because v # x has been assumed, we may choose w such that 21 # x* holds as well. From the very construction of Di and its arc labeling, and because v # x*,we can draw the following conclusions:

(il) q ( D i )n x ( D r )= 0 for i

# r;

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.ll

Figure IX.4. Case (al). Transforming D into Di for 20 = y (thus x = x*). Possibly u = y. Note that if T‘ E 7 g ( D i ) , the corresponding T E I , ( D ) does not begin with a j .

(i2) Every T E ‘T,,(D) corresponds to a unique T‘ E ‘T,,(Di)for some i, and rT = rT,; (i3) ll,(D) I=

C l T ( D i )1,

where the sum contains ;d(w) terms.

The theorem applies vacuously to every disconnected D i , and it applies to every weakly connected Di because of (14a1) and (13). This together with (il), (i2), (i3) implies the validity of (Z), a contradiction to the choice of D. (a2)v = z, and therefore y # z . Set 20 = y and construct the digraphs Diand their arc labelings as in case (al). However, we now consider 7 g ( D i )instead of 7 v ( D j ) .Similar to (il), ( i 2 ) ,( i s )we now have:

(jl) 7 , ( D i ) n 7 J D J = 0 for i

#T

;

IX.12

IX.Eulerian Trails - Bow Many ?

( j z ) Every T E 7y(D) either corresponds to a unique T' E 'Ty(Dj)for some i, in which case 7rT = rT,,or else starts with a j = (y,z).

In the latter case, we can express Taj:= T in the form

where T(z,y) is the corresponding subtrail joining x and y in D. This ( D ) := {Taj E % ( D ) / T a starts , with a j } yields a bijection between and Tz(D), which is defined by

zj

TajE 7aj(D)++ T,

= T(z,y),aj

E 7z(D).

(j3)lq,(D) I=II-/,(D)l=l%(D) 1 - C Il,(Di) 1, where the sum contains $(y) terms. However, the above bijection implies

(note that rTzr;h: = ( 1 , 2 , 3 , . . . ,q ) and compare this with (13b) and the arguments following it). Moreover, (14a,) is valid by construction; and since case (a,) has been handled already we conclude the validity of the equations

( j 2 ) ,(j3) and (14Uz),imply which, together with (jl),

This contradiction to the choice of D finishes the proof of case (a2)and thus of case a) as well.

b) 1 {z,y, z } - {v} I= 1. If follows that y = z =: t , and either v = t or v = z. Denote by C the cycle of D containing z.

IX.1.As Many As - Parity Results for (Mixed) (Di) Graphs

IX.13

(b,) v = x. Suppose first that A, C A,. Because of Remark 1 X . l . a ) we assume w.1.o.g. that E A:, uq E A;, and form D, := D - x which satisfies (12) with D, m place of D (note p D = = p D - 1, qD, = qD - 2 ) Whence the theorem applies to D,. Moreover, a bijection between ‘&(D,) and I,(D)is given by

Ti E qD,)* T,

:= a,-l,T,,a,

E %(D)

which implies for the corresponding eulerian permutations TTt

=

1,.. . ,q - 2

1, 2 , 3 , . . . , q - 1 , 4 . . ., zq--2, q >

TT= = ( Q - 1, i,, i,,

’

that for

= ( l , 2 , 3 , . . . , q - 2 , q - 1)

,

from which follows that S g n ( K T , ) = s!?n(T;t)= sgn(TT,)

That is, D satisfies ( 2 ) for contradicts the choice of D.

*

= x since D, does so for v = t. This

Supposing now A, $tA, we form Dj,i and Dias in case ( a l ) for the explicit choice w = t , a j = (x’,t)E A ( C ) , ai E A$ - A ( C ) . Note that x # x’ by the assumption of this case; thus d,(x) = d D i ( x ) = 2, x* = x’ (see Figure I X . 4 ) . Since the conclusions (zl),(i,),(i3)of case ( a l ) are also valid in the present situation, we again obtain a contradiction by concluding that D satisfies ( 2 ) . This settles the case v = x.

(b2) v = t. Again, form D ilj and Dias in case ( a l ) for w = t , a j = (t,”),a; E A , - A(C) (see also Figure IX.4). The conclusions to be (&), ( j 3 )of case drawn in the present situation are a modification of (jl), ( a 2 ) ;namely:

(kl) = (jl), (k,) = ( j 2 ) (with t in place of y). ( k 3 ) I T ( D ) I=I 7aj(D)I +C I 7 ( D i )1, where the sum contains i d ( t ) terms, and where 17aj(D)1=17,(D) 1.

IX.14

IX. Eulerian Trails - How Many ?

We also note that (14a,) holds in the present situation. Arguing now as in the case (a,) we conclude the validity of

Moreover, since case (b,) has been handled already, we conclude from ( 1 4 ~ that ~)

This, the preceding equation and (kg) imply the validity of the theorem for = t . This repeated contradiction shows that D has no 2-valent vertices.2)

D is a loopless 2-regular digraph .

(15)

Observe that for p D = 2 the corresponding eulerian permutations are the elements of the Klein group, two of which are even permutations and the other two odd. Hence, the theorem holds for pD = 2. It follows from the choice of D that p D >_ 3. Consequently and by the same token, an arc (y, z ) E A ( D ) exists such that y # # z # y. Denote

(see Remark 1X.l.a)). Possibly A , n A, 3 ( a 4 } which has no bearing on the following consideration.

Form the eulerian digraph 0; for i = 1 , 2 as follows: split y into two 2-valent vertices in one of two possible ways such that a4 and a; are adjacent, i = 1 , 2 ; suppress the 2-valent vertex incident with u4 and attach a loop ( z , z ) at z . Label the arcs of D; by letting b, := a , for m # 4, i, define b, := ( z , z ) , and let bi denote the unique element of 2, In [SWAN63a, SWAN69al it is assumed erroneously that x has no 2valent neighbors. While this error does not seriously matter in case a), it restricts the consideration of case b) to the first part of case (bl). Th'1s was discovered by Mr. Erich Wenger, [WENG82a], who settled the remaining cases in a way similar to me (by reducing them to either (al)or (a,)).

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.15

v Y

Df

D

Figure IX.5. The transformation of D into 0:.

( A ( D )U { ( z , ~ ) } )i , = 1 , 2 . This transition from D to DT is described in Figure IX.5, where b, := al denotes the arc adjacent to a,; hence {i,Z) = {1,2}.

A(D:)

-

We observe the following facts:

7-,(Df)n I,(D,*) = 0; (1,) Every T E I , ( D ) corresponds to a unique T' E 7 , ( D t ) for i = 1 or i = 2, such that TT = X T , ; (11)

(1,) p D y = p , , q D f = q D . Whence the theorem applies to because of (14), (15).

D:,i = 1,2,

Unfortunately, the correspondence expressed in (1,) is not a bijection; for it does not cover precisely those elements T* E 7,(D:), i = 1,2, which are of the form

T*= . . . , b,, b,, b,, . . . , s E {6,7}.

(15*)

In order to take care of this discrepancy we let w = z , z* = z , j = 4, i = T, T = 6,7, and form D4,r,D, respectively, as in the proof of (14), case

lX.Eulerian Trails - How Many ?

IX.16

(al). Label the a c s of D, analogously to our defining the arc labeling of DT (with T in place of i), and let s be such that ( T , s} = {6,7} (see Figure IX.6).

'i

Y \I

a4

D

Dr

Figure IX.6. The transformation of D into D,. Now we observe the additional facts:

(Z4) There is a bijection between those T* E q ( D T ) which are of the form (15*), and the elements TI' E 'Tv(DV)such that rT*= 7rT,,; (Z5)

pDT=p D ,

qD,

= q D , and the theorem applies to D, because of

(15).

By ( E , ) and (Z4) on the one hand, by (Z3) and (Z5) on the other hand, we have

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.17

That is, D satisfies the conclusion of the theorem, another contradiction.

I

Having shown that I7,"(D) !=I 7 t ( D ) in all cases possible with regard to the choice of D,we may conclude that no counterexample exists. This finishes the proof of Theorem IX.2. w e note that the condition qD 2 2pD cannot be weakened in the statement of Theorem IX.2. This can be seen from the eulerian digraph D, obtained from the digraph of Figure IX.l by subdividing one arc. For the conclusion of Theorem IX.2, applied to D,, fails if is 2-valent since in this case I 7,(Dl) I= 1 (it also fails if d(v) = 4 since 7:(D,) = 8 or 7,(D1)= 8 depending on the arc labeling of Dl); and go, = 2pD, -.1. In fact, D,can serve as the basis for the construction of an infinite family of counterexamples D to the conclusion of Theorem IX.2 which satisfy q D = 2pD- 1 (see [SWAN63a]). All one has to do is to attach an oriented digon at the 2-valent vertex of D, and continue this operation step by step. Every D thus obtained satisfies q D = 2pD - 1 and has precisely one 2-valent vertex which is the root of precisely one eulerian trail in D. Thus, the conclusion of Theorem IX.2 fails for D. However, there is another point to be made. For in general, Theorem IX.2 does not tell us much concerning odd and even elements of 7 ( D ) , where 7 ( D )is defined in accordance with Definition VII.1 (see Remark 1X.l.a)). Also, 7 ( D ) # 7 , ( D ) may very well hold (see the preceding paragraph and statements (*), (**) preceding Remark IX.1). However, if D satisfies the hypothesis of Theorem IX.2 and has a 2-valent vertex z1, then 7 , ( D ) = 7 ( D )in the sense of Definition VII.l, and by the conclusion of the theorem, 1 2 17~) I=IW) I .

i=im~)

In other words, for a given root

z1

and a given initial arc a , = ( v , t )

Theorem IX.2, in general, does not tell us whether I 7tl(D) I=I 7; (D) I holds, although 7 ( D ) = 7 a l ( D )in the sense of Definition VII.l and statement (*) (D),la", (D),74 (D) are defined as corresponding sets of eulerian trails starting with a,).

(zl

The preceding considerations, however, lead directly to a corollary to

IX. Eulerian Trails - How Many ?

IX.18

Theorem IX.2 which is due to M.P. Schiitzenberger and which was proved in 1958.3)

+

Corohry IX.3. Let D be a digraph satisfying q = qD 2 2pD 1, where A(D) = {al,. . . ,a,}. D has as many even eulerian trails starting with a, as it has odd eulerian trails starting with al.

Proof. W.l.o.g., D is a weakly connected eulerian digraph; otherwise, the theorem holds vacuously. For w $! V ( D )and a l = (x,y) form

D,

:= ( D - {Ul))

u {w, (2,4 (w,Y))

which is also a weakly connected eulerian digraph. Define the arc labeling of D , by b; := a;, 2 = 2 , . . . , q , b, := (w,y), b,+, := (2,w) . Since 40, = q D 1 2 2pD 2 = 2 p D , we can apply Theorem I X . 2 to D,; hence I7z(D,)I=I 7,"(Dl)I. However, the correspondence

+

+

-

T, = b,, . . , bzq,b,+l defines a bijection between la, ( D ) and 7,(Dl) such that Ta ]

= a 1, * .

. , azq

*

s9n(Ta1)= s 9 4 V . The corollary follows from this and the preceding equation. Corollary I X . 3 implies a parity result on the number of eulerian trails in graphs, digraphs, or mixed graphs. Its proof is left as an exercise.

Corollary IX.4. Let H be a graph or digraph or mixed graph. If q H > 2p,, then I I ( H ) 1 is even. Remark IX.5. The proof of Theorem IX.2 is very long when compared with the proofs presented in t h e cited references. However, t h e proof of this theorem is a typical example of the ease with which special cases

are overlooked. Therefore, I worked out all the details and even included all the explicit permutations in order to show how certain operations on digraphs influence the signs of eulerian trails. This approach, however,

3, See the footnotein [BERG66a, p.1701 and [BERG73a, Chapter 11, Theorem 91. The latter reference contains a proof which coincides in large parts with Swan's and which also deletes the consideration of certain cases (see the preceding footnote). The same mistake can be found in [BOLL79a, $5, Lemma 151. As a. matter of historical record we note that Swan acknowledges Schutzenberger's result in a footnote of his paper [SWAN63a].

IX.l.As Many As - Parity Results for (Mixed) (Di) Graphs

IX.19

made it desirable to separate the following considerations from Theorem IX.2 rather than to present a unified theory of even and odd covering trails in (mixed) graphs and digraphs.

In generalizing the preceding considerations of this section let H now denote a graph, digraph or mixed graph; let G := G , denote the underlying graph (G = H if H is a graph). Label the edges and/or arcs with b,, . . . ,b,, q = q H . Suppose G has a (closed or open) covering trail T G starting at v E V ( G )= V ( H ) ,and consider the corresponding arc/edge sequence T in H . Let 7,(G,H) denote the set of all such arc/edge sequences starting at v. Of course, just as in the case of digraphs, T G corresponds to a permutation

define sgn(TG) correspondingly. However, this definition of the sign function does not adequately describe T since in T an arc (5,y) may be traversed from y to 2,i.e., in the opposite direction. Let oT denote the number of arcs of H in T traversed in the opposite direction and define

sgn(T) = (-l)OTsgn(T,)

.

Correspondingly, we speak of odd (even) T if sgn(T) = -1 (+1) (note that T may be odd (even) if T G is even (odd)), and we denote by I,", H ) ( T ( G ,H ) ) the corresponding sets of odd (even) T.4)

Remark IX.6. a) Just as in the case of closed or open covering trails in digraphs we also note in the case of T E 7v(G,H ) that sgn(T) depends on the choice of w and on the choice of the labeling of the elements of E ( H ) U A ( H ) . However, if I T,"(G,H ) \=I I , ( G , H ) I for one labeling, this equation holds for any labeling { b J i = 1,.. . ,q } of E ( H ) U A ( H ) . For, oT is a parameter independent of any labeling. Whence the equations established in Remark 1X.l.a) also have their correspondence in the present more general case. Consequently, in the following results we need not always explicitly mention such labeling. In the following we call 4, Joan P. Hutchinson, [HUTC74a, HUTC74b, HUTC75a], speaks of negative (positive) E paths, E circuits respectively.

Ix.20

IX.Eulerian Trails - How Many ?

v E V ( H ) equalizing if I 7,"(G,H ) I=I 7 t ( G , H ) 1; we call H equalizing if every v E V(H)is equa~izing.~)

b) If H is equalizing, then H' := ( H - a) U { a R } is also equalizing: for if we compare any T E X ( G ,H ) with the corresponding T' E q ( G ,H I ) , sgn(T) = -sgn(T') since TG = T& and I oT - oTf I= 1. Whence we may conclude by induction that if H" is obtained from H by reversing the orientation of some arcs, H" is equalizing if and only if H is also equalizing (note that E ( H ) = E ( H " ) in this case). c ) If X(b) > 1 for some b E E ( H ) or b E A ( H ) then H is equalizing. This follows by virtue of the same type of argument used in the proof of Theorem IX.2 in establishing (1).

d) If H contains two vertices of odd degree, 2 and y say, H is equalizing if and only if 5 or y is equalizing (it is left as an exercise to work out the details of a proof of this statement - see also [HUTC74b, Proposition 21 for the case of graphs). In any case, Theorem IX.2 implies the validity of the next result (we assume automatically that a labeling of E ( H ) U A ( H ) is given). Lemma IX.7. H is equalizing if qH 2 2p,. Proof. In order to have 'ZU(G,H)# 0 for at least one v E V ( H ) (otherwise, H is trivially equalizing), we assume that G := G, is connected and either G is eulerian or G has precisely two odd vertices u , w . Let 6 E ( G ):= { H I , .. . , H k } , k 2 2, be the set of eulerian orientations of G if G is eulerian; otherwise, let it be the set of orientations of G such that Hi U { u ~ , is~ eulerian } for i = 1,.- ., k, where uw,u fZ A ( H i ) is an arc joining 'toto 21. By Theorem IX.2, I 7,"(Hi) I=I 7,"(Hi) I# 0, where v is an arbitrary vertex of H if G is eulerian, and v is the odd vertex of G for which i d H i ( v )= odHi(v)- 1, i = 1,.. . , k, otherwise. For any other choice of 21 in the latter case, 'Tv(Hi)= 0 and Theorem IX.2 holds vacuously.

On the other hand, every T E 7 v ( G H , ) induces precisely one orientation HiE b E ( G ) ,i E { 1,. . . ,k}, which results from H by orienting the edges of H and reversing the orientation of some arcs of H . Noting that every 5, In [HUTC74a, HUTC74b, HUTC75aI such vertices, (mixed) graphs respectively, are called null in the case of digraphs or mixed graphs and cancelling in the case of graphs.

uC.1. As Many As - Parity Results for (Mixed) (Di) Graphs

T' E 7 , ( H i ) ,i = 1,...,k, corresponds to

IX.21

a unique T E I , ( G , H ) we

conclude that k

However, by the first part of Remark IX.6.b) and since for H" = Hithe same arcs of H have been reversed for every T' E 7 , ( H i ) , we conclude that 7,"(Hi) corresponds to a subset 7*of 7 , ( G , H ) lying entirely in 7:(G, H ) if and only if 7:(Hi) corresponds to such a subset lying entirely in T ( G ,H ) ; and I* n'7;"(G, H ) # 0 if and only if I*C I , ( G , H ) . This, the preceding equation and the above application of Theorem IX.2 yield

i= 1

{ e , o}, i = 1, . . .,k, and yi = e (= 0) if and only if I A ( H ) - A ( H i )If 0 (rnod2) (E1(rnod2)). This finishes the proof of the lemma.

where

{ri,Si} =

Corollary IX.7.a. If q H 2 2p, + 1, then for every 21 E V(H)and every b E A, U E,,7:(G, H ) has as many elements starting with b as there are such elements in 7 : (G, H ). Corollary IX.7.a. follows from Corollary IX.3 in the same way as Lemma IX.7 follows from Theorem IX.2; its proof is therefore left as an exercise. Note that even for the case of graphs, Lemma IX.7 does not impZy Corollary IX.4 since for T E 7 , ( G , H ) one also has the inverse sequence T R E %(G, H ) .

In the case of graphs we have another condition sufficient for H to be equalizing, which is independent of any (non-trivial) lower bound on q.

Proposition IX.8 ([HUTC74b, Proposition 11). If G is an eulerian graph satisfying qG 2 or 3 (mod4) then G is equalizing. Proof. The proposition being vacuously true if G is disconnected, we assume for H = G that I , ( G ) := 7 , ( G ,G) # 8 for an arbitrary but fixed

Ix.22

IX.Eulerian Trails - How Many ?

vertex v. Associating with every T = e i , , . . . ,ei, E T ( G ) , q := qG, the inverse sequence T R = e i q ,. . . ,eil we obtain

where S E {0,1} satisfies S q (mod2). It follows that rTRrG1 is an odd permutation if and only if = 1(mod2), i.e., if and only if q E 2 (mod4) or q G 3 (mod4). Thus if q 2 or 3(mod4), then sgn(TR) # sgn(T) and the above permutation T ( G ) ++ T ( G ) defined by T H T R becomes a bijection 7 t ( G ) H 7 z ( G ) . This and the arbitrary choice of v E V ( G )imply the validity of the proposition.

9

For the next proposition but also for further discussing the concept of equalizing (mixed) (&)graphs, we need a certain classification of the elements of 7 , ( G ,H ) . From now on, denote 7,(G) := ‘irv(G,G ) and let TG E 7 , ( G ) be the object corresponding to T E 7,(G, H ) .

Definition IX.9.Let H be a weakly connected mixed graph all of whose vertices have even degree, and let 21 E V ( G ) . Suppose the elements of E ( H ) U A ( H ) are labeled b,, . . . , b,, q = q H . For T ,T‘ E 7,(G, H ) we say that T and T’ are rotation equivalent if r T G ( j ) = ii implies x T & ( j )= i j + k for some k E (1,. . . , q } and j = 1,.. . , q (we set j + q = j , j = 1 , 2,.”, 4). It is plain to see that in fact the concept of rotation equivalence is an equivalence relation RE; its proof is therefore left as an exercise. Note that for d(v) = 2r, where 21 and H are as in Definition IX.9, the equivalence classes of 7,(G,H ) under RE contain precisely T elements each. For, if T and T‘ belong to the same equivalence class, then X T = X,,, while the converse is not generally true (see Exercise IX.3). We denote by PPRE(v)the partition of 7,(G,H) into equivalence classes induced by RE.

Lemma IX.10. Let H , RE and P P R E ( be v ) as in Definition IX.9 and the paragraphs following it. Suppose q = q H is odd. For every C E P R E ( v ) and every T,T’ E C it follows that sgn(T) = sgn(T’). Proof. First of all it follows from the very definition of T and T‘ being rotation equivalent that oT = oT,, since T and T’pass the same set of

IX.1.As Many As - Parity Results for (Mixed) (Di) Graphs

IX.23

arcs in the opposite direction. Whence it suffices to show that sgn(TG) = sgn(T&). By definition of RE, if rTG-

1

- (i,:

TTATFt

=

(k:l,

2, k+2,

2, i,,

... ' ' ) , then .. ., i,

..., ...,

k+1, 2k+1,

...

..., k

for some k E { 1 , . . . ,q } . It follows from the very structure of rTAr?; that its expression as a product of disjoint (permutation) cycles consists of q / s cycles of length s, where s E N is the smallest number such that s - k 0 (modq). Since q is odd by hypothesis it follows that s is odd and so each of the q / s (pairwise disjoint) permutation cycles can be expressed as a product of s - 1 0 (mod 2) transpositions. Hence rTtr?; G can be expressed as the product of an even number of transpositions, i.e., sgn(rT, T+:) = 1. That is,

=

G

from which the lemma now follows.

Corollary I X . l l . Let T E 7 v ( G , H ) TI , E I , ( G , H ) such that G is eulerian, X , = X,, , and TG and T&induce the same eulerian orientation of G. If qH is odd, then sgn(T)= sgn(7").

Proof. Note that in the proof of Lemma IX.10, we did not make use of the fact that T and T' start at the same vertex. Moreover, the hypothesis of the corollary also implies for r T G ( j )= i j that r T A ( j = ) i j + k (see Definition IX.9). Thus, the corollary follows from the proof of Lemma IX.10. Proposition IX.8 and Lemma IX.10 are the basis for the next result (see [HUTC74a, Lemmas 3,4], [HUTC74b, Proposition 41, [HUTC75a, Lemmas 1,2]).

Proposition IX.12. Let H be a (mixed) (di)graph with odd q = qH and having no odd vertices. Let 7v,rep 'irv(G,H ) be a set of representatives with respect to PPRE(v), where v is a fixed vertex of H (we assume a labeling b,, . . . ,b,, q = qH, of A ( H ) U E ( H ) given). The following statements are equivalent.

IX.Eulerian Trails - How Many

IX.24

1)

TJ

2)

I {T E ?;,rep/sgn(T)

?

is equalizing. =

11I=I {T E ?;,rep/sgn(T)

= -1)

I*

3) H is equalizing.

Proof. W.1.o.g. H is weakly connected; thus 7 , ( G , H )# 0. Suppose / 7 : ( G , H ) [=I 7 : ( G , H ) I and consider an arbitrary C E P R E ( v ) . By hypothesis ( q = 1(mod 2)) and thus by Lemma IX.10, C n 7;6(G, H ) # 0 implies C T'(G,H ) for any 6 E { e , 0). Moreover, I C I= $(v) in any c which denote the respective subsets of case. For 7;rep,7;rep even and odd arc/edge sequences, we thus have

Hence 1) implies 2); the converse follows by the same token and by multiplying the last equations by i d ( . ) . In order to finish the proof of the proposition it suffices to show that 1) and 2) imply 3) since 3) implies 1) anyway. W.1.o.g. p H > 1 since 1) and 3) are equivalent if p H = 1. Whence let w E V ( H )- {TJ} be chosen arbitrarily. For every T E 7 , ( G , H ) consider T R ,the inverse sequence, and let TG, Tg E 7 , ( G ) be the corresponding eulerian trails in G. Since, by definition, sgn(T) = (-l)"'sgn(TG),

sgn(TR) = (-l)"TRsgn(TGR)

,

we conclude from Proposition IX.8 and its proof that sgn(T) = -sgn(TR) if and only if either I E ( H ) 1- 1(mod 2) and qH 3 (mod 4) or I E ( H ) I=

=

O(mod 2) and qH 1 (mod 4). For, in the first case oT G oTR (mod 2) and sgn(T,) = -sgn(Tg), while in the second case oT $ oTR (mod2) and sgn(TG) = sgn(Tg). Arguing dong lines similar to those in the proof of Proposition IX.8 we may conclude that I 7 , ( G ,H ) I=I 7,"(G,H ) I for That is, w is equalizing. Since TJ is equalizing by any w E V ( H )- {TJ}. assumption we conclude that H is equalizing in the cases considered. Thus we are left with the consideration of cases where qH = 3 (mod4) and I E ( H ) I O(mod2),as well as qH 1( m o d 4 ) and I E ( H ) I 1(mod2). Now consider C 7 , ( G ,H ) , a system of representatives with respect to PRE(v). W.1.o.g. we may assume that every element of 7,,Tep starts or ends with a fixed f E E, U A , - A, (such f exists since H is weakly

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.25

connected). Denoting by 7u(f, 1)('TU(f,q ) ) the subset of 7u,rep whose elements start (end) with f , we have

By definition, for T E %(G, H ) the inverse sequence T R and T belong to different equivalence classes. We conclude from the assumption concerning 7u,rep that

(note that if T,,T, belong to different classes of P R E ( v ) , then so do T p , T t ) . Since sgn(T) = sgn(TR) (which follows from the assumption concerning q H and E ( H ) I), we may conclude that the bijection cp : %(f,1) ---t % ( f , q ) defined by cp(T) = T R is sign-preserving (that is, sgn(T)= sgn(cp(T))).This, (l),(2) and the validity of 2) imply

I

where 7:(f,l) 1).

w,

(7t(f,1)) denotes the set of even (odd) elements of

Now we consider 7 w ( G , H )again. The correspondence T t+ T R for T , T R E 7 , ( G , H ) and the fact that, under the given assumptions, sgn(T) = sgn(TR),define a sign-preserving bijection T) : 'T; t I,,where 11(1, is) the set of those elements of 7 , ( G , H ) in which the traversal of f starts (ends) at TI.Since 'irw(G,H ) = 'T; U I,,it thus suffices to prove that

1'7;"1=lTl

(4)

7

where 7 : (7:) denotes the set of even (odd) elements of 71.For we will then have I I=! I as well, and since 7 nI, = 0 (because f $ A ( H ) ) , we can then conclude

We consider P R E ( w ) the , corresponding partition of 7w(G,H ) into equivalence classes with respect to RE. By Lemma IX.10 and the definition of RE we conclude that C nq6# 0 implies C g for every 6 E { e , o} and every C E PRE(w). Thus is the union of certain classes of P R E ( w ) . Since each of these classes contains precisely ;d(w) elements, in order

T6

IX.26

IX.Eulerian Trails - How Many

?

to prove (4) it sufiices to find a particular 7 i ; ; e p C ‘T; having as many even elements as it has odd elements. Given the definition of RE, each C E P R E ( w )satisfying C 7’contains precisely one element Tc, such that f belongs to the first segment (of Tc,f)starting and ending at w (i.e., starting at w ,a walk through Tc,f passes f before reaching w again). Let

._ .By definition of 7u(f, 1) and 7 , ,(1) r e p , we obtain the following bijection between these two sets: for arbitrary Tc,! E 7 i t r ) e p let S(w,v)be the longest possible subsequence of Tc,! starting at w, ending at v and not containing f nor any segment starting and ending at w. Thus

T C , f = S ( w , .),T(v,

4 ,

where T ( v ,w) is the corresponding complementary subsequence of Tc,f. By definition of Tc,f and S(w,v), the first arc/edge of T ( v , w ) is f. Whence T := T ( v , w ) ,S(w,v)E I,(f, 1) . Conversely, for every T’ E Tu(f, l), if we denote by S’(w, v) the subsequence of T’ starting at w ,ending at ZI with b,, (where xTb(q) = iq) and containing precisely one element of A , U E,, then we have

T’ = T’(v,w),S’(W,V ) and T&,f:= S’(W,v ) , T‘(v,w)E

7itr)ep,

where T’(’u, w) denotes the corresponding complementary subsequence. In particular, the definition of the above S(w,v) and S’(w, v) imply that if T = T’, then S(w,v) = S’(w, v). It follows that

Tc,f = S(w,v), T ( v ,w)c-) T = T ( v ,w),S(w,v) defines a bijection between 7 w(1) , r e p and ?;(f, 1) which is sign-preserving by Corollary IX.ll. It now follows from ( 3 ) that 7 i : ; e p contains as many odd elements as it contains even elements; hence (4) holds as well. That is, w is equalizing. Since w had been chosen arbitrarily in V ( H )- {v} and since v is equalizing by assumption, it follows that H is equalizing. This finishes the proof of Proposition IX.12. Following the proof of the preceding proposition we introduce some notation.

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.27

Definition IX.13. Let H be a mixed graph with underlying graph G, and suppose a fixed labeling of the edges and arcs of H be given. Let f = bi E A, U E, and assume f is not a directed loop. Finally, let 7,(f, 1) denote the set of all arc/edge sequences of H starting at v along f and corresponding to an open or closed covering trail of G, and let 7E(f,l), 7t(f,1) denote the respective subsets of even and odd elements 1). Denote of 7,(f,

(see equation (1) of the proof of Proposition IX.12). Correspondingly, call f an equalizer with respect to v if and only if t"vf = tz,f, and call H f-equalizing if and only if f is an equalizer for every f E A, UE, and every v E V ( H ) ,provided f is not a directed loop (see equation (3) of the proof of Proposition IX.12). The requirement in this definition that f not be a directed loop, is of no relevance to our further considerations. This follows from the next lemma whose proof is left as an exercise.

Lemma IX.14. Let H be a mixed graph, A ( H )U E ( H ) = { b l , . . . ,b q } . If H contains a directed loop, then H is f-equalizing. Thus, in the sequel we may always assume that H has no directed loops. We also observe that if G, the graph underlying H , is eulerian and if tE,f = tz,f for f = vw or f = (v,w), where v # w ,then we also have For, the bijection cp : T t) T R , T E T(f,l),is either iL,f = sign-preserving or sign-reversing (i.e., sgn(cp(T))= -sgn(T)), regardless of the size of q, and it is a bijection between 'T,,(f, 1) and 7,(f, q ) (see equation (2) of the proof of Proposition IX.12). Moreover, there is a bijection 11, : 7,(f, q ) -,7 w ( f , 1) defined by

where b j l , b i 2 , .. . , b j E 'T,,(f,q), f = b j q . It follows that T,!J is signpreserving if and onfy if Q is odd, and sign-reversing otherwise. Whence 1) -, 7w(f, 1) is a sign-preserving or signwe conclude that T,!Jcp : 7v(f, reversing bijection so that tz,f = tz,f implies tL,f = t"wf,and vice versa. Thus for an edge/arc f to be an equalizer is independent of the incident vertex with respect to which f is being considered. We note that if H has odd vertices, any edge/arc incident with only even vertices is automatically an equalizer. For in this case a covering trail of

IX.Eulerian Trails - How Many ?

IX.28

G = G H exists if and only if G has precisely two odd vertices, and that trail has to start and end at odd vertices. Finally, we observe that H being f-equalizing implies that H is equalizing. In particular, Corollary IX.7.a says that if q H 2 2pH 1, then H is f-equalizing.

+

For brevity's sake we introduce some additional notation.

Definition IX.15. Let H be a mixed graph, and let i E {0,1,2,3}, j E {O,l}, k E { 0 , 2 } . We say that H is of type (i,j;k) if and only if q H = i(mod4), I E ( H ) j (mod 2), and H has k odd vertices. The following results are direct consequences of Definition IX.15 and H), preceding considerations (see the definition of sgn(T),T E 7u(G, and the proofs of Proposition IX.8 and Proposition IX.12); their proofs are therefore left as an exercise.

Corollary IX.16. Let H be a weakly connected mixed graph having 0 or 2 odd vertices, and let A ( H ) U E ( H ) = {bl,. . . ,b q } . Consider the bijection cp : 7 u ( G , H ) 7z(G, H ) defined by cp(T) = T R ,where x = v if G is eulerian, and x = w if {v,w} are the odd vertices of H . The following statements are true for any choice of k E {0,2}. --f

1) cpissign-reversingifHisofeithertype (l,O;k),(3,1; lc),(O,l;k),

(2,O; k). 2)

'p

is sign-preserving if H is of either type (1,l;lc), (3,O;k),

(070;k),(231; k). As a consequence of this corollary we also have the following. Corollary IX.17. Let H satisfy the hypothesis of Corollary IX.16. The following statements are true. 1) If H is of type (0,l; 0), ( 1 , O ; 0 ) , (2,O; 0), or ( 3,l; 0), then H is equalizing. 2) If H is of type (1,l;0) or ( 3 , O ; 0), then H is equalizing if and

only if H is f-equalizing. We note explicitly that the preceding two corollaries are valid independently of any lower bound f ( p H )for qH. Next we extend almost all of Corollary IX.17. We automatically aSsume that the arcs and edges of the mixed graphs considered are labeled.

Proposition IX.18. Suppose for some fixed q E N that every weakly connected mixed graph H' of type (1,l;O) or ( 3 , O ; O ) is equalizing if

IX.1.As Many As - Parity Results for (Mixed) (Di) Graphs

IX.29

qH, = q and IE(H') IS 1. Let H be a mixed graph such that q H = q and I E ( H ) 1s 1. If H is of either type (1,l;2), (3,O;2), (1,O; k), k E {0,2}, then H is f-equalizing.

Proof. A) Suppose fist that H is either of type (1,l;2) or of type (3,O;2). Denote by 2 and y the odd vertices of H , and consider A,UE, = { b l j / j = 1,. . .,d'}, where loops are counted only once in this set; so, d' = d ( y ) if E, contains no loops ( A , contains no loops by the assumption following Lemma IX.14). Now construct H ( j ) , j = 1, . . .,d' from H as follows: if d' > 1,

,

V(H(j)= ) V ( H ) ,A ( @ ) U E ( H ( j ) )= A ( H ) U E ( H )

the incidence functions of H ( j ) and H coincide on A ( H ) U E ( H ) { b l j } , blj E A, U E C A ( H ) U E ( H ) is incident with z E V ( H ( j ) )but !not with y f V ( H ( J )if) blj is not a loop; otherwise, it becomes an edge incident with both z and y (this construction is demonstrated by Figure IX.7). If d' = 1, then let H(') be obtained from H by identifying z and y. Set H' := H ( j ) ,j = 1,.. .,d', in any case and carry the edge/arc labeling of H over to H'.

I

\

H

*'

H':= H (i)

Figure IX.7. Constructing H' from H by letting blj become incident with x instead of y, and without changing the orientation of b l j . Since blj is not a directed loop, this construction is uniquely determined. If blj is a loop in Ev, then the two possibilities of changing the incidence function yield the same result.

IX.30

IX.Eulerian Trails - How Many ?

Note that H’ is uniquely determined for every btj E A , U E,. For the following considerations, however, a determining factor is whether blj appears as a loop in H’. It follows from the assumption and from the construction of H’, though, that H‘ is of type (1,l;0) or (3,O;0). Consequently, since q H l = q H it follows from the hypothesis that H’ is equalizing. By Corollary IX.17 it even follows that

H’ is f-equalizing .

(1)

That is, for f = b l j we have in H’ in particular

On the other hand, since there is a sign-preserving bijection between 1) c H ) (G := G H ) provided f is not a directed loop in H‘, we obtain from (2) the validity

<(f,1) C <(G’, H’) (G’ := G H l )and

T(f,

T(G,

of

G>f= t;,f

(3)

which has to be read as equation in H (Note that T E 7,(f7 1) written as an arc/edge sequence, can also be interpreted as T E 1) since we do not change the labels of any arc/edge in constructing H‘ from H ) .

%(f,

In fact, the above bijection 7,(f7 1) t) ‘T,(f, 1) yielding (3), no longer holds if f = blj is a directed loop at 2 in H‘. This can happen if and only if 0 # A, n A, & A ( H ) . In this case we define

and observe that H” is of type ( 0 , l ; 0) or of type (2,O; 0). Assume now w.1.o.g. that f = blj = b, (see Remark IX.G.a)), and that H“ is weakly connected. By Corollary IX.16.1) we have s g n ( T ) = -sgn(TR) for every T E 7=(G“,H ” ) , G” = G H l .l Since GI’ is eulerian we also have T R E 7,(G”, 23”).Thus we have

On the other hand, we claim that the bijection $ : T, E 7,(G”, HI’) + Ty:= f,T, E

%(f, 1)

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.31

is sign-preserving or sign-reversing depending on the orientation of f. For, if T denotes the eulerian trail of GI' corresponding to T, and if rT = " 27' ' ' ? - is the eulerian permutation associated with T,then

'>

the open covering trail To of G' corresponding to

Ty, where $(T,)= T,,

( 1,, 27 .3,...,q

has as its associated eulerian permutation rTo= Moreover, we have for r; :=

'7

'7'*.7

21, '2,-.-,

q-

'7

29-1,

47 q 9

'17

22,"*,iq-1

the

of

sgn(?rT)= sgn(r$). Thus we have

which is an even permutation since q is odd. F'rom this and the fact that oTV = oT, if f E A:, and oTV = oT, 1 if f E A;, we conclude that $ is sign-preserving if f E A:, and sign-reversing, otherwise. Thus our claim is correct which together with (4) implies the validity of (3) even if f E A, n A, C A ( H ) ;whence we conclude the validity of (3) for every f E A, U E, C A ( H ) U E ( H ) . Using a symmetrical argument we may conclude

+

t:, f = t:, f for every f E A, U E,

(3')

A ( H ) U E ( H ) . Since

for every f E A, U E, and w E V ( H ) - (5,y}, we may conclude from (3), ( 3 9 , (3") that H is f-equalizing if H is of type (3,O;2) or of type (1,l;2). This finishes the first part of the proof of the proposition (note that if any H" or any H' is disconnected, then (3) holds trivially).

B) H is of type (1,O; k) for some k E {0,2}. E ( H ) = 8 follows. Suppose first that k = 2. For an arbitrary f E A,, where y is an odd vertex, let Hl be obtained from H by replacing f E A ( H ) with an edge ef to which we assign the same label carried by f. Hl is of type (1,l;2). It follows from the proof of case A) that Hl is f-equalizing. In particular, we have in Hl tE,er = t ; , e t . (3"')

IX.Eulerian Trails - How Many ?

IX.32

However, there is a bijection w between the elements of 7v(ef,1) C %(Gl,Hl) ( G , = GHI) and those of % ( f l l )E I , ( G , H ) which is defined for Tef:= e f , T * by = fl T*

UPef)

*

Thus, w is sign-preserving or sign-reversing depending on whether f E A; or f E A;. This and (3”’) imply the validity of (3) for every f E A . Analogously, we conclude the validity of (3’) and (3”) as in case Aq. Whence H is f-equalizing if k = 2. Finally suppose that k = 0 and consider an arbitrary y E V ( H ) and an arbitrary f E A,. Forming H , as above we deduce the validity of (3”’) and thus of (3) from the hypothesis that H , is equalizing since it is of type (1,l;0) which in turn implies that H , is f-equalizing by Corollary IX.17.2). This finishes the proof of the proposition. We also have the following lemma.

Lemma IX.19. Let H be a mixed graph without odd vertices and such that q = q H is odd. Suppose an arc/edge labeling b,, . . . ,b, given and that H has a 2-valent vertex v; denote H’ := H - ZI. If H’ is equalizing, then H is equalizing in which case the elements of A, U E , are equalizers.

t,}.

Proof. We assume w.1.o.g. that A, U E, = {b,-,, Denote N ( v ) = (5,y}; possibly z = y. Suppose now that H‘ is equdzing. It follows that

[ c ( G ’ , H’) [=(I,(G’,H’) I=[ T ( G ’ ,H’) / = l q ( G ’ ,HI) I

.

(*)

-,

W.l.o.g., b, E A, U E,. For T i E 7z(G’, H’) we have T := bq,T’,b E 7 v ( GH , ) . Denoting by TG,and TGthe respective elements of ‘T,(G;fand I,(G)we claim that

.

sgn(TQ) = sgn(T,) In fact, if nTG,=

(.

1, 2, ..., q - 2

.

21, 22,

1, 2 ,”., 4 - 2 , q 2 1 , 22,

*

-

1

2,-21

4

--

- 1,q

* 1

*

nTGI and

nGGf :=

Zq-2

have the same sign. Moreover, by definition

of rTGfand T we have rTG= obtain

then

(**,

-1

-

rTG,nTG -

‘ ” 1

41 i,,

i2,-

* l

4-11 i,--21

(1,2l.--lq-1lq)

4 . Thus we 4- 1

IX.l. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.33

which is an even permutation since q is odd by hypothesis. This implies the validity of (**). Since b, and b are passed by T in the same direction regardless of the choice of T4-1we have

where c is a constant independent of the choice of T', and c E {0,1,2, }. Thus sgn(T) = (-l)"sgn(T') which implies that the above relation between T' and T := b,, T',bqd1 defines a sign-preserving or sign-reversing bijection cp between 7, (GI,H ' ) and T ( G ,H ) . This and (*), however, imply for f = b,

Thus f = b, is an equalizer, and using a symmetrical argument (with y in place of z) we may conclude that b,-l is an equalizer as well. However, since q is odd and H has no odd vertices, equation (* * *) can be interpreted as an equation concerning a system of representatives of

%(GHI7 hence we conclude from (* * *) and since cp is sign-preserving or signreversing that 7v,rep has as many odd elements as it has even elements. By Proposition IX.12, H is equalizing. The lemma now follows. Looking at Proposition IX.18 and Lemma IX.19 it becomes apparent that there is something special concerning mixed graphs of type ( 3 , l ;2). This will become clear after the next result.

Theorem IX.20. Let H be a mixed graph with a given arc/edge labeling b,, . ,b,, q = q H ' For p = p H , suppose that either q 2 2p - 1 > 1 and I E ( H ) I< 1, or p = q = 1 and IE ( H ) I= 0. Suppose further that every mixed graph H* of type (3,l; 2) with q* := q H * 5 q and I E ( H * ) I= 1, is equalizing. Then H is equalizing. o

.

Proof. By Lemma IX.7, the theorem is true if q > 2p - 1. Whence we assume q H = 2pH - 1. Proceeding indirectly let H be a counterexample to the theorem having as few vertices as possible.

Ix.Eulerian Trails - How Many ?

Ix.34

If p = 1, then H is a digraph with a directed loop in which case Lemma IX.14 implies the validity of the theorem.6) Whence we may assume p > 1, and since the theorem holds vacuously if H is disconnected or has more than two odd vertices, we assume that H is weakly connected and has at most two odd vertices. Suppose, p = 2. If H contains two parallel arcs, then H is equalizing by Remark IX.6.c). Consider the case A ( H ) = ((v, w), (w, v)}, where V ( H )= {v, w}. However, replacing (w, v) with (w, v ) R (i.e. by a second arc joining v to w), we obtain HI having two parallel arcs. HI is equalizing by Remark IX.6.c). It follows from Remark IX.6.b) and d) that H is also equalizing. Thus H contains a directed loop since q = 3, p = 2. By Lemma IX.14, H is equalizing in this case. It follows that H is equalizing whenever p = 2.7) Whence

Suppose G := GH is a connected eulerian graph. It follows from

that d H ( v ) = 2 for at least one v E V ( H ) . For one such v form HI := H - v . WehavepH, =p-1, qHr = 2pHr-1 > l,andtriviallyIE(H'))< 1. We conclude from the choice of H that HI is equalizing. It follows from Lemma IX.19 that H is also equalizing. This contradiction to the choice of H , as well as Lemma IX.14 and Remark IX.Gb),c) imply that H is weakly connected, has n o directed loops n o r two arcs joining the same pair of vertices, and it has precisely two vertices of odd degree. (2) Since q is odd we conclude from the hypothesis and (2) that H is of either type ( 1 , O ; 2), (1,l;2), (3,O;2)) (3,l;2). We have seen that H* is equalizing if G* := GH* is eulerian and q H * = q, in which case H* is of either type (1,l;0), (3,O;0) (if H* is of type ( 1 , O ; 0) or (3,l;0), then 6, J.P. Hutchinson does not consider this case; she rather assumes automatically p > 1 which is no loss of generality in view of the algebraic applications of this section. Note, however, that the conclusion of the theorem fails for

p=q=/E(H)I=l. 7,

loops.

J.P. Hutchinson considers a priori only those H that have no directed

IX.1. As

Many As - Parity Results for (Mixed) (Di) Graphs

IX.35

H* is equalizing anyway by Corollary IX.17). Moreover, I E ( H * ) 15 1. Invoking Proposition IX.18, or the hypothesis that H * is equalizing if H* is of type (3,l; 2 ) , we may now conclude that H is equalizing in any case. This finishes the proof of the theorem. In order to prove that any mixed graph H satisfying I E ( H ) 15 1 and qH = 2pH - 1 > 1 is equalizing, it follows from Theorem IX.20 that it suffices to do so if H is of type (3,l; 2 ) . In fact, this is the central part of the proof of [HUTC75a, Theorem 31 (see Case IV of that proof, where it is combined with treating digraphs of type (1,O; 2)). Hutchinson’s proof is very complicated (it rests on various technical lemmas). Therefore, I restrict myself to outlining this proof extensively so that details can be filled in by the reader.8) Again we assume automatically that the elements of A ( H ) U E ( H ) have been labeled with b, , . . . ,bqH.

Theorem IX.21. If H is a mixed graph of type (3,l; 2 ) satisfying qH = 2pH - 1 and I E ( H ) 15 1, then H is equalizing. Outline of proof. As before, one proceeds indirectly. The theorem holds for p H = 2 in which case it must have a directed loop or two arcs joining the same pair of vertices. Let x and y be the odd vertices of H . First, it is proved that 5 = m i n { d ( z ) , d ( y ) } 2 5: it is trivial to see that S > 1; otherwise, w.1.o.g. d ( x ) = 6,N ( x ) = {u}, and there is a signpreserving or sign-reversing bijection between 7 , ( G , H ) and 7 , ( G - x , H x ) ; and H - x is equalizing by Lemma IX.7. Suppose now 3 = d(z) 5 d(y). If x is adjacent to a loop I, then {I} = E ( H ) (see Lemma IX.14); thus H - x is of type (1,O; k ) , k E { 0 , 2 } . Invoking for H - x Corollary IX.17 if k = 0 and Theorem IX.20 if k = 2 , and observing that there is again a sign-preserving or sign-reversing bijection between 7,(G, H) and I,(G- x,H - x), {u} = N ( x ) ,we may conclude that H is equalizing because H - x is equalizing (note that Theorem IX.20 holds for q < q H ) . Thus x is incident with three different edgeslarcs. Form Ho = H U { b q + , } , where bq+, is an undirected loop at x . H , is equalizing by Lemma IX.7. We have for A , U E , := {bq-2, bq--l, bq} c A ( H )u E ( H ) and Go := G H o, P

%(GO,HO)

Z(bj,1) -

= % ( b q + l , 1) u

(1)

j=q-2

I have tried without success for several weeks to develop a less constructive proof, i.e., a proof which follows more along the lines of the preceding results.

IX. Eulerian Trails - How Many ?

IX.36

Every Tj E l , ( b j , 1) is of the form

T. 3 = b j , . . . , b k , bq+l, b,, . . . , where { j , k,E} = { q - 2 , q - 1,q } ; w.1.o.g. assume j = q - 2. Form

where b’g-l is incident with the initial and end-vertex of the section b k , bq+l, b, of T j ,and where bb-l is an edge if and only if precisely one of b k and b, iS arc. Suppose first that b&l is not a loop. Then there is a sign-preserving or sign-reversing bijection between the elements of 7 2 ( b q - 2 , 1) and those of 7(Gq-2H , q - 2 ) , Gq-2 := G H q d 2The . same is true with respect to the elements of x ( G q - 2 Hq--2) , and those of 7,,(Gq-2- z, Hq-2 - z), where {v} = N Hr - 2 (x) (note that z is an end-vertex of Hq-2 since d H ( z ) = 3). By construction and the choice of H , Hq-2 - z is equalizing; hence tz,bq-2

tirbq-2*

Suppose now that bbWl is a loop. For the above Tj we then have T’ := b j , . . . ,b,, b q + l , b k , . . . , which is also an element of x ( b j , 1). Thus the correspondence Tj T’ defines a bijection of 72(bj,1) onto itself which is sign-reversing if and only if bb-l is an arc. In this case we have immediately t z , b q - 2 -- 0t 2 , b q - 2 ( j = q - 2 by assumption). If, however, b& f-,

is an edge, then the above correspondence Tj w 2’ is sign-preserving, and there is no longer a bijection between 7 2 ( b n - 2 ,1) and 72(Gq-2, Hq-2). However, if we replace 7 2 ( b n - 2 ,1) with 7#q-2, 1) C 7 z ( b q - 2 , l), where T E q ( b q - 2 , 1) if and only if T = b q - 2 , . . . ,bp-l, bq+l, b,, . . ., then we can argue as in the case where b&l is not a loop; and using the fact that T’ t)T j is sign-preserving we also obtain t z , b q - 2 - tz,bq-2.

By relabeling the elements of Ho we obtain for each set in the second term of (1) the equation tz,bj = t z , b j . This and (1) as well as the fact that Ho is equalizing imply

which in turn implies

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.37

since there is a sign-reversing bijection between 7z(b4+1, 1) and 7z(G, H) (note that q is odd). It now follows from the choice of H that 6 2 5 must hold. Now form

H* = H

u (2,

( Z , X ) , (2,

Y)), z

@V(H)

H* is of type ( 1 , l ;0 ) , and qH* = 2pH, - 1; so if it is equalizing, then it is f-equalizing by Corollary IX.17. For b,+, := ( z , z ) and b,+, := (z,y), there is a sign-preserving bijection between 7,(b,+,, 1) and 7%(G, H ) . It follows that if H* is equalizing, then ( z ,x) is a n equalizer and H is equalizing.

(4)

H contains a 2-valent vertex (see the argument following (1)in the proof of Theorem IX.20). If H* contains two adjacent 2-valent vertices, u and v say, then H , := H* - u is of type ( 3 , l ;2) if E, = 8, otherwise it is of type (3,O;2). Moreover, pH, = pH and qH, = qH. However, H , has v as an end-vertex, so H , - v is equalizing by Lemma IX.7. Consequently, H , is equalizing (see the beginning of the proof). It follows from Lemma IX.19 that H* is also equalizing. This and (4) contradict the choice of H ; whence V 2 ( H * ) the , set of 2-valent vertices of H * , is an independent set. We have in any case,

A

For H := H* - V 2 ( H * )we also have qa = 2pa - 1 (this comes about precisely because V 2 ( H * )is an independent set). However,

which implies 6(k)5 3. This inequality as well as (5) and the fact that V,(H*) is an independent set imply that

H* contains a vertex v E V ( H * )- V,(H*) which is adjacent to at least d H * ( v )- 3 etements ofV,(H*). (6) (see [HUTC75a, Lemma41). Let v be chosen such that IN H * ( v ) n V 2 ( H *I ) is a maximum. Assume the elements of NH* (v) labeled in such a way that

IX.Eulerian Trails - How Many ?

IX.38

N H .( 2 ) ) = { q , . . , 2 ) d } , d = d H ,( 2 ) ) (possibly 2); = vj for some 2 u,E V,(H*) for i = 4,. . . ,d. Let

# j ) , and

a d note that pH = P H , q H = 48. Suppose first that I NH*(ud)I= I. It follows that H is of type ( 3 , l ;0) or of type (3,O;0), and q H = 2 p -~1. In the first case is equalizing by Corollary IX.17. In the second case H is equalizing because of Lemma IX.19 following an application of Theorem IX.20 (note that H contains a 2-valent vertex in any case). Thus H is equalizing anyway, and a renewed application of Lemma IX. 19 proves H * to be equalizing which in turn implies that H is equalizing because of (4). This contradiction to the choice of H implies

I NH*(vi)I=

2 for every uiE N H . ( v )r l V 2 ( H * ) .

(7)

Thus, we have t o consider the case where H is of type (3,O;2) or ( 3 , l ;2). However, if H is of type (3,O;2), we transform H into various digraphs of type (3,O;0) as in the proof of Proposition IX.18, each of which contains a 2-valent vertex which allows a reduction to a digraph of type ( 1 , O ; 2). We may apply Theorem IX.20 to the latter digraphs and then Lemma IX.19 to see that the former are equalizing as well. Then we conclude as in the proof of Proposition IX.18 that H is equalizingg) It follows that

H

as

of type (3,1;2); its odd vertices are v and w := wd,

(8)

wi E N H *(ui) - {u}. We conclude as before that r n i n { d g ( v ) ,d g ( w ) } 2 5 ; so, by choosing u (see (6)) we have

IN&)

n v 2 ( H )12 d - 4 .

(9)

Denote A, U E, = { e l , .. . , e d - l } C A ( H )U E ( H ) such that ei is incident with v; , 1 5 i 5 d - 1 (see Figure IX.8). -We have for G := G R ,

'1 Here and in the subsequent discussion we always assume that the labeling of A ( H ) U E ( H ) is adapted, if necessary, when transforming H into a smaller mixed graph. This is no loss of generality because of Remark IX.6.a).

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.39

V

Figure IX.8. The mixed graph I?; for simplicity's sake, the arcs are drawn as edges (we do not know whether E, U

::u;

Ev; = 0).

For i = 4,.. .,d - 1, we conclude that t: e . = t;,,; since there is a sign-preserving or sign-reversing bijection between q ( e i ,1) and IWi (G vi,H - vj) where wi E NB(vi) - {v} (possibly wi = w . Note that v is an even vertex in H - vJ; and Theorem IX.20 is applicable to H - 21;. Thus, in order to show that <(G, H ) contains as many even elements as it contains odd elements, it s f i c e s to show 9

3

3

i= 1

i=l

s

Consider a fixed i E { 1,2,3}. Every T E I v ( e i ,1) defines a set of transitions X,(v) in H - e ; . We distinguish between two cases (see [HUTC75a, Lemmas 5 and 61). Case 1. ( e i , eh, ek}-{e:} 6 X,(v). Note that in this case every { e ; , e ; } E X,(v) corresponds to a subsequence of T whose end-vertices are v j and w k : IliUIlely, s j , k := e j , e k , f k . Replace s j , k with V j w k # E ( R )if and O d y if Sj,k contains no edge, and with ( v j ,wk) g' A(@ otherwise. Repeat this

IX.Eulerian Trails - How Many

IX.40

?

replacement procedure as long as possible; call the new mixed graph H. This is a generalization of the replacement procedure developed in the discussion following (1). Similar to this discussion, if we assume that A ( @ U E ( d ) - ( A ( H )U E ( H ) ) contains no loops, then there is a signpreserving or sign-reversing bijection between the set of those elements induce X,(v), and F,,(ei, l), the corresponding set - of of T(ei,1) which sequences in H starting at t~ with ei. We have qfi. = 2pfi - 1, but I E ( H ) I> 1 may hold. However, d f i ( w ) = 1; thus H := H - v satisfies qfi = 2pfi implying that H is equalizing because of Lemma IX.7. Since there- is-a sign-preserving or sign-reversing bijection between T,,(ei, 1) = Ef) and I?) (G = G B , G = GG),it follows that H is equalizing. Since H varies as X,(v) varies we conclude for

x(G,

xi(G,

IT; := {T E '7;(e;, I)/{e{, e & } g X,(v>,

{i,1,m}= { 1 , 2 , 3 ) )

that

5

contains as many even elements as it contains odd

.

(12)

One reaches the same conclusion if A ( & ) u E ( f i ) - ( A ( R ) u E ( 8 )contains ) at least one loop; one simply has to apply the corresponding part of the discussion following (1). Case 2. {ei,eL,e$} - {ei} E X,(v). Let

u 3

7' := {T E

x ( e ; , l)/{e;, e;} E X,(v), { z , j ,k} = {1,2,3}}

.

i=l

We wish to show that

7' contains as many even elements as it contains odd.

(12') this purpose, observe that X,(v) induces a partition X & ( v ) of . . . , eLV1} into transitions (note that d - 1is odd). We shall again dethe replacement procedure performed at the beginning of the proof in Case 1 (but now somewhat differently). Whence we assume that the mixed graphs arising by this procedure have no loop other than that of H (if H has a loop at all); for, the case where new loops arise can be treated in a way analogous to the previous considerations. r

We proceed by induction on I X&(v) I starting with IX & ( v )I= 0 in which case (12') holds since it implies d H ( v ) = 3. Whence suppose ( X $ ( v ))>0.

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.41

Now consider

T

E (4,. . . ,d - 2). We claim that 7 ' ( e r ,e d - l )

contains as many even elements as it contains odd, (13)

for every T = 4, . . .,d - 2. To this end we assume w.1.o.g. that It follows that every T E 7 ' ( e d - 2 ,e d - l ) is of the form

T

= d - 2.

or of the form

where gj E Awd-l U Ewd-l - { f d - l ) , and i E {1,2,3) (see Figure 1x3). In any case, apply the replacement procedure first to { f d - 2 , ed-2,e d - l } ; this set is correspondingly replaced by an edge/arc f'. Then replace the set {f', f d - 1 , gj} correspondingly by an edge/arc f" (a set of three arcs or of one arc and two edges is replaced by an edge, two arcs and one edge are replaced by an arc). Thus each of the above T is transformed into

T It = e ; , . . . ,f" ,... which is an element of 7v(Gy,HY), where Gy = GHf,,for 3

and where f " is incident with wd-2 and xj (see Figure IX.8). It follows that qH!l = 2pHy - 1, and - more importantly - IE(H,!')I= 1. Thus H,!' is 3 of type ( 3 , l ;2) and pHf,#< p H .HY is equalizing by the choice of H (Note J that applying the above replacement procedure may first yield a mixed graph of type (1,O; 2) but having two edges). IX&,(u) I=[X & ( u )I -1 by construction, where XkIf( u ) refers to the partition of { e ; , . . .,e&--3}(Note that TI' starts with some e i , i E {1,2,3}). Defining with respect to H,!' in the same way as we defined 7' with respect to H , and applying

3''

IX.42

IX.Eulerian Trails - How Many ?

induction we may conclude that ?;" has as many even elements as it has odd. However, because of the above replacement procedure, the same conclusion holds true for 'T/ C 7'corresponding to $If. Moreover, m

This and the arbitrary choice and the equation

T =d -

u

2 imply the validity of (13). This

d-2

7' =

7'(e,,ed-l)

7

r=4

where the union is again disjoint, imply the validity of (12') which finishes the considerations of Case 2. We summarize: since 3

(J7&, i= 1

3

1) = 7' u (J?;

,

i= 1

where any two terms on either side of this equation are disjoint, since (12) and (12f) imply that the right-hand side of (14) contains as many even elements as it contains odd, it follows that the same holds true for the left-hand side of (14) which in turn implies (11). Whence 21 is equalizing, and therefore is equalizing. By Lemma IX.19, H* is equalizing, and therefore by (4),H is equalizing. This final contradiction to the choice of H finishes the outline of the proof of the theorem. With the help of Theorem IX.20 and Theorem IX.21 we are able to prove the next result. This proof differs from that of [HUTC75a, Theorems 4 and 51 since we make use of the fact that for certain types of mixed graphs we can conclude that they are f-equalizing if they are equalizing.

Corollary IX.22. If H is a digraph or mixed graph of either type (0,o; k), ( 2 , o ;k), (0,1; k), k E { 0 , 2 } , satisfying QH = 2 p -~ 2 > 0 and

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

Lx.43

I E ( H )I< 1, and having an arc/edge labeling b, ,. . . ,b, (q = qH), then H is equalizing. Proof. If k = 0, let

TJ

be an arbitrary vertex of H . Form the digraph

,

H'=HU{f}

where f 4 A ( H ) is an arc incident from w and to TJ, where w E V ( H ) - { v } is chosen arbitrarily. If k = 2, then denote the odd vertices of H by TJ and w; form H' as above. In either case assign to f the labe1 b,+,. w e have qHl = 2pHt - 1 anyway, and H' is of either type (1,O; k), (3,O;k), ( 1 , l ; k), k E {0,2}. Since Theorem IX.21 holds, we can apply Theorem IX.20; thus H' is equalizing. By Corollary IX.17 and Proposition IX.18, H' is even f-equalizing. In particular, in H' we have

On the other hand, there is a bijection p between 7w(f, 1) C 7w(G',H') and 7,,(G,H),defined for arbitrary T := b,+,, bi,, . . . ,bi, E 7 w ( f , 1) by p(T) = b i l , . . . ,bi, E ?-,(G,H ) , q = qH. Since q + 1 is odd, it follows that p is sign-preserving (note oT = o d T ) ) . This and (*) yield

I T X G ,H ) I=I I,"(G, HI I Thus, if G is eulerian, then H is equalizing since TJ had been chosen arbitrarily; and if G is not eulerian, then being equalizing implies that H is equalizing (Remark IX.6.d)). That is, H is equalizing in any case: this implies the validity of the corollary. We note that the essence for the validity of Corollary IX.22 lies in the fact that H' (constructed from H ) is f-equalizing. On the other hand, we were not able to prove that a mixed graph of type ( 3 , l ;k), k E {0,2}, satisfying qH = 2pH - 1 and I E ( H )I= 1, is f-equalizing. The theoretical reason for there being no relatively simple proof that such mixed graphs are equalizing, and which would be similar to the proof of Theorem IX.20, lies in the following result which we state without proof (see [HUTC75a, Theorems 6,7,8]).

Theorem IX.23. Let p 2 2 and 0 following statements are true.

<

q

5 2p - 2 be integers. The

1) If q < 2 p - 2, then there exists a weakly connected mixed graph H with at most two odd vertices, satisfying pH = p and qH = Q such that

IX.44

IX.Eulerian Trails - How Many ?

H is not equalizing, for any choice of I E ( H )

{0,1).

In particular, there is an infinite number of digraphs of type (1,O; 2 ) and of type (3,O;0) which are of size q = 2p - 3 and not equalizing.

2 ) If p is even and q = 2p - 2 , then there exists a mixed graph H of type ( 2 , l ;2 ) satisfying p H = p , qH = q, I E ( H ) I= 1, such that H is not equalizing. Theorem IX.23,- 2 ) explains why, in general, a mixed graph H satisfying and I E ( H ) I= 1, and being of type ( 3 , l ; O ) (and thus as well) cannot be f-equalizing. For if every such H were f-equalizing, we could delete an arbitrary arc to obtain a mixed graph H' satisfying the hypothesis of Theorem IX.23.2). By an argument analogous to the proof of Corollary IX.22, we could then conclude that H' is equalizing, thus contradicting Theorem IX.23.2). We also note that Theorem IX.23.1) explains why we cannot conclude in Corollary IX.22 that H is generally f-equalizing. Thus Theorem IX.20, Theorem IX.21, Corollary IX.22 and Theorem IX.23 describe precisely the various types ( i , j ; k ) of mixed graphs H for which we may conclude that they are equalizing, and where qH is as small as possible, and I E ( H ) I< 1, with the exception of type ( 2 , l ;0), where qH = 2pH - 2. In fact, the loopless mixed graph on two vertices, one arc and one edge is not equalizing. As we shall see later, there is an infinite number of non-equalizing mixed graphs of this kind. However, it might be of interest to determine those H of type ( 3 , l ;k), k E { 0 , 2 } , which satisfy qH = 2pH - 1 and I E ( H )I= 1, and which are f-equalizing. Looking at mixed graphs H satisfying q H > 2pH -1, we know already that H is equalizing if qH = 2 p H , and f-equalizing if qH 2 2pH 1 (Lemma IX.7 and Corollary IX.7.a). In these cases we have no restrictions on I E ( H ) I. On the other hand, comparing Theorem IX.2 with Theorems 1 x 2 0 and 1 x 2 1 and observing that qD 2 2pD and qH = 2pH - 1 a e the best possible conditions, one is led to conjecture that any mixed graph satisfying q H = 2pH and I E ( H )15 1, is f-equalizing. This conjecture is proved next. Again we assume automatically that an arc/edge labeling b,, . . . , b, is given.

+

Corollary I X . 2 4 . If H is a mixed graph having at most one edge and satisfying qH 2 2pH, then H is f-equalizing. Proof. W.1.o.g. H is weakly connected and has at most two odd vertices. Because of Corollary IX.7.a it suffices to deal with the case qH = 2pH

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.45

only.

1) Suppose that G = GH is eulerian. Consider an arbitrary f E A ( H ) U E ( H ) ;w.1.o.g. f = b, . H' := H - { f } satisfies qHl = 2pHI - 1 and is equalizing by Theorem IX.20 and Theorem IX.21. Let z and y denote the odd vertices of H' if f is not a loop; otherwise, set z = y, since f E A, U E, . There is a bijection cp between 7 v ( f , 1) 7v(G,H ) and %(GI, H'), G' = GHt . cp is sign-preserving or sign-reversing depending on the orientation of f if f E A ( H ) ; cp is definitely sign-reversing if {f}= E ( H ) (note that q is even). In any case,

s

tE,f = G,f

follows from

[ I,"(G', H') [=I c ( G ' ,H') [ . Since f E A ( H ) U E ( H ) had been chosen arbitrarily we conclude that H is f-equalizing. 2 ) If H has two odd vertices, we denote them by x and y. For any f E ( A , U E,) n (All U Ell),we form H' as above and conclude as above that f satisfies (*). Note that in this case G' is eulerian . For any other f , we form H' as in the proof of Proposition IX.18 (see Figure IX.7). G' is eulerian, qHl = qH = 2pH. However, because of the solution already achieved in case l), it follows that G' is f-equalizing, so f satisfies (*). Thus, in any case f is an equalizer, i.e., H is f-equalizing. The corollary now follows.

In fact, Corollary IX.24 can be proved in other ways. One can even improve most of this corollary. One only has to employ fully Theorems IX.20 and IX.21 as well as Proposition IX.18. For this purpose, we define a concept which is even stronger than that of being f-equalizing. Definition IX.25. An f-equalizing mixed graph H (or digraph, or graph for that matter) with a given edge/arc labeling b, , . . . ,b,, q = q H , is called transition-equalizing (t-equalizing for short) if and only if for every v E V ( H ) (if GH is eulerian) or for every v E {z,y} (if z and y are the odd vertices of H ) , and for every f E A, UE, and every g E A , UE, - { f} (w # 21 if f is not a loop, f E A , U E,,,), the number of even elements T E ?-,(G,H ) of the form T = f , 9 , . .., equals the number of odd such elements. We then write for short tElfg = With the help of the concept of vertex-splittings, we can prove the following result.

IX.Eulerian Trails - How Many ?

IX.46

2

Corollary IX.26. Let H be a mixed graph without loops satisfying q H 2pH and I E ( H ) 15 1, and having an arc/edge labeling b,, . . .,b , q qH. If qH = 2pH and H is of type (o,o,IC) or of type (2, j, k'j, j {O,l}, k E (0,2}, then H is t-equalizing. If qH > 2pH then H is

= E t-

equalizing regardless of its type. Proof. L e t v E V ( G ) , f E A , U E , , g E A , U E , - { f } f o r w # v a n d f E A , U E,, be chosen arbitrarily. W.1.o.g. f = b,, g = b,. Consider

where b', @ A ( H ) U E ( H ) is an arc incident from v and to 2, where z and g are incident and z = v if and only if g E ( A , U E,) n ( A , U E,). We have qHl 2 2pHl - 1, and H' has at most one edge. In fact, if {f,g} n E ( H ) # 0, then H' is a digraph by definition. By the very definition of a mixed graph being t-equalizing, we now assume that H has at most two odd vertices and that v E { u , y } if H has u and y as its odd vertices. Moreover, we assume H to be weakly connected and that w has been deleted in forming H' if d H(w) = 2. Thus H' is weakly connected. Redefine b, := b', E A(H'). It follows from the hypothesis that H' is not of type (3,1, k), k E { 0 , 2 } if qHl = 2pEp',- 1. Thus we conclude from Corollary IX.17, Proposition IX.18, Theorem IX.20, Theorem IX.21 and Corollary IX.24 that H' is f-equalizing. That is, we have in H' e

t v,bl

-

- t v,bl O providedb, is not a loop

.

(*>

Suppose first that b, E A(H') is not a loop. Then there is a signpreserving or sign-reversing bijection 'p between 7,(b1, 1) 5 7v(G',H') and those T E 7,(G, H ) which are of the form T = b,, b,, . . . . It depends on the orientation of f, g respectively, and on the parity of q whether cp is sign-preserving or sign-reversing. However, since we assumed that b', = b, is not a loop it follows from (*) that we have in H

(**> However, if b', = b, is a loop, then let

H" := H' - { b , }

.

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.47

Since H‘ is not of type ( 3 , l , k ) it follows that H” is not of type (2,1, k), k E {0,2}. Hence H“ is equalizing by Corollary IX.22 if q H # l = 2pHll - 2, or otherwise by Theorem 1x.20 and Theorem IX.21. That is, we have for G“ = GH,,

However, we also have

H” = H

- {f,g}

and there is a sign-preserving or sign-reversing bijection cp between T* E 7,(G”, H”) and T E T ( G ,H ) of the form T = f , g, T * ;i.e., cp(T*)= T = f , g, T*. This and (* * *) imply (**), even if 13; is a loop. Thus (**) holds in any case. Since TJ E V ( H ) had been chosen arbitrarily, respectively E (u,y) if u and y are the odd vertices of H , the corollary now follows. Let us have another look at Theorem IX.20, Theorem IX.21, Corollary IX.22, and Corollary IX.24. On the one hand, the two corollaries follow from the two theorems. On the other hand, in the proof of Theorem IX.21 the consideration of a mixed graph of type ( 3 , l ;2) was first reduced to the consideration of a mixed graph of type (1,l;0) which was then reduced to another mixed graph of type (3,l; 2). Thus the following questions arise. Are there other ways of proving the results quoted ? To which extent are there equivalences between various types of rnized graphs ? That is, given a proof that any mixed graph H of type (i,j;k) satisfying qH = 2pH - 6 , 6 E {O,I,2}, is equalizing or even f-equalizing, does this imply that any mixed graph H’ of type (2,j ’ ; k‘) with analogous constraints o n qH is equalizing, f-equalizing respectively ? These questions are answered by the following theorem for which we assume a labeling b,, . . . ,b, of the arcs/edges of H given, q = qH. Also, we recall that any mixed graph of either type ( 0 , l ; 0), ( 1 , O ; 0), (2,O; 0), ( 3 , l ;0) is equalizing (Corollary IX.17.1)).Moreover, we omit in the sequel the considerations of mixed graphs of certain types ( i , j ;2). They can be shown to be f-equalizing, t-equalizing respectively, if and only if this is true for mixed graphs of type ( i , j ; O ) ; we leave it as an exercise to do this. Finally, for brevity’s sake the symbol H ( i ,j ; k), stands for the statement “An arbitrary mixed graph H of type ( i , j ; k ) of size q = qH such that (E(H)I 5 1”. In the case qH = 2pH - 1, however, we omit this subscript. Moreover, we shall use formal equations H = H ( i ,j; k), in the proof of the next result; such equations will be self-explanatory in the context of their use. Also, in

IX.48

IX.Eulerian Trails - How Many ?

various instances the subscript q in the symbol is used to indicate the size of the graph under consideration.

Theorem IX.27. Let 'H be the class of mixed graphs H having at most one edge and no directed loops and satisfying qH = 2pH - 6,6 = 0 , 1 , 2 , p H 2 2. Let p := p H , q := q H . Consider H ( i , j; k ) ) pE 'H. Then we have:

(I) Any two of the following statements are equivalent. (1.1) H(1,O;0) is f-equalizing and H(0,O;O)2p-2 is equalizing.

(1.2) H(1,O; 2 ) is f-equalizing. (1.3) H(1,l;0) is equalizing. (1.4) H(1,l;2) is f-equalizing. (1.5) H(3,l;2) is equalizing.

(1.6) H(0,O; k)2p-z is equalizing, k = 0,2. (1.7) H(0,l;2)2p-z is equalizing. (1.8) H(0,l;O)2p is f-equalizing. (1.9) H(2,O;0)2pis t-equalizing. (1.10) H(2,I;O ) 2 p is t-equalizing. (11) Any two of the following statements are equivalent. (11.1) H(3,O;0) is equalizing. (11.2) H(2,O; 2)2,-2 is equalizing. (11.3) H(0,O;0 ) 2 pis t-equalizing. (11.4) H(2,O;0 ) 2 pis f-equalizing. Moreover, the statement

(111) H(O, 0; O)2p-2 is equalizing is implied by any of the statements (Li), 1 5 a 5 10, and in turn implies any of the statements (ILj), 1 5 j 5 4. Finally, since (1.5) is a true statement, all of the statements ( I i ) , (II.j), 1 5 i 5 10, 1 5 j 5 4, and (111) are true.

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.49

Proof. In accordance with the classification presented in the statement of the theorem we proceed in three steps. We always assume that the mixed graphs H from which we start are weakly connected.

I) We show the validity of the following chains of implication (1.3) + (1.4) + (1.10) 4 (1.7) 4 (1.8) + (1.3), (1.6) 4 (1.1) + (1.3), (1.4) + (1.2) + (1.9) (1.8) t) (1.5) . ---f

Once these implications have been proved, we may observe that we have a strongly connected digraph D defined by

V ( D )= {(I.i)/I 5 i 5 lo}, A(D) = {((I.i),(I.j))/(I.i)-+ (I.j), 1 5 i , j 5 10, i # j ) , where the implication (1.i) + (1.j) is taken from any of the above three chains. However, D being strongly connected is equivalent to saying that any two of the statements (1.1) - (1.10) are equivalent.

(1.3) + (1.4). This implication follows from parts of the proof of Proposition IX.18. (1.4) + (1.10). Consider in H = H(2,l;O)2p an arbitrary transition t = {a',b'}. W.1.o.g. a = b, and suppose first that a E A ( H ) , where a is the arc corresponding to a'. H' := H - a is of type (1,I;2) which is f-equalizing by assumption (note that A ( H ) contains no loops); let x and y denote the odd vertices of HI. In particular, since b E A , U E, for z = x or z = y, we obtain in HI

where z is chosen such that a E A, and t is viewed as a transition at z. However, for T, E 'T-,(b,1) and T, := a, T, E l u ( a ,1) (where u is the vertex incident with a other than z ) we obtain a bijection cp between 7 , ( b , 1) and the set of those elements Ta,bof 7u(G, H ) which are of the form Ta,b = a, b, . . . . Whether cp is sign-preserving or sign-reversing depends on the orientation of a. In any case, though, (1) implies the validity of t:,ab = t t , a b (2) The same conclusion can be drawn if E ( H ) = { a } and a is not a loop; for in this case we replace b in H' = H - a with an edge eb to obtain H"

IX.50

IX.Eulerian Trails - How Many ?

which is also of type ( 1 , l ;2). However, the validity of ( 1 ) with respect to H" and eb yields the validity of (1) in H' as well; consequently, the validity of ( 2 ) also follows. Finally, if a is a loop, then H' = H - a is of type (1,O;0) since H has no directed loops. In this case, we redefine H', H' := H - b, to obtain H' = H ( 1 , l ;2) whose odd vertices we denote again by x and y such that { a } = E,. Redefine the edge/arc labeling of H by exchanging the labels of a and b. By assumption we then have in H' tS,a = t i , = . (3) Depending on the orientation of b this yields a sign-preserving or signreversing bijection between '&(a, 1 ) C_ 'TZ(G',H') and the set of elements T E 7 , ( G ,H ) which are of the form T = a, To,b, and where To E 7,(Gf a , H'-a). However, T R = a, b, 7': (note that a is an vndirectedloop), and the relation T f-f T R defines a sign-preserving or sign-reversing bijection between {T E T ( G ,H ) / T = a, To,b } and {T E 7,(G7H ) / T = a, b, Tf}. This and (3) yield the validity of ( 2 ) for u = x. Whence ( 2 ) holds regardless of the pattern oft. Since t was an arbitrary transition and since H is f-equalizing by Corollary IX.24, it follows that H is t-equalizing. This finishes the proof of the implication.

(1.10) + (1.7). Let 2 and y be the odd vertices of H = H(0,l;2)2p-2. Since q = 2p - 2 = 0 (mod4) and p 2 2, it follows that p 2 3. Let z E V ( H )- {x,y} and form H' = H U {a,., a x y } ,where a,. and a z y are arcs not in H , uzz joins x to z and azy joins z to y. By construction, H' = H(2,1;0)2p.,Extend the labeling of H to H' by letting b,+, = azz, b,+, = uzy. Since H' is t-equalizing by assumption we have in H f t z , b q + l bq+z

However, the relation T'

f-f

- tz,bq+lbq+a

.

(4)

T defined for

T' = bq+l, bp+27T E %(GI, H ' ) and T E q ( G ,H ) is sign-preserving (owing to the orientation of b,+,, b,+, and independent of the parity of a). This and (4) imply that y is equalizing in H ; thus x is equalizing as well (see Remark IX.G.d)), and thus H is equalizing. The validity of the implication now follows.

(1.7) + (1.8). For an arbitrary f E A ( H ) U E ( H ) , H = H(0,l;0 ) 2 p , introduce vf 4 V ( H )and define

H f = ( H - f ) u {vf,af}

;

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

Ix.51

af is incident with wf and 21, where w is one of the end-vertices of f , and af is an arc if and only if f is an arc. Assigning to uf the same label carried by f in H , we obtain a sign-preserving bijection between 7w(f, 1) C 7w(G,H ) and 7 (uf,1) E Kf(G’, H’), where w E V ( H ) is ?, incident with f and w # w if and only if f is not a loop (w.1.o.g. we assume af incident from vf if and only if f is incident from w). H’ is of type (0,1;2) and q‘ = 2p’ - 2 by construction; it is thus equalizing by assumption. That is, we have in H‘

which, by the above sign-preserving bijection, is tantamount to saying that we have in H tL,f = tL,f

This in turn implies that f is an equalizer in H . Since f had been chosen arbitrarily, it follows that H is f-equalizing: this finishes the proof of the implication. (1.8) --t (1.3). Consider a 2-valent vertex w in H = H ( 1 , l ; O ) (w exists since q = 2p - 1). Assume w.1.o.g. that A, U E, = {bq-l, b,}. In any case, form H’ := ( H - v) u {f} , where f is an edge joining 2,y E N ( v ) if E, # 8, while f is an arc otherwise. Set f = b,-l. Note that f is a directed loop if and only if E, = 8 and IN ( v ) I= 1 (Observe that N ( v ) # 8 since p > 1). However, H’ is of type (0,l; O)2p in any case, and it is f-equalizing - either by assumption or by Lemma IX.14. Consider a fixed z E (z,y}. We now distinguish between two cases. Case 1) f is not a directed loop. Then we have in

H’

However, there is a bijection cp between % ( f , l )C T ( G ’ , H ’ ) and %(bqm6, 1) C Z ( G , H ) , for some 5 E (0,I}, where 5 = 0 if and only if b, $! A, U E, C A ( H ) U E ( H ) . Observing that cp is sign-preserving or sign-reversing depending on the orientation of the elements of A, and of f , we conclude from ( 5 ) that b,-6 is an equalizer of H . By the proof of Proposition IX.12 and Corollary IX.17.2) it follows that H is f-equalizing (Note that g is odd in this case).

IX.52

IX.Eulerian Trails - How Many ?

Case 2) If f = ( z , z ) E A(H’), { z } = N ( v ) C_

V ( H ) ,then we form

which is of type ( 3 , l ;0) since H‘ is of type (0,l; O ) 2 p . H“ is equalizing by Corollary IX.17.1); hence we have

from which we may conclude the validity of (5) in which case we assume f to be passed in a fixed direction (according to its orientation, for example). Arguing now as in Case 1, we may again conclude that H is f -equalizing. Thus H is f-equalizing in any case which finishes the proof of the implication; it follows that any two of the statements (1.3), (1.4), (1.7), (1.8), (1.10) are equivalent.

(1.4) 41.2). This implication follows from part B) of the proof of Proposition IX.18. (1.2) -, (1.9). This implication follows from the proof of the implication (1.4) ---t (1.10).

(1.9) + (1.6). Denote by x and y the odd vertices of H = H(0,O;2)2,-2 and let z1 := ~t:= y be arbitrary if H = H(O,O;O),,-,. Observe that we may assume p > 2 because of this type (the case H(0,O;Ic)2 is impossible since 2 $ 0 (mod 4)). Let z E V ( H ) - {x,y} be chosen arbitrarily. Introduce new arcs bq+l,bq+2 joining y to z and z to x, respectively. The digraph H’ thus obtained is of type (2,O; 0) and is t-equalizing by assumption. Let t = {bb+,,bb+,}. Then we have2Pin H‘

However, the relation Tq+l,q+2

._

--bq+,, bq+2, T,

t t

T,

,

T, E 7 , ( G , H ) , defines a sign-preserving bijection between the set of H ) . This and elements of %(GI, H I ) starting with b q + l , b,+, and 7,(G, (6) as well as Remark IX.6.d) classify H as equalizing which finishes the proof of the implication.

E l . As Many As

- Parity Results

for (Mixed) (Di) Graphs

IX.53

(1.6) + (1.1). Deleting an arbitrary arc b in H = H ( l , 0;0 ) we obtain H' = H(0,O;2),,-? which is equalizing by assumption (w.1.o.g. b = bp). Arguing in a way similar to the proof of Lemma IX.19, we may conclude that b is an equalizer. Since b had been chosen arbitrarily, it follows that H(1,O; 0) is f-equalizing which had to be shown.

(1.1) + (1.3). Suppose f i s t that f E E ( H ) , H = H ( 1 , l ;0 ) , is a loop xx. H' = H - f is of type (0,O; O ) 2 p - 2 . W.1.o.g. f = b,. Since H' is equalizing by assumption, it follows that C , f

=C

, f

since

I7,"(G',H') I=! 7,"(G',H') I

.

That is, f is an equalizer of H ; hence H is equalizing by the proof of Corollary IX.12 arid thus f-equalizing by Corollary IX.17.2). If, however, f is not a loop, replace f with an arc a f to obtain a digraph H' of type (1,O; 0) which is f-equalizing by assumption. Thus we have in H'

which is tantamount to saying that

holds in H . Thus, f is an equalizer of H in this case as well. Concluding again that this is sufficient for H to be f-equalizing we have finished the proof of the implication.

(1.8) and (1.5) are equivalent. Consider first H = H(O,l, 0 ) It is ??' equalizing by Lemma IX.7. By assumption H' := H - f is equalizing for every fixed f E A ( H ) since H' is of type ( 3 , l ;2) and because we may assume w.1.o.g. that f = b,. As in the case (1.6) + (1.1) we conclude that f is an equalizer of H . Hence every arc of H is an equalizer. Since H is equalizing anyway we have for every 21 E V ( H )

For g = zy E E ( H ) (possibly x = y) we thus have

Ix.Eulerian Trails - How Many ?

Ix.54

i.e., g is also an equalizer. Thus, (1.5) implies (1.8). Now let H = H(3,l;2), and let uZy $! A ( H ) be an arc joining the odd vertices z,y of H . Form H‘ = H U {uzy} and assign to uZy the label b,+,. H‘ is of type (0,l; O)2p; it is f-equalizing by assumption. For f = uzy we have C,f = C,f * (7) Since there is a sign-preserving or sign-reversing bijection (depending on the orientation of f) between 7z(f, 1) & 7=(G’,HI) and 7y(G, H ) defined bY

-

f,Ty Ty, TyE q G H ) we conclude from (7) and Remark IX.6.d) that y is equalizing and, therefore, H is equalizing. Having proved all implications listed at the beginning of I), we have finished the first step of the proof.

11) We prove the validity of the chain of implications (11.1) t (11.3) t (11.2)

---f

(11.4)

+

(11.1)

.

(11.1) t (11.3). Consider H = H(O,O;O),,. Let t = ( U ~ , , U ~be ~ }an arbitrary transition; possibly z = y. W.1.o.g. uzu E A;, uuy E AS and a z u = b q 4 , uuy = b, (see Remark IX.6.a), b)), where u z u , a u yare the respective arcs corresponding to the elements oft. For an arc uxy $! A ( H ) joining x to y, form

where we assume u $! V ( H ’ ) if d H ( u )= 2. H’ is of type (3,O;0) or of type (3,O;0)2p,+lby construction (p’ = p H - 1 if d H ( v ) = 2). Assign to azy the label bq-l. Suppose first x # y. H’ is f-equalizing by assumption and Corollary IX.17.2), or by Corollary IX.24 respectively. Thus, we have for f = azy E 4 H ’ ) t L Y - t:,azy . (8) However, for

TL := uzyTyz E %(axy,1) C TJG’, HI)

IX.1. As Many As - Parity Results for (Mixed) (Di) Graphs

IX.55

and

Tx := a x v , a v y , TuxE T ( G ,H ) we define a bijection 'p between 7x(axy,1)and those elements of 'Tx(G,H ) which start with axv,avy, by

cp is sign-preserving by construction.

This and (8) imply

If x = y, then form HI'

:= HI - {azy} = H

- {azy,a v x )

which is of type ( 2 , 0 7 0 ) 2 p - 2or of type ( 2 , 0 , 0 ) z p , ,(if d H ( v ) = 2). By Corollary IX. 17.1), H" is equalizing in any case. Thus we have

For TL E I,(G",H") and Tx := u,,,,u,,,T~ E T.G,H), we define 'p as above and conclude also in this case that 'p is sign-preserving. This and (10) imply the validity of (9) even if x = y. Having proved (9) in all possible cases, we conclude that H is t-equalizing since t had been chosen arbitrarily.

(11.3) + (11.2). Observe first that H = H(2,0;2),,-, must satisfy p > 2; otherwise, H would have a directed loop which is impossible by the hypothesis of the theorem. Let z and y be the odd vertices of H ; choose z E V ( H )- {z,y}. Introduce new arcs u z z , u r yjoining x to z , z to y respectively, form

H' := H U { u z z , u z y }

,

and extend the arc labeling of H to H' by defining b,+, := u z r , b,+, := u z y .HI is of type (0,O; O)2p by construction and t-equalizing by assumption. In particular, we have in H'

IX.56

IX.Eulerian Trails - How Many

?

However, the relation

is a sign-preserving bijection by the construction of HI and the way we extended the arc labeling to HI. We conclude from this and (11) that

I q ( GH ) I=I I,"(G, H ) I which together with Remark IX.6.d) implies the validity of the implication.

(11.2) + (11.4). Let uZyE A ( H ) , H = H(2,O;0 ) 2 p be , chosen arbitrarily. For a vertex 2, 6 V ( H ) define HI by

V ( H ' ) = V ( H )lJ{%I, Aza = {%yh A $ ( H 1 ) = A $ ( H ) , A,(H') = A , ( H ) if

E V ( H ) - {x}

(that is, we split off uzy at z). Retain in HI the arc labeling given in H (note that A ( H ) = A ( H ' ) ) . H' is of type ( 2 , O ; 2)2p-2 and equalizing by assumption. This and the construction of H' yield

That is, uzy is an equalizer in H . Since uzy E A ( H ) was chosen arbitrarily, the implication is true.

(11.4) -+ (11.1). H = H(3,O;0) has a 2-valent vertex v since q H = 2pH - 1. Proceeding now dong the same lines pursued in the proof of the implication (1.8) -+ (1.3) (but now neglecting any considerations of edges), we conclude that (11.4) --t (11.1) holds. One only has to change appropriately the types of (mixed) (di)graphs obtained, e.g., H ( 3 , 0 , 0 ) is reduced to HI of type ( 2 , O ; O),,, or to HI1 of type ( 1 , O ; 0). In the latter case, however, HI' is also equalizing by Corollary IX.17.1). This finishes the second step of the proof of the theorem. 111) In fact, (1.1) implies (111) trivially; so, any statement (I.i), 1 5 i 5 10, implies (111) since these ten statements are pairwise equivalent by I). To see that (111) implies (II.j), 1 5 j 5 4, it suffices - because of 11) - to prove the implication (111) + (11.2). To this end, introduce z # V ( H ) , H = H(2,O;2 ) 2 p - 2 , and form

H' := H

u { z , ( z , 4,(Y,4)

IX.1.As Many As - Parity Results for (Mixed) (Di) Graphs

IX.57

where 5 and y are the odd vertices of H . Extending the arc labeling of H to II' by defining b,+, = (2, z), b,+, = (y, t),and observing that H' is of type (0,O; O),-, we conclude from the assumption that we have in

H' tz,bq+i

+ tz,bq+a - t:,bq+i + tzrbq+a

'

(12)

However, the relation T t+ T Rf or T ,T R E %(GI,H'), is sign-preserving by Corollary IX.16.2). Observing that d H , ( t )= 2 implies

T E q((t, z), 1) if and only if T R E T((y, z ) , 1) we now conclude

Consequently, (12) and (13) classify the elements of A, as equalizers. Now, the bijection

defined by cp((z,

4,T,, (Y,4 )= T, E %(G,H )

is sign-preserving since q is even. Thus,

tz,bq+l

- t z , b q + l implies the valid-

ity of I I,"(G,H ) I=I 7:(G, H ) I which, by Remark IX.6.d), is tantamount to saying that H is equalizing. This finishes the third step of the proof of the theorem whose validity is thus established. True enough, the proof of the implications (1.i) + (II.j), 1 5 i 5 10, 1 5 j 5 4, does not explicitly require the proof of the implications (1.i) 3 (111) t (I1.j) as we did for i = 1 and j = 2. This is evident from the implication (1.9) + (11.4) which holds by definition. On the other hand, (111) is implicitly contained in the statements of I): namely, it appears as part of (1.1) and (1.6). Interestingly enough (111) + (1I.j) holds for j = 1,2,3,4. Apart from the cases omitted in the statement of Theorem IX.27 which are the subject of Exercise IX.7, further statements could be introduced to generalize Theorem IX.27. Some of them can be extracted from the proof of the implication (111) + (11.2) (see also the proof of Lemma IX.19).We formulate them as corollaries, the proofs of which are left as an exercise.

IX.58

IX.Eulerian Trails - How Many ?

Corollary IX.28. Let H be a mixed graph of either type (0,O; O ) 2 p - 2 , (2,O; 0)2p-2, (3,l; 0). Suppose H has no directed loops. H is equalizU E,) are equalizers, for any given ing, and all elements of UuEv2(H)(Au arc/edge labeling b,, . . . ,b,. Similarly, from the validity of statement (11.2) in Theorem IX.27 we may conclude the validity of the following (observe that the validity of (1.1) is secured by earlier results).

Corollary IX.29. If H is a mixed graph of type (3,l; 0) and {f} = E ( H ) , then H is equalizing and f is an equalizer. In fact, Corollary IX.29 can be extended to mixed graphs H of type ( 3 , l ; 2). It is left as an exercise to show this. However, Corollary IX.28 cannot be extended t o include all mixed graphs of type ( 0 , l ; 0)2p-2precisely because not all mixed graphs H' of type ( 2 , l ; 2)2p-1 are equalizing (see Theorem IX.23.2)). To construct such mixed graphs H , take any H' of type ( 2 , l ; 2)2p-2, introduce a new vertex z and join it with an arc to each of the odd vertices of H'. The mixed graph H thus obtained is of type (0,l; 0 ) 2 p - 2 .It is equalizing by Corollary IX.17.1); however in order to satisfy the full conclusion of Corollary IX.28, H' would have to be equalizing which is not generally the case. As we have noted before, Theorem IX.23.2) also explains why mixed graphs of type ( 3 , l ;k), k E {0,2}, are not always f-equalizing. This in turn implies that mixed graphs of type ( 0 , l ; O),? we not generally t-equalizing. However, the following is true (we leave its proof as an exercise).

Corollary IX.30. Let H be a mixed graph of type ( 2 , l ; O ) 2 p - 2 . Suppose an arcfedge labeling in H is given. If H is equalizing, the conclusion of Corollary IX.28 also holds. Unfortunately, not all mixed graphs of type (2,l; O)2p-2 are equalizing (we argue this fact in a sketchy way and leave the details as an exercise). Consider a digraph H of type (1,O; 2)2p--3 whose odd vertices be denoted by x and y. Consider H , and H2 constructed as follows: H , is obtained from H by adding an edge f joining z and y, while H , is obtained by adding a directed path P of length three joining z and y, where P n H = {z,y}. H , is equalizing if and only if H2 is equalizing (extend a given arc labeling of H onto H I ,H , respectively, and see the discussion between (11) and (12) in the outline of proof of Theorem IX.21). However, H , is of type ( 2 , l ; 0)2p-2, while H2 is of type

IX.l.l. An Application to Matrix Algebra

Ix.59

(0,O;0)2pa--4.Assuming H , is equalizing and using the fact that the correspondence T * T R is sign-preserving in H,, one deduces the validity of the conclusion of Corollary IX.28 which in turn implies that H is also equalizing. Thus, for every non-equalizing H of type (1,O; 2)2p-3 (such H exists by Theorem IX.23.1)) the corresponding Hl of type ( 2 , l ;O),-, is not equalizing either. By a similar argument one can show that every such H gives rise to a non-equalizing digraph of type (3,O; O)2p-3. Thus, in Theorem IX.23.1) it sufEces to have non-equalizing digraphs of one type to deduce the existence of non-equalizing digraphs of the other type (cf. Lemma IX.19). The above Corollaries IX.28, IX.29 and IX.30 can be used to extend Theorem IX.27; we leave it as an exercise to do this. Nevertheless, these corollaries do not suffice to show that all statements of Theorem IX.27 are equivalent. For, while it is true that the new statement "H(3,l;0) satisfies the conclusion of Corollary IX.28 implies (1.3), we cannot automatically conclude the validity of this new statement from (11.1) which in turn implies Corollary IX.29. It might be an interesting research topic to develop additional tools which possibly lead to a proof that any statement (II.j), 1 5 j 5 4,implies any statement (Li), 1 5 i 5 10. ))

IX.l.l. An Application to Matrix Algebra In the sequel we use some of the results on equalizing mixed graphs but also Swan's Theorem (Theorem IX.2) to establish results in matrix algebra. In fact, this is the area in which the preceding considerations of this chapter have their roots. In order to abbreviate the respective proofs, we start with some general considerations. Let n 2 1 be a fixed integer, let M, denote the vector space of all n x n matrices over the field of complex numbers, and let any m > 1 elements A,, . . . , A , E M, be given.") As before let S, denote the symmetric group on rn objects, the integers 1 , . . . m,for example. Define )

sgn(a)Ail . . . ,Ajrn

[Al,A,, . . . , A , ] :=

)

(A)

WES, lo) In most of the references in this subsection IX.l.l, the authors consider fields of characteristic 0 or commutative rings or finite fields.

IX. Eulerian Trails - How Many ?

1X.60

We say that the standard identity of degree m is satisfied in M, if and only if

[A,, . . .,A,] = 0 for any choice A , , . . . , A , E M,

(S)

(0denotes the zero matrix).l') Let us now specialize the choice of A,, . . . ,A , E M, by considering the standard basis B, of M,,

B,

:= { E i , j / u j ,= j 1, a k , l = 0 if ( i , j ) #

(k,Z), 1 5 i , j , k,Z 5 n }

where ar,s denotes the ( T , s)-th entry in Ej,j.Consider a collection Em := { E i k , j k1, 5 k 5 m } of m elements of B, and construct a digraph D := D(Em) as follows:

where the arc u i j , j kcorresponds to E i k r j kE Em and joins vik to v j k .We note that since E j k i j k= E i l , j lmay hold for 1 5 k < Z 5 m, D may have multiple arcs; it also may have loops since i , = j, can hold. In particular, the multiplicity of an arc in A ( D ) equals the number of times which the corresponding matrix appears in Em. In fact, some authors do not state explicitly how, in the definition of [A,, . . . , A , ] , the sequence i , , . . . . ,i,. and 7r are interrelated. In fact, some seem t o assume that 7r = ( a , , . . .),,2 (which differs from our definition of K ) , or have not concerned themselves with this question at all. However, since (1,2) = (2,1), this question is not irrelevant (note that in the above interpretation, neither the identity nor products of permutation cycles and would occur.) On the other hand, ( i , , i , , . . . , i m ) = 2 1 , 2 2 7 * * * -_

i,, i,, . . . , a ,

1 , 2 , . . . ,m

.

i , , i,, . . . , a , i,, i,, . . . , z1

.. , m . i , , i,, . . . ,z, 1,2,.

)-l

)

= (1,2, . . . )m ) .

Thus, the two different ways of interpreting 7r are related in the case where K is a cycle of length m; and this relation is independent of the actual choice of 7 r .

IX.l.l.An Application t o Matrix Algebra

IX.61

Next we consider special subspaces of M,, namely M: and M;, the respective subspaces of symmetric and skew-symmetric matrices in M,. A basis of M : is given by

B :

:= {ECj/a+,j= aj,i = 1, ak,l = 0 if {i, j} # { k , l } , 1 5 i

(note that ECj = E&, implying E+ E bY

B : ) ,

5 j 5 n}

while a basis of M i is given

B; := {E,;jlai,j = -aj,+ = 1, ak,l= 0 if {i, j } # { k , l } , 1 5 i < j 5 n}, in which case we have Es<j = -Ej; and thus Ei,; = 0 for i = 1,.. . ,n. Rephrase (S) by replacing M, with M:, or M i , or any other subspace M: C M,, and denote the corresponding statement with (S+), (S-), (S*) respectively. Introduce analogous symbols with respect to the definition of (A). Let EL be a collection of rn elements of mixed graph H := H(EL) by

B:

:= B : U

B; and define a

V ( H )= {u,,. . 9%) 7 E A ( H ) joins vj to vj if and only if EaYjE E& n B; *

f i , j E E ( H ) joins vi and vj if and only if ECj E EL n B :

l
, (HI

(where the multiplicity of an arc/edge is determined by the number of appearances of the corresponding matrix in EL). Returning to the definition of [A,, .. . ,A,] we observe that [A,,.. .,A,] is multilinear (compare this with the classical way of evaluating a determinant). That is, in studying [A,,. . . , A,] for some choice A,, . . .,A , E M, (M:, M;, M: respectively), we can reduce it to the consideration of a sum, each term of which is of the form T . [ E l ,. . . , Em], where T is a complex number and El,. . . ,EmE B, (Bi, B;, BL respectively). For example, if we consider arbitrary matrices A,, . . .,A , E M,, then we can write n

(k)

a+,jE;,j,

A, = i,j=1

k = 1,.. . , m , where { E i , j / i j, = 1,. . . ,n} = B, .

IX.62

IX.Eulerian Trails - How Many ?

Thus we obtain the following equation which we denote by

(R).

One proceeds analogously if A,, . . . ,A, E M i ( M i , ML respectively) by using Bt (B;, BL respectively) instead of B, (with the only distinction that i, 5 j , (ik < j,, respectively), Ic = 1,.. . ,m, has to hold in the above sum if A, E Mi(M; respectively)); denote the corresponding equations by (R+)((R-), (R*) respectively). With the above technical considerations and definitions we are now able to present short proofs of the results of this subsection. These proofs rest on the use of Swan's Theorem as well as on the main results concerning equalizing graphs (except Theorem IX.27). We stmt with a theorem which is the essence of [AMIT5Oa] and is the point of departure for the theory developed in all of IX.l.

Theorem IX.31 (Amitsur-Levitzki). Let n E N be given. The standard identity of degree rn, (S), is satisfied if and only if rn 2 2n. Proof. In view of equation (R) it suffices to prove [Eil,jl> * ..> E a.m r i .m

] = O

(E)

for an arbitrary (but fixed) choice Eil,jl,.. . ,Eirn,jrn E B,. Construct the digraph D as described in (D). By definition, we have for any Eq,,, E,,, E B, the equation

Eq,7. . J%,t = b , S Eq,t (0) where S,,, is the Kronecker delta. That is, looking at an arbitrary term of the sum in (A),s g n ( x ) Eilk, j l k say, it follows that *

JJF=,

n m

k=l

Ei,k,j,k# 0 if and only if j l k =

k = 1,.. . ,rn- 1 , (1)

IX.l.l. An Application to Matrix Algebra

IX.63

where { ( i l k , j l k ) / k= 1,.. . ,m} = ( ( i k , j , ) / k = 1,.. .,m}. Moreover, we have to define A, := Eik,jk,k = 1,.. . ,m7 so that the factor sgn(lr) is defined by setting 1,2, . . .,m lr := ' l , ' Z , * . * , 1m > '

(

in accordance with (A). However, (1) is tantamount to saying that

nill:= nr=,EiIk # 0 corresponds to a uniquely determined covering ,j,k

trail ' ; I , = aill , j z l vertex vill ,

7

ajil ,ji,

7

.

3

~ji,,+~ ,Jim .

of D = D(E,) rooted at the

(2)

where Em := ( Ea.k * J, k/k = 1,.. .,m}. This follows from (1) and (D). Moreover, this correspondence is bijective because of (1) and (2). However, whether Till is an eulerian trail or just an open covering trail depends on whether jlm= ill or jlm# i l l . Furthermore, the definition A, := E j k , j kk, = 1,.. . ,m, induces an arc labeling of D by associating the label bk with the arc u j k , j k ,k = 1,. . . ,m. It follows from this very definition that V(Til1)= w-44 . (3) It follows from the hypothesis rn 2 2n (note that p D = n, q D = m) that

I c ( D )1=1q(D) I

E V(D)

for every

by Theorem IX.2. That is, for v := vill and

,

T := T% . we have

However,

IT.111 = E 111 . 931,.

*

This, (3) and (4) imply that the non-zero terms of (E), the associated covering trails of which are all rooted at v = vitl, sum up to

IX.64

IX.Eulerian Trails - How Many ?

Now, if such D is eulerian, then %(D)# 0 for every v E V ( D ) ,and each such 7,(D)corresponds to a set of terms in (E) whose sum = 0 by (5). On the other hand, if such D is not eulerian, then 7,(D) # 0 for precisely one 21 E V(D);in this case we define sgn(T,) = sgn(T) = 0 for every w E V(D)- {v}. Thus, if we put E D , , := E '11 . 331,. ' 21 = 21.211 ' the terms of (E) sum up to

because of ( 5 ) . Thus, by using (1) and (6) we obtain the equation

) E .'11 731,.

where {II,. . . ,Zm} = (1,. . . ,rn}. Finally, if rn = 2n - 1, then for every n E N an eulerian digraph D, of order n and size rn exists, which has precisely one eulerian trail rooted at its unique 2-valent vertex v (see Figure IX.l and the paragraph following the proof of Theorem IX.2). Denote V(D,)= {u,, . . . ,v,} and consider precisely those Ei,j for which (vi,vj) E A ( D , ) . Since D, has just one eulerian trail T, rooted at v it follows that any eulerian trail T of D , results from a cyclic permutation of T, (viewing T and T, as closed arc sequences). Since rn = 2n - 1 is odd, it follows from Corollary IX.ll that sgn(T) = sgn(T,). These considerations imply that (E) does not vanish denoted arbitrarily as Ale).Whence for these m matrices (we assume E,,j we may conclude that the standard identity of degree rn = 2n - 1 is not satisfied in M,. This finishes the proof of the theorem.

In order to improve the lower bound of Theorem IX.31, the range of matrices has to be specialized for which the standard identity of degree rn should be satisfied. In fact, if we consider (S+)instead of (S)then rn =

IX.l.l. An Application to Matrix Algebra

IX.65

2n is still best possible. For, if A,, . . . ,A, E , : B the corresponding G as defined by (H) is a graph, and there is an infinite number of connected graphs G satisfying q = 2p - 1, which are not equalizing.In order to see this, let P be a path on p vertices u,,. . . ,up, such that u, and up are the end-vertices of P; attach a loop at every vertex of P. The graph G thus obtained satisfies q = (p - 1) p = 2p - 1, and it has precisely one covering trail starting at either u, or up. Whence G is not equalizing with respect to any edge labeling. On the other hand, E(G) corresponds to Eo := {Eiti, E,ti+,/i = 1,.. . , p - 1) U {E:,) C B;. Set A,*-, := E:, i = 1,.. ., p , A,; := i = 1,. . ., p - 1. However, with this notation which also induces an edge labeling of G we obtain

+

Eli+,,

+s9n

1 , 2 ,..., 2 p - 1 2 p - 1 , 2 p - 2 , ..., 1

(

) A,,4A2,4 *

*

A,

*

Moreover, the choice of A,, . . . ,A,,-, and equation (0)in the proof of Theorem IX.31, as well as the fact that E[j = Ei,j Ej,i whenever i # j and = E i , j ,imply that (C) reduces to

+

Eli

But even if we require G to be eulerian, there are infinitely many sets E& c UnEN B;, such that m = 2n - 1 and (S+) does not hold. To see this, we just look at the above G and form

Gl := G u ~

~ p + l ~ ~ l ~ p + l ~ ~ p ~ p + l ~

For p, = p + l , G, satisfies rn = q1 = 2p, - 1 = 2n - 1 , and it is eulerian. The corresponding set of matrices is

EL:= Eo

"

{Jqp+17

E,+;,+,l

a

Now, IqJG,)

I= 4,

1L i

+

L P,

lqp+l(G,)I= 2

,

but for every i = 1,. . . , p 1, we only have two classes in PPRE(uU;). Because of Lemma IX.10 (q, is odd) the elements of each particular class have the same sign. Assuming now p l = 1(rnod2) we have rn = q1 = 2p, - 1 1(mod4). Now it follows from the proof of Proposition IX.8

IX.66

IX. Eulerian Trails - How Many ?

that every element of 7vi(G1) has the same sign. By Corollary IX.ll, all elements of Ur=’,’x i ( G l ) have the same sign (We do not distinguish between two possibilities of traversing an undirected loop; hence such loop is considered to be orientable in only one way, i.e., we do not distinguish between the two half-edges of this loop). It is positive if we extend the ._ labeling in the previous example by defining A,, := ELp+,, .E z p + l . Using these considerations and (4z) we obtain

(Note that E,,,E,,, . . .Ep+l,l = El,,.. .EP+,,,E,,,, a.s.o., and that sgn(7r) = 1 for all those T E S, for which the associated product of matrices does not vanish).

As we have seen above, G and G, always contain loops. So the question arises: are there loopless graphs of order p and size 2p - 1 which are not equalizing ? This question has been asked by J.P. Hutchinson, [HUTC74b, Question 11; she also quotes such examples for p = 6,7 but does not explicitly exhibit them. It has ben observed in [HUTC74b] that if (A) (or (A+) or (A-) or (A*)) does not vanish, it might be that the trace of this matrix is zero. This yields the following generalization. Call a mixed graph H , V ( H )= (q,. . . ,v,}, weakly equalizing12)if and only if

It now follows that the trace in (A*) vanishes for some choice of A , , . . . ,A , E B: if and only if the associated (mixed) (di)graph is either weakly equalizing or it contains odd vertices (Exercise IX.12). We quote some results in this direction, the proofs of which are left as an exercise (see also [HUTC74b, Propositions 5,6,7]).We assume arc/edge labelings to be given. Proposition IX.32. Let H be a mixed graph having at most two odd vertices and an even number of arcs. The following statements are true: J.P. Hutchinson defines this concept for graphs only: she speaks of “totally cancelling graphs”.

IX.l.l.

An Application to Matrix Algebra

IX.67

1) If GH is not eulerian and if H is not equalizing, then H is weakly equalizing if and only if either q H 2 (mod 4) or q H = 3 (rnod4).l3) 2 ) If G H is eulerian and Q H = 0 (mod 2 ) , then H is weakly equalizing.

=

3) If GH is ederian and q H 1 (mod 2 ) , then H is weakly equalizing if and only if H is equalizing. One can also prove by analogous arguments the following.

Proposition IX.33. Let H be a mixed graph having an odd number of arcs. The following statements are true. 1 ) If GH is not eulerian and if H is not equalizing, then H is weakly equalizing if and only if qH o (mod 4) or q H G 1 (mod 4).14)

=

2 ) If H is of either type ( 2 , 1 ; 2 ) , H , ( 3 , 0 ; k ) , H . , k E { 0 , 2 } , then H is

weakly equalizing if and only if H is equalizing. 3) If H is of type ( 2 , l ;O ) q H , then H is weakly equalizing.

We return to matrix algebra and the problem of specializing the range of matrices in order to obtain the standard identity of degree m for some values m < 2 n . Contrary to the case of (S+)we have the following result in the case of (S-).

Theorem IX.34. Let n 2 2 an integer. The standard identity (S-) of degree m is satisfied if and only if m 2 2 n - 2. Proof. In view of (R-), the equation analogous to (R), it suffices to prove (E- 1 [Ei,,j,1 * * Eii,j,] = 0 for an arbitrary (but fixed) choice Ea&,,.. . ,E a i , j mE i, < j,, k = 1,.. .,m and that

B,. Note that

.

(->

E 2-$3. = E .1J. - E j , ; ,

l j i <j j n

Let D be the digraph corresponding to

1 3 ) In the proof of [HUTC74b, Proposition 51 one has to replace ‘the same sign’ with ‘opposite signs’. 14)

In the cases where

H

from Corollary IX.17.1) that

is of either type (0,l; 0),, (1,O; 0),, it follows

H

is equalizing regardless of the size of Q.

IX.68

IX. Eulerian Trails - How Many ?

and defined in (H) with respect to E ; (see the discussion preceding Theorem IX.31). w e may assume that rn = q D < 2 p D = 2n; otherwise, we could reduce the whole problem to the more general case treated in Theorem IX.31 in which case (S-) holds. Consequently, D is of either type (a,0; k), (p,0; k ) 2 p - 2 where p = P D = n, IC E (0, 2}, a E {l,3}7 P E {0,2}. In each of these possible cases D is equalizing by Theorem IX.21, Corollary IX.22 and Theorem 1X.20. Also, by setting A , = Et:,jk, k = 1 , . . . ,rn, we induce an arc labeling of D. Next we turn to the consideration of an arbitrary non-vanishing term of

(see the proof of Theorem IX.31). Look at two consecutive factors in (n); for simplicity's sake set ilk = i, jlk- j , - T , j l k + , = s. Observing that i < j and T < s and using (-) we then have

(compare this with equation (0)in the proof of Theorem 1x31).In order to have (n)satisfied, it is necessary and sufficient that (0)is satisfied for any two consecutive terms in the product in (II):this is the case if and only if at least one of the Kronecker deltas in (0)is positive. That is, either i < j = T < s, or i 3. 2> s , o r r < i = s < j .

<j

= s and i : ~ or , i =r

< j and

This implies in any case that the two arcs b,, and blk+l (corresponding to Ez,3,ELs respectively) must be adjacent. However, we can conclude more than that: namely, if both b,, and blk+l are passed according to their orientation or both are passed in the opposite direction (i < j = T < s or T < i = s < j), then (0)reduces to Ei,s # 0, Ej,v # 0 respectively. In the other two cases we have -E+ # 0, -Ej,s # 0 , -Ei,+- E j , j # 0 respectively, which means that precisely one of these two arcs is passed according to its orientation. Moreover, the indices in the non-vanishing term(s) of (0)indicate how blk and blk+l are passed.

IX.l.l.An Application to Matrix Algebra

IX.69

Summarizing this discussion of (0)and its relation to (n),as well as observing that Ej = Ei,jEj,B, Ei,, = Ei,jEj,v,a.s.o., we obtain

where { T , s} = { i , j } (in fact T = i, s = j holds if and only if blk is passed according to its direction). Moreover, oT is the number of arcs passed by T in the opposite direction, where T corresponds to the covering trail TG as induced by I'Ir=1Eq, ,sik - Enl , s l , in G := G, (see the discussion following (1) in the proof of Theorem IX.31).15) Finally, because of the way in which we defined the arc labeling in D , we also have sgn(T,) = 1, ...,m sgn ( I , ,...,1-1 * Thus,

and (1) reduces to

Again, every term of (E-) satisfying (11) corresponds to a unique T E Uf=l 7,,;(G, 0 ) whose initial (end-) vertex is vn 1 (v,,,) if T corresponds to rill as expressed by (3). We conclude as before that this correspondence is bijective. Classifying the terms of (E-) satisfying (11) in accordance with (3) (i.e., two terms belong to the same class if and only if their respective reduction (3) contains the factor Er,l r s I r n ), putting ED,, = ErI, ,sI, for v = v,,,, and using the fact that D is equalizing, we obtain basically the same equations (4),( 5 ) and (6) as established in the proof of Theorem IX.31, by an analogous argument. The only exceptions are that sign(T) has to be stated in terms of (2) (see above), and if G = G, is not eulerian then there are precisely two vertices IC and y (the odd vertices of D),to which non-vanishing terms in (E-) correspond. So In the definition of B, we had assumed i < j for E,Yj E B,. Thus, if T starts in wj, dong uiIl, j I l , then will contain the factor E,T,,jIl rather than 15)

. This explains why we did not use the notation Theorem IX.31, but rather It is the matrix E,,,

EjT,

,

rill.

information on the initial vertex of

T.

n. as in the proof of "1

which contains the

IX.70

IX.Eulerian Trails - How Many ?

we define sgn(T) = 0 for every w E V ( D )- ( 5 , ~since ) 7 , ( G , D ) = 0 in this case. Thus we obtain an equation analogous to (7) in the proof of Theorem IX.31,

Thus it has been shown that ( S - ) holds for rn

2 2 n - 2.

To see that (S-) does not hold in general for rn 5 2 n - 3, we invoke Theorem IX.23.1) by which there is an infinite number of non-equalizing digraphs D of type ( l , 0 ; 2 ) 2 n - 3or (3,0;O)2n-3, n E N . On the other hand, in the discussion immediately preceding this subsection IX. 1.1 we remarked that every non-equalizing H of type (1,O; 2)2n-3 gives rise to a non-equalizing digraph of type (3,O;O)2n-3, and vice versa. By Proposition IX.33.2), however, a non-equalizing digraph D of type (3,O;O)2n-3 is not even weakly equalizing. Labeling the vertices of D with v,, . . . ,v, and assuming w.1.o.g. that E A ( D ) joining vi to v j implies i < j (see Remark IX.G.b)), we associate with every such the matrix EzLi E B, . Setting EL := {Et;j/ai,j f A ( D ) } it follows from (H) that D = H(Ek). Labeling the elements of EL with A , , . . . ,A,, m = 2 n - 3 , we conclude that [A,,. . . , A,] # 0 precisely because D is not equalizing. If one attaches a directed path P of length p , ending at some v E V ( D ) , and which satisfies V ( P )n V(D)= {v}, then the resulting digraph D, is equalizing if and only if D is equalizing. Whence D, is not equalizing implying that (S-) is not satisfied for

This finishes the proof of the theorem. We note that for even n, Theorem IX.34 appears already in [KOST58a, Theorem 3.51. There B. Kostant uses cohomology theory to deduce Theorem IX.31 from the formula of Dynkin. As a spin-off, so to say, he then obtains his Theorem 3.5. Moreover, he shows that Theorem IX.31 is equivalent to a theorem of Frobenius on the characters of the alternating group A, C S , . As a consequence of Kostant's partial solution of Theorem IX.34 it was conjectured in [SMIT72a] that Kostant's result can be extended to all n > 1 and that rn 2 2n - 2 is best possible. In fact, independent work by J.P. Hutchinson and L.H. Rowen (who published his

IX.1.1. AQ Application t o Matrix Algebra

IX.71

result about a year earlier) solved this conjecture, [HUTC74a, HUTC75a, ROWE73a, ROWE74a1, while F.W. Owen, origindy announcing the same results in [OWEN73a], apparently published only the result that (S-) does not hold for rn < 2n - 2, [OWEN75a].16) In each of these papers, graph theory plays a prominent role. However, J.P. Hutchinson’s work is closer to pure graph theory, so I relied mainly on her work to develop the material of this section IX.l.17) More recently, L.H. Rowen reproved [KOST58a, Theorem 3.51 by different methods which yield additional results related to the standard identity, [ROWE82a]. It should also be noted that [ZALE85a] contains an analogy to the Theorem of Amitsur-Levitzki for the case where the matrices have their entries from a field of characteristc p > 0. Another important result of [HUTC75a, ROWE74al can be obtained by further exploiting the previously established theory of equalizing mixed graphs. Since its proof adopts arguments analogous to those developed to prove Theorem 1X.34, we leave it as an exercise.

Theorem IX.35. Let n > 1 be an integer. Consider the standard identity (S*)of degree rn for any choice of A , , . . . ,Am-, E M i , A , E M i . The following statements are true. 1) If n is odd, then ( S * ) holds whenever m 2 2n - 2.

2) If n is even, then ( S * ) holds whenever m 2 2n - 1. Moreover, the lower bounds on m stated in 1) and 2) are sharp. In fmishing section IX.l we note that the other results developed for equalizing graphs, in particular Theorem IX.27, can also be utilized to state further results on polynomial identities (or, rather, to define polynomial identities with additional properties, and to show the extent to which they exist). It is left as an exercise to do this.

16) According to a letter by J.P. Hutchinson, F.W. Owen’s proof of all of Theorem IX.34 contained a flaw (she refers to a remark by L.H. Rowen).

17)

A considerable amount of it is new.

IX.72

IX.Eulerian Trails - How Many ?

1x2. The Number is ...- A First Excursion Into Enumeration Our starting point for the following considerations is Theorem VI.33. Recall: given a spanning in-tree D, with root w in a (weakly) connected eulerian digraph D , an eulerian trail T of D exists starting at w such that for every w # w, T leaves w for the last time dong the unique element of A$ nA(D,). Conversely, if we mark at every w # w in a given eulerian trail T of D starting at w the last arc by which T leaves w, then the marked edges induce a spanning in-tree rooted at v. Thus we can partition I,(D)into classes, such that each class contains precisely those elements of I,(D)which - by the above arc-marking process - define one particular tree. However, we still have to define a notion by which we consider two eulerian trails of D to be different. For this purpose we return to Definition VII.l by which we consider two eulerian trails TI,T2 of D to be different precisely if the systems of transitions defined by TI and T, satisfy the inequality

Thus, the choice of an initial vertex, i.e., the consideration of 7 , ( D ) as well as the choice of an initial arc in A:) poses no restriction. Note the contrast between this point of view and the one from which we started our considerations in the preceding section. Consequently, we are faced with two questions: 1) How many spanning in-trees, rooted at a fixed vertex, does a given eulerian digraph D have ?

2) How many Do-favoring eulerian trails exist in D if D - Do is a spanning in-tree ? These questions are the basis for the next two subsections.

IX.2.1. The Matrix Tree Theorems

IX.73

IX.2.1. The Matrix Tree Theorems For the next theorem we rely on the proof as presented in [KAST67a].18) Denote by T,(D) the number of spanning out-trees of D rooted at v. Moreover, given the adjacency matrix A(D) = of D define the Kirchhoff matrix A*(D) = (aTj), 1 5 i , j 5 p as follows (we assume the vertices of D labeled as vl, . . . ,up, p = p D ) :

Furthermore, let A ; , k ( D )be the (k,k)-cofactor of A * ( D ) . Finally, call D, C D a basic subgraph with root vi if and only if id(vi) = 0 and id(vj) = 1 for j # i. Basic subgraphs are characterized by the next lemma whose proof is left as an exercise.

Lemma IX.36. A basic subgraph D rooted at v is weakly connected if and only if it is an out-tree. Moreover, if D is disconnected, then every component D‘ for which v @ V ( D ’ ) contains a cycle. Now we are going to prove what has become known as the Matrix Tree Theorem for digraphs.

Theorem IX.37. For any digraph D , T,;(D) = det A;,+(D). Proof. First we study the determinant of the Kirchhoff matrix corresponding to a basic subgraph D o , say. W.1.o.g. let Do be rooted at ul. Thus, the first column of A*(&) is the zero vector; this is also true for those columns for which the corresponding vertices are incident with a loop, while every other column contains precisely one entry which is 1 and precisely one entry which is -1, whereas the other entries of this column are 0. This follows from the fact, that i d ( v j ) = 1, 1 < j 5 p , implying 18) This proof is already contained, essentially, in [TUTT48a] and [AARD5la]. We note, however, that Kasteleyn and others attribute Theorem IX.37 (see below) to Kirchhoff, [KIRC47a]. A careful study of this article shows that, in fact, Theorem IX.37 is implicitly contained in it (see also [BIGG?6a, p. 1331 -- unfortunately, the authors of this wonderful book fail t o point out this fact). On the other hand, Harary and Palmer (see [HARA73a,p.25]) attribute it to R. Bott and J.P. Mayberry and claim that the Matrix Tree Theorem for graphs (see Theorem IX.38 below) is implicit in Kirchhoff’s work. For a shorter proof of Theorem IX.37 we refer t o [ZEIL85a] which also contains references to other proofs.

IX. Eulerian Trails - How Many ?

IX.74

= 0, otherwise, a;,j = 1 and - u ~ ,=~ -1 if (vi,vj) E A(Do),and = id(vj) - X “j = 0 = a: 93.. provided i # j; otherwise, Suppose now that Do is disconnected. By Lemma IX.36, Do contains a cycle C which we express as an arc sequence in the form

1 < ij loop.

5 p,

j = 1,.. . , k. Thus we assumed implicitly that C is not a

Denoting the column vectors of A * ( D o )corresponding to vij, by bij , we have the following non-vanishing components of bij .

15 j

5 k,

setting k

+ 1= 1

.

Therefore, k

j=1

(from now on, 0 denotes the zero vector). That is, the column vectors of ( D o )are linearly dependent, hence det A;,l(Do)= 0

.

(1)

We observe that if Do has a loop at v j , we reach the same conclusion since bj = 0 ( j # 1). Suppose now that Do is weakly connected. By Lemma IX.36, Do is an out-tree rooted at u l . We label the vertices of Do according to their distance from q ,i.e., d(v,,vi) < d(wl,vj) implies i < j . ” ) Thus, (vr,u s ) E A ( D o )implies r < s which in turn implies b i l j = 0 for i > j . That is, AT,l(Do)is an upper triangular matrix; hence, considering Do with its new vertex labeling, we obtain det A;,l(Do)= 1 . ‘’1

Note that the path P ( v , , vk) from

k = i,j.

zil

(2)

to v k is uniquely determined,

IX.2.1. The Matrix Tree Theorems

Ix.75

However, the relabeling process is tantamount to reordering simultaneously rows and colums in A*(Do);i.e., if the r-th column becomes the s-th column, then the r-th row becomes the s-th row. So, relabeling the vertices of Do has no impact on equation (2) which, therefore, can be read as an equation on Do together with its original vertex labeling. Consider now A*(D). By its very definition, aj := P

fies

C

. . .,

satis-

= 0. That is, again by the definition of A*(D), aj can be

i= 1

expressed as

i= 1

such that the components of ):a

satisfy

k=l

,d T .

I

(3)

Assuming w.1.o.g. that we want to determine T , ~ ( D we ) thus obtain by repeated application of (3) (note that the first equation in (3) holds if we delete the first components)

where Diz is obtained from D by deleting all but one arc of A; & A ( D ) , Diz,i3is obtained from Diz by deleting all but one arc of A13 C A ( D i z ) ,a.s.0. That is, Dizrit ,...,i,is a basic (loopless) subgraph of D with root u1 , for any choice of ij E (1,. . . , dJT}, 2 5 j 5 p . Whence, the right-hand side of (4) is nothing but the number of basic subgraphs Do of D having det ATtl(D0)# 0; i.e., by (l),(2) and by Lemma IX.36, the multiple sum in (4) is nothing but T,,(D). This and (4) as well as the fact that simultaneously changing rows and columns does not alter a determinant, finally establish the theorem (note that basic subgraphs with loops are disregarded in (3) and (4) which is no loss of generality because (1) holds in this case as well). We note in passing that a generalization of the Matrix Tree Theorem (called the All Minors Matrix Tree Theorem) has been published in

IX.Eulerian Trails - How

IX.76

Many ?

[CHA182a]20),by which one can enumerate special types of forests in digraphs. This result is then extended to signed (di)graphs (i.e., where each arc/edge is given a positive or negative sign) and applied to matroid theory. On the other hand, Theorem IX.37 itself has already interesting applications apart from the one presented in the context of this book. In fact, it can be used to solve certain systems of linear equations. This has been done by the biophysicist T.L. Hill for homogeneous systems and by S.R. Caplan for inhomogeneous systems, [HILL77a, CAPL82al. Conversely, it is shown in [CAPL82a] that Theorem IX.37 follows by solving a certain system of linear equations.

In fact, there is also a Matrix Tree Theorem for graphs G. 21) Similar to Theorem IX.37 one starts with the adjacency matrix A(G) = ( a i j ) and defines A*(G) = by setting arj = -ai,j for i # j , and a *,t. = d ( v ; ) - A , , ; i,j =I,...,p.

Theorem IX.38. For any graph G, the number T ( G )of spanning trees of G satisfies T(G) = det Af,;(G), i = 1,.. . , p . Proof. In order to derive a recursive formula for the number of spanning trees in G we first observe that it does not matter which vertex we choose as a root since we are dealing with edge sets instead of arc sets. That is, the set of spanning trees rooted at zr E V(G) is the same if we choose w E V(G) - {w} as a root instead. However, in order to establish a recursive formula for T(G) = .,,(G), 21; E V(G), and to apply induction, we assume for the €allowing w.1.o.g. that the labeling of V(G) has been chosen such that e := 01v2 E E(G), and recall that A, denotes the set of edges parallel t o e . Following the procedure developed in the proof of Theorem IV.7, we also consider G - A, and G,. Label the vertex of 1'

This theorem had been published before (see the references of

S. Chaiken's article), but the proof presented in [CHAI82a] is apparently more elementary. 21) The roots of this theorem go back to 1860, [BORCGOa]. In fact, Borchardt's proof resembles that of the Matrix Tree Theorem for digraphs, although the term tree does not appear in that paper; nonetheless Borchardt It should distinguishes between cyclic and non-cyclic products of terms be noted, however, that [KONI36a] does not mention this article. We also refer to J.W. Moon's work [MOON70a] which also contains several references and a historical account of this theorem.

Ix.2.1. The Matrix Tree Theorems

Ix.77

G, obtained from the identification of v1 and

21, with 21,. We proceed in this manner simply because we can classify the spanning trees T of G as satisfying either E ( T ) n A, = 0 or E ( T ) n A, # 0. The spanning trees of the first type are in l-l-correspondence to the spanning trees of G - A,, while every spanning tree of the second type corresponds to a spanning tree T' of G,; and there are A(e)(= a,,, = a,,,) such trees in G, corresponding to the same T' in G,. Thus we obtain

We reinterpret (*) in terms of matrices and determinants. For purely practical purposes assume p 2 2 and observe that for any G on 2 vertices 21,

*,

7,(G) = al,, = a,,, = a;,, = a,.,, = det A;,,(G) = det A;,,(G)

. (**)

Whence we deal with the case p > 2. Now set A;,,(G,) := (M) (Note that the 'first' vertex of G, is v2), and observe that with this notation we have

Now, a determinant is a multilinear function. That is,

where bh, b;, b,, . . . ,b, are the row vectors of the respective matrices. Applying this fact to (* * *) and (* * * *) we obtain det A;,, (G) = det A;,, (G - A,)

+

det A:,l (G,)

.

(4

IX.Eulerian Trails - How Many ?

IX.78

P

Next observe that the definition of A*(G,) implies

C a; = 0 , where i= 1

{ai/i= 1,. . . , p } is the set of row vectors of A*(G,); i.e., det A*(G,) = 0 since the row vectors are linearly dependent. This implies, however, that det A;,,(G1) = 0 if G, is disconnected: for in this case we can rearrange the rows and columns of A*(G,) (i.e., relabel the vertices of G,) such that w.1.o.g.

")

A*(G,)= (A1 , 0 A, and where (A,) = A*(G,) for a component G, C G,. Thus the row vectors of (A,) are linearly dependent implying det A;,,(G1) = 0. This consideration and (**) permit us to apply induction for the particular choices of G, = G - A, and G, = G, respectively, in order to obtain

T ( G- A,) = det A;,,(G - A,), (oo), (0) and

T(G,) = det A;,,(G,)

.

(00)

(*) now yield T(G) = det A;,,(G)

.

This finishes the proof of the theorem. In principle, the method used in proving Theorem IX.38 can be modified to yield a proof of Theorem IX.37 just as Borchardt's proof of Theorem IX.38 follows along the lines of the proof of Theorem IX.37. Namely, look at v1 E V ( D )and a fixed v2 E V(D)such that there is an arc a E Atl nAZ2 (if no such a exists, then 7 J D ) = 0 anyway). For D,D,,D - A, an equation analogous to (*) holds as well (see the proof of Theorem IX.38). The subsequent considerations on matrices and determinants can also be used since we did not employ the fact that A*(G) is symmetric if G is a graph. The only real distinction is that in the proof of Theorem IX.37 21, is the explicit root, whereas in the proof of Theorem IX.38 q plays implicitly this role. This also explains why in adapting the proof of Theorem IX.38 t o a proof of Theorem IX.37 one has to index the various terms in (*) and ( 0 0 ) with q ,v2 respectively.

However, this possibility of adapting a proof of Theorem IX.37 to a proof of Theorem IX.38, and vice versa, as outlined above does not mean that A;,;(D) = A;,j(D) for i # j (just take an orientation of K 2 ) .We shall see below, however, that such an equation holds if D is an eulerian digraph. Moreover, the proof of Theorem IX.37 is relevant to the discussion of eulerian orientations of a graph.

IX.2.2. Enumeration of Eulerian Trails in Digraphs and Graphs IX.79

IX.2.2. Enumeration of Eulerian Trails in Digraphs and Graphs In this subsection we only consider (weakly) connected eulerian (&)graphs in order to determine I 7u(D)1, I 7,(G) I respectively. We first determine II,(D) I and then apply the corresponding result to obtain a formula for I7,(G) I as well. Observe that any two elements of 7 , ( D ) (I,(G)) start at along the same element of A;f (B,,). By Theorem VI.33, every T E 7 u ( D )determines a unique in-tree D, c D induced by the ‘last’ arc of T at every w E V(D)- {v}, and in turn D, determines a whole set 7 D 1 c 7 , ( D )of ( D - D,)-favoring ederian trails, each inducing the same in-tree D, . Consequently, in turning to the second question which precedes subsection IX.2.1, we proceed as follows. Suppose we arrive at w E V ( D )- {v} for the first time in the course of constructing some T E 7D1. It follows that there are od(w)- 1possibilities to leave w without using a E A$ n A ( D , ) . Having arrived at w for the second time, there are still od(w)- 2 such possibilities to leave w, etc. In general, at the k-th arrival at w we still have od(w)- k choices to leave w without using a, provided k 5 d(w) - 1. Consequently, there are

(od(w)- l)!

(1)

possibilities to construct some T E 7 D 1 for w E V ( D )- {v}. The same is true, however, if w = v , since in v the initial arc is fixed for every T E 7 , ( D ) . Thus, after returning to v for the k-th time, k < id(v) = od(v), we still have od(v) - Ic possibilities to construct some elements of 7D1. Whence, for V ( D )= {v,, . . . ,u p } , v = q,we have

for every in-tree D, rooted at v = v,. This answers the above question 2) and shows at the same time that l I D l I is independent of the actual choice of the in-tree D,. Whence, if we denote by r’ ( D ) the number 91 of spanning in-trees of D rooted at vl, we obtam

IX.80

IX.Eulerian Trails - How Many

?

In order to apply Theorem IX.37 we observe that

defines a bijection cp :

(D)+ 7;,( D R )which in turn yields

because of (3). Applying now Theorem IX.37 and using (3) and (4) we obtain P

I I v 1 ( DI=) detAT,,(D) n ( o d ( v i ) - l)!

.

(5)

i= 1

Noting that the choice of an initial vertex w1 and initial arc a E A:] has been a matter of convenience only, i.e.,

(and thus any vertex vi E V(D) could have been chosen as a root) we deduce from (5) the following classical result. Theorem IX.39 (BEST-Theorem). Let D be a weakly connected eulerian digraph, V(D) = { q , . . . ,wp}, and let A * ( D )be the Kirchhoff matrix of D. For any i E { 1, . . . , p } , P

II(D)I= det A:,i(D) n ( o d ( v j ) - l)! . j=l

€+om this theorem we have an immediate consequence.

Corollary IX.40. If D is a weakly connected eulerian digraph, then its Kirchhoff matrix satisfies AZ,;(D) = A;,k(D), 1 5 i, k 5 p D . The name of Theorem IX.39 relates to the authors of [AARD5la] and [TUTT4la]: N.G. de Bruijn, T. van Aardenne-Ehrenfest, C.A.B. Smith and W.T. Tutte. In fact, [TUTT4la] represents only a partial solution of the BEST-Theorem. We note that the BEST-Theorem and the concept of a bieulerian digraph can be used to deduce a parity result on I I ( D ) I. Namely, i f D is a n eulerian digraph and V,(D) = 0, then I I ( D ) I s 1(mod 2 ) if and only i f D is 2-regular and bieulerian, [BERM78b, Theorem 3.21. The proof of this result rests on the fact, that

IX.2.2. Enumeration of Eulerian Trails in Digraphs and Graphs IX.81 in a 2-regular digraph D the parity of I I(D)I equals the parity of the number of 1-factorizations of D, [BERM78b, Theorem 3.11. This, Corollaries VI.30 and VI.31, plus the BEST-Theorem yield the above parity result. We now turn to the enumeration of eulerian trails in graphs. For practical purposes, we assume that the graphs and digraphs considered in the sequel are loopless, unless stated otherwise. In view of the BEST-Theorem it is tempting to proceed as follows. Consider v1 E V(G)= (u,, . . .,up}, e E E V 1for , a connected eulerian graph G. Let D,, - ..,D, be the eulerian orientations of G such that a, E A$ n A ( D , ) , i = 1,.. . ,k, and apply the BEST-Theorem t o each D i . Thus

(note that odDi( u j ) = $ d G ( v j ) regardless of any particular eulerian orientation of G). This idea is not new; even for the case of K2s+lit has been noted, e.g., in [PALM78a] that “ a solution of this problem m a y depend o n the determination of all the even orientations of K2s+111. This is not surprising: for, while the product in (+) is a constant independent of Di, i = 1 , . . . , k, the terms of the sum in (+) are not. This has been illustrated already by Figure IV.3 (see also the discussion preceding Theorem IV.ll), where different eulerian orientations of the double triangle yield different numbers of rooted out-trees. Thus, by Theorem IX.37 and Corollary IX.40, det A:,,(Di) varies, in general, as i E { 1,. . .,k} varies. It is not even clear how this determinant varies; i.e., how Zarge is min{det A;,,(Di)/i = 1,.. . , k},

maz{det A;,?(Di)/i = 1,.. .,k}

,

. k

1

det A;t,,(Di) ?

i=l

This is one problem only. The other problem is the number k. That is, how does one determine the number (if not the set) of eulerian orientations of G ? These questions will be dealt with later. For the time being we simply accept these problems as given facts. In trying to find a formula for I 7 G I we have to take these facts into account. The aim is to

IX.82

IX.Eulerian Trails - How Many ?

find an operator w such that w ( G ) = ( I ( G )[ or, at least, w(DG)= ( I ( G )I for some fixed orientation D, of G. We permit w to depend on numerical parameters of the actual input G. This has been done in [FLEI83a], where the starting point is equation (+). However, in the next step one considers all orientations of G containing a given arc a , E A:l (see the discussion preceding subsection IX.2.1). There are precisely 2q-l such orientations, q = qG, most of which are not eulerian, of course. For any weakly connected digraph D define now 'D=(

1 if D is eulerian 0 otherwise .

,

Numbering the various orientations of G by D,, . . . ,D,, m = 2 4 - ' , such that a , E A& n A ( D , ) , i = 1,.. . , m, we can transform (+) into

We have thus reduced the problem of finding w to the problem of determining a method which produces numerically A * ( D i )for an arbitrary orientation Diof G, and of finding a formula for 6 , ; . The second problem is quite easily solved. Let

i=l

where {eJz = 1,.. . , p } is the standard basis in the euclidian pspace EP, and assume w.1.o.g. that the vectors ej are written as column vectors. Thus j, is a column vector all of whose components are 1. Reinterpreting j, as a p x 1-matrix we write J, instead of jp. Looking at A*(D)for any weakly connected &graph D (loopless or not), it follows from the very definition of this matrix that the r-th row sum equals zd(u,) - od(v,). Furthermore, we let I x I denote the length of x E EP, and use sgn for denoting the sign function defined on R. Thus we obtain the following formula whose proof is an easy exercise.

Lemma IX.41. 6, = 1 - sgn I A*(D)- J,

I.

Whence we are left with the question of how to generate the above orientations D,, . . . ,D,, m = 2 4 - l . For this purpose we consider the incidence matrix B(D) of the digraph D.

IX.2.2.Enumeration of Eulerian Trails in Digraphs and Graphs IX.83

If D, is a digraph such that G D 1 = G D , then

D, = (D- A , ) u A? for some subset A, A ( D ) . This equation has its correspondence with respect to B(D,) and B ( D ) . Namely, if a k E A, joins vj to vj, {v;,vj} V(D),then the k-th column vector of B(D) satisfies

while in B(D, ) the corresponding entries satisfy

Thus, if A , = { a i l , .. .,a;#},then B(D,) is obtained from B(D) by multiplying the i,-th, . . . ,is-th column with -1 and all the other columns with +l. Consequently, if we define the p x q matrix C ( A , ) = ( q j )as having identical row vectors and such that the entries satisfy c 14,. =-1 for r = l , . . . ,s ,

c,,~ =~1 , otherwise,

and using the Hadamard (elementwise) product of matrices B and C (written as B o C ) , then the above transformation of D into D , is expressed by B(Dd = 0 7

W,)

where A , A ( D ) is the same as used in the actual digraph transformation. We also observe that the i-th row sum of B(D’) is od(vi) - id(v,), i = 1 , . . . , p , where D’ is an arbitrary digraph and p = p D , . This observation and the preceding considerations axe summarized in the following lemmas.

Lemma IX.41.a. S, = 1- sgn IB(D) J,

1.

Lemma IX.42. Let D = Do be a digraph, A ( D ) = {a,,. . . , a q } , and denote ? ( { a 2 ,..., a,}) = { A , ,..., A m - , } , A, = 8. Furthermore, let D; , i = 1 , . . . ,rn - 1, rn = 2 9 - ’ , be the digraphs obtained from D by reversing the orientation of the elements of A ; . We have

B(Di) = B(D,) o C ( A ; ) , i = 0,.. . ,rn - 1 and j, is the fist column vector in each C ( A i ) .

,

IX.Eulerian Trails - How Many ?

IX.84

However, the application of the BEST-Theorem requires the use of the Kirchhoff matrix rather than that of the incidence matrix. The next lemma shows that this obstacle can be overcome. Whether D is loopless or not, A**(D)denotes the matrix obtained from A*(D) by letting the entries of the main diagonal be od(vi) - A, instead of id(vi) - A,;. Note that A * * ( D )= A * ( D )if and only if D is eulerian.

Lemma IX.43. Let D be a digraph, and consider A**:= A * * ( D ) ,B = B(D). We have 1 A * * = - ( B o B + B ) . B ~. 2

Proof. Observe first that

BoB+B=(B+I)oB

,

where I is the p x g matrix all of whose entries are 1. Thus the (i, k)-th entry i i , k of $(B o B B) - BT satisfies

+

For i = k each term of this sum satisfies

Assuming now i

#k

we obtain

(note that the j-th column of B(D) has precisely one entry +1, precisely one entry -1, while the other entries are all 0). That is, a term in the above sum does not vanish if and only if a j E A$ n A;k. This and the value of each non-vanishing term yield

IX.2.2. Enumeration of Eulerian Trails in Digraphs and Graphs IX.85

Therefore, the equation claimed by the lemma is valid. Combining Lemma IX.42 and Lemma IX.43 we express for

Di= (Do -Ai)UAR, B := B(Do), C j := C ( A j ) , i = 0 , . . . , m - 1 , the matrix A**(Di)in the form

A**(D;)= jj(B(D,) o B(D;) + B(D;)) B(Di)T 1

*

1

+ B o C,)- (B o C i ) T = ((B o B) - (B o C i ) T+ B . BT) 2 = -(B o B 2

(observe that the Hadamard product is commutative and C j o C j = I, and that B(D,) . B(D;)T = B * BT because of b ; , j c i , j b k , j C k , j = b i , j b k , j ~ ; , j = ‘i, j bk,j

1.

To finish the construction of the operator w we are left with the tasks to define Cj = C(AJ without relying on any arc sets and to produce B := B(Do)- an initia2 orientation of G - by relying only on B(G), for example. For the first task consider c ( i ) ,the first row of C, (recall that any two rows of C, are equal). c ( i )depends on the choice of A; A ( D ) - { u , } , or rather on the indexing of the elements A, of the power set P ( { u 2 ,u 3 , . . . ,u , } ) . We observe that c ~ ,the ~ j-th , component of c ( i ) ,satisfies c1 . = 33

-1

{ +1

if and only if u j E A;, otherwise .

Now suppose w.1.o.g. that the index i has been chosen in such a way that its binary expression b(i), written as a q-dimensional row vector, satisfies ( b j ( i ) denotes the j-th component of b(i))

It now follows from the definition of c ( i ) and b(i) that

~ ( i=)b(2Q- 1 - i) - b(i)

.

IX.Eulerian Trails - How Many ?

IX.86

Thus we have found a way of defining C i , i = 0,. . . ,2"' considering the actual sets Ai C_ A(D,) - (al}.

- 1, without

As for the second task, i.e., to produce an initial orientation B := we consider B(G) = (6i,j) and define B(G)= (6j,j) by B(D,) = -

b,,j=O

j = l ,. . . , q i-1

k= I Now define

B := B(G)

+ B(G) o B(G)

.

to an orientation of G, observe first To see that B in fact corresponds that bi,j = 0 if and only if bi . = 0. For, bi . = 0 implies bi,j = 0 by '3definition of B, while bi,, = 0 # b,,, implies 11

an obvious contradiction. Next we look at the non-vanishing terms b i l , j and bi2,." in the j-th column of B; assume i , < i,. The definition of bi,j yields bil,j = 0 and thus bil,j = bil,j = 1, while = -2 and thus b.2 2 3 3. = 1 - 2 = -1 = -b. 12,3.. That is, the non-vanishing entries of B occur at the same places as in B(G), and every column of B has an entry which is 1 and an entry which is -1. That is, B is the incidence matrix of an orientation of G. We call this orientation the basic orientation of G. Summarizing the discussion following (++) we have thus obtained the formula

. det ((B o B) - (B o C i ) T+ B - BT)

. 131

(+++>

Thus, for p E N , given even integers d, (= d ( v j ) ) , j = 1,.. . , p , and given q E N , the right hand side of (+++) can be viewed as an operator

IX.2.2. Enumeration of Eulerian Trails in Digraphs and Graphs IX.87 w whose input B can be any p x q matrix. For a graph G of order p and size q and having degree sequence (d,, . ,$), it yields the number

..

+

of eulerian trails of G for B = B(G) B(G)o B(G). We note, however, that G has to be assumed not to have any loops. To extend the validity of (+++) to graphs with loops we observe that Lemma IX.43 remains valid if we restrict the incidence matrix B to those arcs which are not loops (note that loops are also disregarded in the definition of A*(D), A**(D)respectively). That is, we have to start with B(G - A(G)) to define B which we use to express A**.Nonetheless, loops are considered in the factors ( ;d(vj) - l)!, and this is all that is needed to include graphs with loops in (+++) (see the BEST-Theorem), provided loops are considered orientable in one way only. We call this the combinatorial point of view. This point of view has been assumed in Definition VII.l (provided the half-edges of a loop are not distinguished). From a topological point of view, however, a loop vjvj can be oriented in two ways. This becomes apparent not only if we look at graphs embedded in some surface (the plane, say), but also if we require homeomorphic graphs to have the same number of eulerian trails. In this case (+++) has to be multiplied with 2' where X =)A(G)I. In order to include both points of view in the extension of (+++) to graphs with loops we define

0 if the combinatorial point of view prevails 1 if the topological point of view prevails .

,

Summarizing the considerations following (+) we state the enumeration formula for I I ( G )1 in the most general case.

Theorem IX.44. Let G be an eulerian graph of order p . Denote

For the incidence matrix B(G - A(G)) = (bi,j) define the p x q matrix B = (6i,j) by &l,j = 0 and b i j =

i-1

-2Cbkfif

i

> 1, j

= 1,..., q

.

k= 1

Set B = B(G - A(G))

+ B(G - A(G)) o B.

F'urthermore, let J, be the q-dimensional column vector whose components are all 1. Also, if b(i) denotes the binary expression of i written

IX.88

IX. Eulerian Trails - How Many ?

as a q-dimensional row vector, i = 0,. . . , 2 q - l - 1, let C i be the p x q matrix with identical row vectors c ( i ) = b(2q - 1- i) - b(i). Finally, let S(c, t ) E ( 0 , l ) be defined as above. The number of eulerian trails of G is given by

.det((B o B ) . (B o Ci)T+ B .B”>

. 131

Let us discuss this formula. The parameters p , q , X , d ( v j ) , 1 5 j 5 p , and the variable matrix Cj, 0 5 i 5 24-1 - 1, are either given explicitly or at least can be easily determined, regardless of whether G is given as a drawing or in the form of a matrix. Furthermore, the determinant of a matrix can be evaluated in O(n3)time if n is the number of rows (in the above case n = p - 1). Thus the formula given by the BESTTheorem is ‘good’ in the sense that the number I 7 ( D )I can be computed in polynomial time. The same can be said for each term of the sum in the formula of Theorem IX.44. However, what makes this formula look ‘bad’ is the number of terms to be computed, namely 2 4 - l . Whence the question arises, whether this number of terms can be appreciably reduced. First of all, it should be noted that this number of terms is, in most cases, smaller than I I ( G )I itself. This follows from the next result (see, e.g., [FLEI76a, Satz 5 ] ) , whose proof is left as an exercise.

Proposition IX.45. If G is a connected eulerian graph of order p and size q, the number of eulerian trails of G satisfies the inequalities

where the lower bound is attained if and only if every block of G is a cycle, while the upper bound is attained if and only if G is a cycle. Thus, if S(G) 2 6 it follows that I I(G)12 2 4 . That is, generally speaking, if 6(G) is large enough, the time needed t o compute I I ( G )I by the formula of Theorem IX.44 is O(l 7 ( G )I) or less. In fact, in computing the terms of the sum of this formula one need not always evaluate the determinants, but rather one determines via the factor 1 - sgn I (Bo Ci)J, I whether BoCi is the incidence matrix of an eulerian orientation of G, i.e., whether

IX.2.2. Enumeration of Eulerian Trails in Digraphs and Graphs IX.89

(Bo Cj)- J,

is the zero vector. This gives rise to the following, [FLEI83a,

Algorithm] .22)

Algorithm IX.46. Let G be a connected eulerian graph of order p such that q =I E(G - A(G)) 1, and having the degree sequence (dl,. . .,d p ) . Let D be the basic orientation of G - A(G); set B = B(D)and X =lA(G) I. Step 0. Set B‘ = B o B, B“ = B BT. Set i = 0,a = 0. Step 1. Calculate C iand B, := B o Ci, x := Bi - J,. Step 2. If x # 0 go to Step 3.

If x = 0 , set a = a + det(B’ - BF + B’’)l,l . Step 3. If i

< 24-’

- 1, set i = i

+ 1. Go to Step 1.

If i = 29-l - 1, cdculate

Step 4. I7(G) I is the number of eulerian trails of G. Thus, the work needed to determine I7(G) I is not nearly as much as a first glance at the formula of Theorem IX.44 would suggest. Nonetheless, the question remains how can the number of terms in the sum of that formula be lowered. One way of achieving this is by observing that if B o C iis the incidence matrix of an eulerian orientation D of G - A(G), it follows that for every j , i < j 5 24-1 - 1, for which the corresponding Aj C A ( D ) satisfy Ai C A j and I Aj - A iI= 1, the orientation arc sets Ai, corresponding to B o Cjis not eulerian. Moreover, if G is simple, then the same conclusion follows even if I Aj - A; 15 2. This implies in terms of the row vectors c ( i ) of C i that those j > i can be disregarded for which c ( j ) - c ( i ) has just one (two respectively, if G is simple) non-vanishing component ( s ) . Another way of restricting the number of terms in the sum of Theorem IX.44 is the subject of the next subsection. 22) The algorithm of that paper differs slightly from Algorithm IX.46. In the first part of Step 3 of that algorithm (Step 2 of Algorithm IX.46) it says “If x # 0 set i = i 1. Go to Step 2.” This is misleading if i = 2q-l - 1.

+

IX. Eulerian Trails - How Many ?

K.90

IX.2.3. On the Number of Eulerian Orientations In Theorem IX.44 we used the basic orientation of G - A(G) as initial orientation, but the approach in establishing that formula shows that we could have taken any orientation Do of G - A(G). For, if D’ and D“ are two orientations of G - A(G) with ael E A(D’)n A ( D ” ) ,then

{B(D’)0 C i / i = 0,.

. . ,2q-l

-

1) = {B(D”)0 c,/i= 0,.

. . , 29-1

1)

-

*

Consequently, since we are interested only in eulerian orientations of G because of the application of the BEST-Theorem, we may just as well use an eulerian orientation of G - A(G) as the initial orientation Do. This Do can be obtained in linear time by following an eulerian trail T of G, simply because T can be obtained in linear time as we shall see in the chapter on algorithms. The question one now faces is the size of { C i / B ( D , )o Cj corresponds to an eulerian orientation of G - A(G)}. To answer this question we first make an observation. Lemma IX.47. Let D , and D, be eulerian orientations of the eulerian graph G. Then Do := B, - A(D,) = ( D , - A ( D , ) ) R , and Do is an eulerian digraph. Therefore, also D, := D , n D, is an eulerian digraph.

Proof. By definition, a, E A ( D o ) e, E E(G), implies a, $ A(D,). Thus, a: E A(&) follows of necessity; that is, D , - A(D,) = (D, - A ( D , ) ) R . We have for every v E V ( D )

i d D l (v) = i d D 3 (21) -k

+

i d D z (V) = idD3(u)

(v) = O O

~

~

(v) D = ~ O d g , (v) -k O d o o (v)

(v) D = ~ odD2(V) = OdD3 (v)

,

id,o (u)

(note that i d D o ( v ) = O d D R ( V ) ) . NOW, i d D l ( v ) = i d D 2 ( v ) = i d G ( v ) implies zdDo(w) = o d D o ( v ) ;therefore, i d D 3 ( v )= o d D 3 ( v ) .Whence both Do and D, are eulerian digraphs. The lemma follows. Thus, determining all eulerian orientations of G containing a given arc a is tantamount to determining all eulerian subdigraphs D’ of an initial eulerian orientation Do such that a # A(D’),and then to forming (DoA(D’))U ( A ( D ’ ) ) R .On the other hand, D’ corresponds to an eulerian subgraph G’ of G, while it is not generally true that an eulerian subgraph of G corresponds to an eulerian subdigraph of Do. We now prove two results which together establish an upper and lower bound for the number O,(G) of eulerian orientations (of G ) containing a prescribed

IX.2.3. Eulerian Orientations of an Eulerian Graph

Ix.91

arc. For the subsequent discussion we assume a loop orientable in two ways (see the definition of 6 ( c , t ) above). We also note that O,(G) is independent of the choice of the prescribed arc a,, e E E(G);this follows from the fact that a digraph D is eulerian if and only if D R is eulerian.

Lemma IX.48 (see also [PALM78a, Theorem 4.41). Let G be a connected graph of order p and size q. The number of spanning eulerian subgraphs of G is 2q-P+l. Proof. Let B be a spanning tree of G; it has p - 1 edges, hence q G - B = q -p 1. Let Eo E(G - B ) be chosen arbitrarily; possibly E,, = 8. Denote G = G,, and let G, be the graph obtained from G by contracting B onto a single vertex and deleting E ( B ) . Thus E(G,) = E(G - B ) , and E, induces a spanning eulerian subgraph of G, since E, is a set of loops in G,. Now let Gp-, be obtained from G, by splitting the vertex of G, into two vertices v and w which are adjacent in B , and add vw E E ( B ) . e E E ( G ) - E ( B ) is incident with v (w) if and only if e is incident with a vertex in the component Cu 2 {v} (C, 2 {w}) of B - {vw}. In GpVl precisely one of E, and E, U {vw} induces a unique spanning eulerian subgraph. Repeating the procedure it follows that E, U E' induces a unique spanning subgraph of G for a uniquely determined E' C E ( B ) (in each step one may have to add an isolated vertex). Since there are 2 4 G - B = 2Q-P+' such subsets E, the validity of the lemma follows.

+

Lemma IX.49. If G is an eulerian graph of order p and size q then 0 G > 3 9-p 23) E ( 1 - (5) *

Proof. Suppose the lemma fails for some G. Among all possible choices G consider one with a minimum number of 2-vdent vertices and, subject to this condition, such that a& := C (d(v) - 4) = 2q - 4p is of such

uEV(G)

minimum. Accordingly, we distinguish between three cases.

1) G has a 2-valent vertex x. If E, C R(G)C E ( G ) ,let G' := G - {x}. Regardless of whether { e } = E, for the prescribed arc a,, we have

OE(G)= 2O,(G')

I

.

The requirementp- V,(G) 12 3 in [FLEI83a, Lemma61 is unnecessary because loops are permitted and assumed to be orientable in two ways. 23)

IX.92

IX.Eulerian Trails - How Many ?

However, because of the choice of G and p’ = p conclude the validity of

- 1,

q’ = q - 1, we

since the lemma holds for G’. We reach the same conclusion in the trivial case B, = A(G) = E ( G ) ,i.e., whenp 2 q = 1. In all other cases suppress x to obtain G‘ which satisfies q’ This and the choice of G yields

U,(G’)

1

(,>,

- p‘

= q -p .

.

However, an eulerian orientation of G‘ corresponds to an eulerian orientation of G, and vice versa. Thus the above inequality can be rewritten so as to satisfy (0). Whence we conclude 6(G) 2 4 because of the choice of G.

G has a 2k-valent vertex wo,k > 2. Consider E&, = {ei, . . . , eLk}. Subdivide el with k - 2 2 1 vertices w l , . . . , such that wiand are adjacent, 0 5 i 5 k - 3, and let e h j , e ; j + , be incident with wj,1 5 j 5 k - 2. Thus wi,0 5 i 5 k - 2, becomes a 4-valent vertex 2)

of the graph G‘ obtained this way. It follows that e 2 k - 2 ,e2k-1,e2k are incident with wo also in G‘ (see Figure IX.9).

Observing now that a&, < CT& and that any eulerian orientation of G’ corresponds to an eulerian orientation of G (although not necessarily vice versa), we conclude from the choice of G that

That is, (0) also holds in the case where G has a 2k-valent vertex for k 2 3. Whence we conclude that a& = 0. 3) G is 4-regular. In this case we proceed by induction on p . If p = 1, then q =I A(G) I= 2, and U,(G) = 2 since we assume a loop orientable in two ways. Thus (0) is satisfied in this case.

Ix.2.3. Eulerian Orientations of an Eulerian Graph

Ix.93

G' Figure IX.9. Forming G' from G by replacing a 2k-valent vertex wo with k - 1 4-vdent vertices wo, wl,. . . ,Wk--2. Suppose now p > 1. Let (v,w) be the given arc belonging to every eulerian orientation of G counted in U,(G), vw E E(G) and possibly v = w. Form G1,2,G1,3,G1,4,where EC = {ei/l 5 i 5 4}, el = ow. Suppress the 2-valent vertices of Gl,j to obtain GY,j, j = 2,3,4, provided contains no component consisting of just one vertex with a loop attached to it. Assume first that Gl,j has this property for every i E {2,3,4}. Thus we conclude that GY,j is 4-regular and pGo = p - 1 for 1,j every j E {2,3,4}. It follows from the choice of G and since q = 2p that

o,(Gy,j) 2

7

j = 2,374

7

where we assume that el,j E E(GYIj),arising from splitting away el, e j E E(G),underlies the prescribed arc in the eulerian orientations of GY,j. Just as a 4-valent vertex of an eulerian digraph can be split in two ways into 2-valent vertices to yield an eulerian digraph, it follows that every

Ix.94

Ix.Eulerian Trails - How Many ?

eulerian orientation of G containing (21, w ) is counted in OE(Gy,jt) and OE(G:,jp)but not in UE(Gy,j3),where {jl,j2,j3} = {2,3,4}. This consideration combined with the preceding inequality yields

that is,

since q = 2p for 4-regular graphs G.

To finish the proof of the lemma we have to assume that at least one j E {2,3,4}, has a component C such that p , = qc = 1. That is, w is incident with a loop in G. If E, 2 A(G) define the 4-regular graph Go := G - {w};otherwise, let Go be the 4-regular graph homeomorphic to G - ( E , n A ( G ) ) . Since pGo = p - 1 in any case we have

On the other hand, since E, n A(G) #

0 we have

U,(G) = 2UE(G0)2

(i)'

regardless of whether ww is a loop. That is, also in this case O,(G) satisfies (0). This finishes the proof of the lemma. The next result is a consequence of the preceding two lemmas.

Theorem IX.50. Let G be a connected eulerian graph of order p and size q and let e E E ( G ) be chosen. O,(G), the number of eulerian orientations of G containing a,, satisfies the inequalities

Proof. The lower bound is nothing but a repetition of Lemma IX.49. The upper bound is a consequence of Lemma IX.48 as follows. It follows

IX.2.3.Eulerian Orientations of an Eulerian Graph

IX.95

from the paragraph preceding this lemma that O,(G) is at most as large as the number of eulerian subgraphs of G not containing a prescribed e E E(G). That is, O,(G) is not larger than the number of eulerian subgraphs of G - e. This latter number is 2(Q-’)-P+l = 2Q-P; whence O,(G) 5 29-P which finishes the proof of the theorem.

In fact, the upper bound of Theorem IX.50 is achieved precisely by those connected graphs all of whose blocks are cycles; we leave it as an exercise to prove this. We note in passing that these eulerian graphs can be characterized by the property that every strongly connected orientation is eulerian, [OELL84a]. The lower bound of Theorem IX.50 is a theoretical justification for our approach to establish the enumeration formula given in Theorem IX.44. For, the number of eulerian orientations of G grows exponentially with respect to the number of vertices if we assume V2(G)= 0. In particular, O,(G) > 2P if 6(G) 2 6 since 2q 2 6p in this case. Returning to E.M. Palmer’s idea (see the discussion following (+) after Corollary IX.40), we obtain from Theorem IX.50

which implies a tremendous growth of I 7 ( K 2 n + l )I as n E N grows. In fact E.M.Palmer’s idea stems from an unsuccessful attempt by V.A. Sorokin to establish a formula for I 7(K2n+l)1, [SORO69a]. A flaw was first pointed out by V.A. Liskovec, [LISK7la]. For small n, however, B.D. McKay could salvage some of Sorokin’s ideas establishing the exact value of I 7(I<2n+l) 1, 1 5 n 5 5, [McKA82a]. To give the reader an idea concerning the growth of I 7 ( K 2 n + lI ) we note that I T(K11) I> 257 - lo2’. Using Liskovec’ techniques B.D. McKay was able to go as far as establishing I 7 ( K I 7 1,) [McKA83a]. In this context we also refer t o the article [WENI89a]. Before continuing our discussion on how to determine O,(G) for an eulerian graph G, we look at the other obstacle one faces concerning a simplification of the sum of Theorem IX.44. As we have pointed out in the discussion following equation (+) after Corollary IX.40, the determinants in the sum of Theorem IX.44 vary as eulerian orientations of G vary. This fact has been noted in [FLEI83a], but also in [McKA82a] where it is mentioned that for n < 7, detAT,i(DKzn+l)is the same for of Kzn+l,while this is not the case if any eulerian orientation DK2n+l

IX.96

IX.Eulerian Trails - How Many ?

n 2 7. Thus, the question arises whether there is some structural invariant (as opposed to numerical invariants such as degree sequence, etc.) common to all eulerian orientations of an eulerian graph. The next result shows that such an invariant exists, [FLEI83a, Lemma 7].24)

Proposition IX.51. For any eulerian orientation D of the loopless eulerian graph G of order p , the number N B ( D )of basic subgraphs of D with root w1 is given by

Proof. We look at equation (4) and the discussion following it in the proof of Theorem IX.37. There, the right-hand side of (4) contains Dia,i3,...,iP which is a basic subgraph with root w1 in any case. Moreid(vi) terms. Thus over, the multiple sum in (4)contains precisely the number of basic subgraphs of D with given root w1 is

nE2

v€V(G)-{vi}

implying the validity of the proposition. That is, the number of basic subgraphs of an eulerian digraph depends exclusively on the degree sequence of the underlying (eulerian) graph.

Of course, there is an exact formula for C?,(G) which can be extracted from Theorem IX.44, namely

but unlike Theorem IX.50 this formula does not tell us anything about the exact growth of O,(G). This also applies to the following considerations (see [KOTZ59a], and [BERM79c] in which K.A. Berman, apparently unaware of A. Kotzig’s work published in Slovak, rediscovered 24) There is a minor flaw in that paper. Namely, i 2 2 must hold (this applies also to equation ( 0 ) on p.193 of that paper), and p has to be replaced by p - 1 in the exponent.

IX.2.3. Eulerian Orientations of an Eulerian Graph

Ix.97

A.Kotzig's results). The starting point is the consideration of the set of trail decompositions 7 D ( G ) of the eulerian graph G; we abbreviate t(G) : = ] 7 D ( G )I. Of course, we assume the trails to be closed.

Lemma IX.52. If G is an eulerian graph, V ( G )= {q,. . . , u p } , then

The proof of Lemma IX.52 follows by observing that this formula for t(G) is nothing but a reformulation of the upper bound of Proposition IX.45. For every Sj E ' T D ( G ) , denote s j =I S j 1, j = 1,. . . , t ( G ) . That is, s j is the number of trails in the trail decomposition Sj of G. In other words, s j is the number of components in the 2-regular detachment of G corresponding to Sj. Each of these components is a cycle and can thus be cyclically oriented in two ways. That is, Sj gives rise to 2'3 eulerian orientations of G. On the other hand, given an eulerian orientation D of G, there exist n:=, id(v,)! systems of transitions of D,each of which defines some trail decomposition of D and thus of G. Thus

where the factor 2 on the left-hand side of the equation stems from the fact that no prescribed arc has been taken into consideration. Consequently, we can summarize these arguments in the following.

Lemma IX.53. For any eulerian graph G

where t = 2-qJ'J:=)=,

and s j =I Sj I for Sj E 7D(G). ' o ) ! d( v i ) !

However, in view of Theorem IX.44 this formula for O,(G) is even worse than the one cited earlier (see the discussion following Proposition IX.51). For, it requires the determination of the numbers s j which can be used directly to determine ('T(G)I: namely, l'T(G) I=! {j/l 5 j 5 t ;s j = 1) I.

IX.98

IX.Eulerian Trails - How Many ?

In fact, Kotzig's idea even gives rise to determining the set of eulerian trails of G.Nonetheless, using s 2 1and the value of t one easily deduces Lemma IX.49 from Lemma IX.53 by first showing that

2 2.3"'

for

k E N. An interesting structural feature concerning the set of all eulerian orientations of a graph was discovered by M. Guan and W. Pulleyblank, [GUAN85b]. They defined the eulerian orientation graph B(G) of an eulerian graph G as follows. V(O(G))is the set of all eulerian orientations D of G (thus both D and D R are vertices of B(G)), and D I D , E E(O(G))if and only if ( A ( D ,nDf))is a cycle (or, equivalently, if and only if (A(DF n 0,))is a cycle). Concerning B(G), the following holds, [GUAN85b, Theorem 4.41, which we present without proof.

Theorem IX.54. If G is a connected eulerian graph, then 8(G) is the graph of a d-dimensional hypercube (and thus hamiltonian), or B(G) is hamilt onian connected.

On comparing Theorem IX.54 with Theorem VII.28 I wonder whether there is a closer relationship between these two theorems. For, a segment reversal in a connected eulerian graph G with respect t o an eulerian trail T can be interpreted as changing the orientation (induced by 7') of an eulerian subdigraph of D,. We note that the authors of [GUAN85b] were led to these considerations when studying the windy postman problem (WPP). In fact, since W P P is an NP-complete problem one can try to find approximations. One way of doing this is to solve the CPP for the given c,(e)), e E E(G), where graph G with cost function c(e) := $(cl(e) c1 and c2 are the cost functions with respect to WPP. Thus one obtains an eulerian supergraph G+. Then one produces a minimum cost eulerian orientation D+ of G+ by invoking the cost functions cl,c2 (see Theorem VIII.100 and the WPT-algorithm following it). We observe, however, that if c(D+)is minimal with respect to all solutions G+ of the CPP and all minimum cost eulerian orientations, c)+ need not represent a solution of W P P for G with respect t o cl, c2. For a W P T in G may use an edge more than twice, while this is not the case for the approximation solutions just described. This fact has its theoretical base in an algorithm whose running time is O ( p . q 2 ) and which produces a minimum cost eulerian orientation of G+, [GUAN85b, p.6611.

+

We finish this subsection by mentioning the papers [SCHR83a, SCHR83b, LASV83a, LASVSSa] which establish lower and upper bounds for O,(G)

IX.2.4. Applications and Generalizations of the BEST-Theorem IX.99

in the case when G is 2lc-regular. In fact, using what has become known as the Martin polynomial, M. Las Vergnas established an explicit formula for O,(G), [LASV83a, ThCorkme 5.2].25)

IX.2.4. Some Applications and Generalizations of the BEST-Theorem In this subsection we discuss a series of papers which deal with various enumeration problems involving or generalizing the BEST-Theorem. In fact, the basis of [AARD5la] (which contains the BEST-Theorem) are certain considerations in dynamics, [MARM34a], and number theory [GOOI46a, REES46a1, and which have applications in telecommunications (see [BERG73a, p.239-2401 and [LIUC68a, p.177-1781). The question studied in these papers is basically the following: given a n afphabet A = { a l , . . . ,an}, does there exist a word W (= sequence), and how many words W are there, such that each k-tuple of elements of A (with or without repetitions) occurs exactly once in W ? This problem can be transformed into the question on the existence of an eulerian trail in a certain digraph (see [GOOI46a, BERG73a, LIUC68al in which a special case is exhibited by the same figure. See also the discussion below). However, in [GOOI46a] the author states that he knows “no simple formula for the number of solution^^' .26) That formula is established by the BEST-Theorem. Questions similar to the above arise in biochemistry if one studies DNA molecules. Owing to their particular structure, those molecules can be treated like words over an alphabet. In [HUTC75b] the following result has been established. Theorem IX.55. Given an alphabet A = { a l , . . . ,a,} and two functions rn : A t N , p : A x A N U ( 0 ) . The number Nw of words of the form W = ail, . . . ,aiq over the alphabet A having the property that each letter appears exactly m i := rn(ai) times in W, i = 1,.. . ,n, and such --f

25) In the case of 4-regular graphs G, relations between the eulerian orientations of G and the Tutte polynomial (if G is plane), the Martin polynomial respectively (if G is embedded in any surface), are established in [LASVBBa, Theorems 2.1 and 5.11. 2 6 ) See also the footnote on p.VI.26 of [FLEISOd].

IX.Eulerian Trails - How

IX.100

Many ?

that ai is followed by a j exactly p i , j := p ( a i ,a j ) times, i , j = 1,.. . ,n, is given by the formula

subject to the conditions n

n

k= 1

k= 1 n

k= 1

where

denotes the Kronecker delta.

1

a=

1,..., n

,

Proof. Consider the digraph D for which V ( D ) = A and in which ai and a j are joined by p i j arcs from ai to a j . Suppose first that ail # a i q , introduce a new vertex a. and new arcs (ai ,ao),(ao,a i l ) ,and denote by Do the digraph thus obtained. It follows from the definition of D that Do is eulerian if and only if the conditions stated in the theorem are met. That is, a word of the form W = a i l , .. . ,uiq exists if and only if the numbers mi and p i , j , i, j = 1,. . . , n, satisfy these conditions. Note the 11-correspondence between eulerian trails of Do starting at a. and the open , covering trails of D starting at a i l . Observing that mil = o d D ( a i 1 )while miq = i d D ( a i ) , it follows that the Kirchhoff matrix A * ( D o )satisfies the equation det Aio(Do)= det(miSij - pi,j):j=l . This and the application of the BEST-Theorem yield in turn

However, if we permute in a fixed T E 7 a 0 ( D 0the ) order in which the pi$ arcs joining ai to a j are passed, we obtain p i , j ! different eulerian trails corresponding to the same word W, expressed by T for fixed i , j E { 1,. , , , n}. Consequently,

IX.2.4. Applications and Generalizations of the BEST-Theorem IX.101

which proves the theorem for the case ail # a;,.

If, however, ail = a '.q 1 then form Do by introducing a. and (ao,a i l ) . Now, Do has an open covering trail starting at a. and ending at ail = uiq if and only if D is eulerian, i.e., if and only if the conditions of the theorem are met; and there is a 1-1-correspondence between the open covering trails of Doand the eulerian trails of D starting at a i l . Treating now Do in the same way as we did with respect to D in the case ail # ai, we finish the proof of the theorem (observe that if ail = a;,, then mil = idDail 1 =

+

1-

idDoai,

Incidentally, the proof of Theorem IX.55 shows how the BEST-Theorem can be reformulated (using the whole Kirchhoff matrix instead of one of its minors), if one considers T,,T, E 7 v ( D )to be different as long as they start with different arcs. Moreover, if we accept a priori the validity of Theorem IX.55 then the BEST-Theorem can be derived from Theorem IX.55 and its proof by using the Matrix Tree Theorem for digraphs (Theorem IX.37). We leave it as an exercise to do this. We note in passing that for I A 15 4, the undirected version of Theorem IX.55 has been discussed in [HUTC75b] as well. Moreover, Theorem IX.55 has been generalized in [HUTC75c], where one starts with a given set W := {wl,. . . ,w,} of (distinct) words over an alphabet A such that each w ihas length at least s 1, s E N , and no wiis the initial sequence . . of wj,z , j = 1,. . .,r, i # j. Furthermore, a function m := W -+ N is given (again, we set m i:= m ( w i ) , 1 5 i 5 r ) . The problem is then to

+

determine the number NW of words W such that wiappears exactly mi times as a subsequence of W, and such that wi and w j are (in W ) either disjoint or they overlap by precisely s letters (possibly i = j if mi > 1). (PI

The digraph D corresponding to this problem is defined as follows. The vertices of D are precisely all distinct sequences vt of length s which appear as initial or final segments of some wiE W ; t 5 2r follows. Now definepi,j = C k E K i , j m k ,where Ki,j is the set of indices for which w k begins with v i and ends with vj. Doing this for all i, j E (1,. . . ,T } , we then introduce in D p r e c i s e l y ~ ; ,arcs ~ joining vi to vj, and for each k E K i , j we attach the label wk to precisely m k of these p i , j arcs. Consequently, a word W as described in (P) exists if and only if D is either a weakly connected eulerian digraph or, at least, D has an open covering trail. Using

IX. 102

IX.Eulerian Trails - How Many ?

Theorem IX.55 and its proof, it is now a straightforward consideration to establish a formula for Nw;it is left as an exercise to do this. We note that the question stated in the introduction of this subsection can be answered as a special case of (P) and its solution. In fact, a circular sequence with exactly one occurrence of every k-letter word over an n-letter alphabet A is commonly known as a k-de Bruijn sequence (see, e.g., [DAYH84a], [JUNG87a, p.411, [EVEN79a, PA]), and

the number of k-de Brui'n sequences over a n n-letter alphabet is (n!)nk-' nk

'

[AARD5la]. The digraph D corresponding to a k-de Bruijn sequence is nothing but a special case of the digraph corresponding to (P):The vertices ofD are precisely the nk-l (g-1)-letter words w ,while the arcs of D correspond to the k-letter words, where u1u2 . . .uk joins u 1 u 2 . . ak-l to u2u3 . . . u l c ,ui E A for i = 1,..., k. It follows that D is a weakly connected n-regular digraph and thus always has an eulerian trail. Following [DAYH84a] we call this D the (k,n)-template digraph (note that u ( D ) = C w E v ( D ) o d , ( w )= n k , in accordance with the fact that there are nk k-letter words). We note that the (k,n)-template digraph is often called the de Bruijn-Good graph or just de Bruijn graph Dk. Reconsider problem (P) and the digraph D constructed from the data contained in (P). In the worst case hfw = 0, namely if D has no (open or closed) covering trail. However, one may ask whether it is possible to obtain n/, > 0 if one starts with an arbitrary function mo : W , --+ N U {0}, where W , W is sufficiently small, and then extends mo to m : W --+ N U (0). This question has been considered in [DAYH84a], where W , is called a set of independently prescribable words and where W is the set of all Ic-letter words over A = {al,..., u n } , k 2 2,n 2 2. Translated into graph theory the problem now is to determine in the (k,n)-template digraph D those arc sets A, = { w i l , . . . ,w i } such that it is always possible to obtain from D a new digraph D, having a covering trail, by the following operations. First replace w i j 6 A, with mij arcs having the same initial and end-vertex as wij, where mjj E N U (0) has been chosen arbitrarily, j = 1,.. . ,r. Call the digraph thus obtained D'. Secondly, by replacing every w E A ( D )- A , c A(D') with m(w) E N u { O } arcs having the same

IX.2.4. Applications and Generalizations of the BEST-Theorem IX.103 initial and end-vertex as w, for certain values m ( w ) ,one then obtains D, having a covering trail. Certainly, every {w}, w E A ( D ) , is a set of independently prescribable words; this follows from the fact that D has an eulerian trail (see above). Consequently, one may ask what the size of a largest set of independently prescribable words is, and how many such largest sets there are. The (nearly complete) answer to these questions is summarized in the next result which we present without proof (see [DAYH84a, Theorem 3.7, Lemma 3.8, Theorem 4.3, Theorem 6.111).

Theorem IX.56. Let A, = { a l , . . . ,a,} be an alphabet on n 2 2 letters. Let A,,, be any maximal set of independently prescribable kletter words and let do,k be the set of all these Ao,k, k 2 2. Denote m,,k := maz{n,k}. The following is true.

‘1

IAO,k!=

{

nk

- nk-1

,k

- ,k-1

+1=3

if mn,k> 2 ifm,,k=2

.

Part of the proof of Theorem IX.56 hinges on the fact that if n 2 2 and S is any (Ic - 1)-de Bruijn sequence, the k-letter words not appearing as sections in S form a largest independently prescribable set of k-letter I=] q ( D )1, where D is the (k- 1,n)words. Moreover, if n > 2 then I template digraph whose number of eulerian trails is given by ( n n! )kn- kl - ’ (see statement (B) above), a special case of the BEST-Theorem. We note in passing that the main part of Theorem IX.56 can be extracted from a more general theorem on hamiltonian digraphs (see [DAYH84a, Theorem 5.11). We also note the paper [PATT73a], where the BEST-Theorem is used more directly as an equivalent mean to enumerate Quenuille’s changeover designs. Finally we mention the paper [DAWS57aJ which combines some of the considerations leading to Theorem IX.55 and the subsequent discussion with probability theory, and [BAUMSSa] which uses the BEST-Theorem directly to enumerate what are called circular patterns in a digraph. Some additional papers dealing with de Bruijn graphs can be found in the bibliography (these papers are marked with the symbol +). Next we discuss a way of generalizing the BEST-Theorem which has been undertaken in [JACS79a], and which starts with considerations on

IX.104

IX. Eulerian Trails - How Many ?

words over an alphabet similar to the above (see Theorem IX.55 and the subsequent discussion). However, the results established in these papers do not use the BEST-Theorem; rather it can be derived - among other enumeration formulas - as a special case of these results. Let the alphabet A be of the form A = (1,. . . ,n } , and let A* be the monoid of sequences over A with concatenation as binary operation A* x A* t A* and E denoting the empty sequence which is the identity in A*. Denote A+ = A* - (e}. Let the functions m and p be defined as in Theorem IX.55. Furthermore, let { w ~ , ~ / Z , = ~ 1 , .. . ,n} and { z i / i = 1,.. . ,n } be sets of indeterminates. For the sake of brevity we also introduce additional notation. For the above sets and functions let x = (zl,. . .,zn),X = d i a g ( z l , .. . ,zn)- the diagonal matrix whose (2, 2)-th entry is xi,and let m and M be defined correspondingly with respect to the function m : A + N . P,V respectively, denotes the matrix whose (i, j)-th entry is p i , j , v i j respectively. Now define n

n

i,j=l

i=l

n

n

Furthermore, for every w E A+ define p(w) : = x m v P

,

where m iand p i , j are the numbers of occurrences of i and ij respectively, in 20, i , j E A. It follows, [JACS79a, Proposition 2.11, that

is the generating function for A+ with m iand p i , j satisfying the conditions stated in Theorem IX.55 if w begins with i, and ends with i, (here, the functions m and p vary as w E A+ varies). In fact, @(x,V) can be written in the form n

IX.2.4. Applications and Generalizations of the BEST-Theorem IX.105

where y = (yl,. . .,yn) corresponds to the unique solution of the system of linear equations yT=xT+XVyT , [JACS79a, Lemma 2-21. The main result of the cited paper, [JACS79a, Theorem 3.11, gives an explicit power series expansion for @(x,V)in the form yr = x x m V P d e t ( M - P ) ( m - J:)!(P!)-'

,

(*I

P

where the summation is over all n x n-matrices P such that

provided there is s E A such that n

m i=

Cpi,j+ bi,,, ,

i = I,. . . ,n

(* * *>

j= 1

(J, E R" again denotes the column vector whose components are all 1). The BEST-Theorem can now be derived from (*) as follows. Observe that (**) and (* * *) classify y, as that part of the power series expansion of +(x, V)which corresponds to the elements of A+ beginning with T E A and ending at some s E A. It follows that the coefficient of xmVP for fixed m and P satisfying (**) and (* * *), is - on the one hand - the number NW of words W over A of the form W = T . . . s for some fixed s which is determined by m and P. However, (*) yields explicitly

NW = (m - JT)!(P!)-' det(M - P) . However, for ai = i, i, = T , i, = s, this formula is nothing but the formula of Theorem IX.55 whose conditions are fulfilled since (**) and (* * *) hold. That is, Theorem IX.55 follows from [JACS79a, Theorem 3.11 without using the BEST-Theorem. As we have noted in the discussion following the proof of Theorem IX.55 (see also Exercise IX.20), the validity of that theorem plus the Matrix Tree Theorem for digraphs yield the validity of the BEST-Theorem. We note that (*) can also be used to determine the number of hamiltonian cycles in a digraph, [JACS79a,

IX.106

IX.Eulerian Trails - How Many ?

Corollary 4.21. However, I.P. Goulden and D.M. Jackson subsequently developed a method to enumerate what are called chromatic trees which, together with the Matrix Tree Theorem for digraphs, yields another proof of the BEST-Theorem, [JACWb]. Extending this method to the enumeration of rooted labeled trees the same authors were able to derive the Matrix Tree Theorem for digraphs, [GOUL82a], which in turn led to yet another proof of the BEST-Theorem and the enumeration of hamiltonian cycles in digraphs, [GOUL8lb].

IX.2.5. Final Remarks We have discussed various ways of enumerating eulerian trails in digraphs and graphs. In this discussion the BEST-Theorem was the cornerstone, so to say. However, in chapter VI we discussed various types of eulerian trails satisfying certain restrictions. Thus the question arises how to enumerate eulerian trails of a special type.

A formula for the number of A-trails in 2-connected outerplane graphs has been established in [REGN76a7Satz 4.1.31; it rests on Theorem VI.87. We present it without proof.

Theorem IX.57. Let G be a simple 2-connected, outerplane, eulerian graph together with a 2-face-coloring such that the outer face F, of G is a 1-face. Let PI, . . . ,F, be the 1-faces of G different from F,, and denote V ( i ):= V ( b d ( F i ) ) .Let P, be the set of all partitions P of {1,2,. . . ,n } . Then

PEP, I € P ' i € I

j U

,

In the case of 4-regular graphs G embedded in a surface S,M. Las Vergnas considers trail decompositions S of G such that X , contains either nonintersecting transitions only, or else X ( v ) is a pair of intersecting transitions, for every w E V(G). In the latter case call an element of S a crossing trail. In [LASV8la, Proposition 5.21 the following result has been established. Theorem IX.58. Let G be a 4-regular graph embedded in a surface S. Let sk(G) be the number of trail decompositions S such that 1 S I= Ic and

IX.107

IX.2.5. Find Remarks

X, is a system of non-intersecting transitions.

h t h e r m o r e , let c r t ( G ) denote the number of crossing trails of G. Then I q G ) I=

( X (- Z )”. . ( G ) - (-1)PG(-2)cr2(G) k> 1

)-

We note that in the case where S is either the sphere or the projective plane or the torus, the polynomialz7)

k>O

and the Tutte polynomial for G , , where G , is defined by M(G,) = G , are closely related (see [LASVgla, Proposition 4.11. - M ( G , ) , the medial graph of a simple graph G, embedded in a surface, is defined as follows: V ( M ( G , ) )= E(G,), and e , f E V(M(G,)) are adjacent in M(G,) if and only if e and f are neighbours in O+(v), where ZI E V(G,) and e , f E E v . ) .

Of course, the number of A-trails in a plane, 3-connected, eulerian graph G equals the number of spanning hypertrees Tiin the associated hypergraph Mi.However, since the problem of deciding whether Mihas such Tiis an NP-complete problem, it may very well be a difficult task to

establish an enumeration formula for II,(G) I (see the discussion following the proof of Theorem VII.14). In any case, nothing similar to the BEST-Theorem can be hoped for (since this formula can be computed in polynomial time).

As for the enumeration of aneulerian trails in a digraph D,this is tantamount to enumerating eulerian trails in GG, the graph underlying DT . In this case, Theorem IX.44 provides an (albeit not very practical) enumeration formula. In contrast, enumerating Do-favoring eulerian trails in a digraph is easy, provided D, = D - Do satisfies the hypothesis of Theorem VI.34. We therefore leave it as an exercise to establish that formula. Finally, enumerating P(G)-compatible eulerian trails is another open problem. 27) In the case of an arbitrary eulerian digraph D, if one considers arbitrary trail decompositions S without restrictions on X,, this polynomial is exactly the Martin polynomial of D. In the case of a graph G , the Martin polynomial of G is defined in the same way, except that -1 is replaced by -2.

IX.108

IX.Eulerian Trails - How Many ?

A method of counting eulerian trails in graphs which differs from Theorem IX.44 has been discussed in [SHISSSa]. As the authors note, their method is a generalization of Tarry’s method of counting eulerian trails in complete graphs (see, e.g., [LUCA94a, p.125-1511 which deals with what has become known as the domino problem). This method amounts to successively replacing wi E V(G),i E (1,. . . , p } , for which d(vi) = 2k, > 2, with k j 2-valent vertices in all possible ways and establishing a system of linear equations whose solution yields II ( G )I. We note that a modification of this replacement procedure will be used in the next chapter to establish an algorithm for determining 7 ( G ) .

IX.3. Exercises Exercise IX.l. Prove Corollary IX.4 (hint: in the case of a (mixed) graph H operate with eulerian orientations of H ) . Exercise IX.2.a) Prove statement d) of Remark IX.6. b) Prove Corollary IX.7.a. Exercise IX.3.Starting from Definition IX.9 prove the following facts: a) Rotation equivalence defines an equivalence relation RE on 7v(G,H ) and thus a pxtition PPRE(v) of 7v(G,H ) into equivalence classes under RE. b) If T ,T’ E C E P R E ( v )then X, = X,, , while the converse is not true in general. c ) Ic(= + d ( v ) for every c E p R E ( v ) . Exercise IX.4.Prove Lemma IX.14. Exercise IX.5. Prove Corollaries IX.16 and IX.17 (for the proof of Corollary IX.17.2) apply Proposition IX.12). Exercise IX.6. Fill in the details of the proof of Theorem IX.21. Exercise IX.7. Prove the following equivalences. a) H(0,l; 0 ) 2 pis f-equalizing if and only if H ( 0 , l ; 2)2p is f-equalizing. b) For j E (0, l},H ( 2 , j ; O ) 2 p is t-equalizing if and only if H(2,j ; 2)2p is t-equalizing . c) H ( 3 , O ;0) is equalizing if and only if H(3,O;2) is f-equalizing. d) H(0,O;0 ) 2 pis t-equalizing if and only if H(0,O;2) is t-equalizing. 2p Note: avoid the use of Theorem IX.27 whenever possible.

IX.3. Exercises

IX.109

Exercise IX.8. Prove Corollary IX.28, Corollary IX.29 and Corollary IX.30. Exercise IX.9. Prove the extension of Corollary IX.29 to mixed graphs of type (3,1; 2). Exercise IX.10. Show that there are infinitely many non-equalizing mixed graphs of type (2,l; 0)2p-2as well as of type (3,O; O)2p--3 (see the discussion following Corollary IX.30). Exercise I X . l l . Extend Theorem IX.27 to include new statements as expressed by Corollaries IX.28, IX.29, IX.30. Exercise IX.12. Show that for A,, . . . , A , E BL, the trace in (A*) vanishes if and only if the associated (mixed) (di)graph is either weakly equalizing or it has odd vertices. Exercise IX.13. Prove Proposition IX.32 and translate each of I), 2), 3) into statements concerning the standard identity of degree m. Do the same concerning Proposition IX.33. Exercise IX.14. Prove Theorem IX.35. Exercise IX.15. Apart from the results stated in subsection IX.l.l, prove statements similar to (S) which take into account results on equalizing mixed graphs different from those used for the proofs of Theorem IX.31 and Theorem IX.34. Exercise IX.16. Prove Lemma IX.36. Exercise IX.17. Prove Lemma IX.41. Exercise IX.18. Prove Proposition IX.45 (hint: apply the Splitting Lemma for the proof of the lower bound). Exercise IX.19. Show that for a connected graph G, O,(G) = 2 Q - P if and only if every block of G is a cycle. Exercise IX.20. Show that the BEST-Theorem can be derived from Theorem IX.55 and its proof by using the Matrix Tree Theorem. Exercise IX.21. Using Theorem IX.55 and its proof establish a formula for NW of the more general problem (P) formulated in the discussion following the proof of Theorem IX.55. Exercise IX.22. Let D, c D,where D and D, are digraphs, and let Do= D - D,.Prove that there is a polynomial time algorithm deciding

IX.110

IX.Eulerian Trails - How Many ?

whether D, satisfies the hypothesis and statement 2) of Theorem VI.34. Furthermore, establish a formula for the number of D,,-favoring eulerian trails of D provided such eulerian trails exist.

x.1

Chapter X

ALGORITHMS FOR EULERIAN TRAILS AND CYCLE DECOMPOSITIONS, MAZE SEARCH ALGORITHMS In this chapter we shall develop several algorithms and discuss their complexity to some extent. Those algorithms developed so far shall not be discussed again.

X. 1. Algorithms for Eulerian Trails Our point of departure is the Splitting Lemma which gives rise to algorithms for special types of eulerian trails. Since these algorithms can be derived from the following general algorithm, we shall only briefly discuss the former and leave their explicit formulation as exercises. The ensuing discussion basically follows along the lines of [FLEI83b, section 61.

Algorithm X . l . Splitting Algorithm. Step 0. Given a connected eulerian graph G of size q > 0, choose an initial vertex vo E V(G)and let To = v0 be the initial trail. Set

H=G, i=O. Step 1. Suppose Ti = vo, e l , v l , . . . , e i , v i has been constructed by a (possibly empty) sequence of splitting away pairs of edges so that Ti appears as a path (in H ) whose inner vertices') are 2-valent in H . If i = 0, choose el E EVoarbitrarily; go to (1.2). If i # 0, set f, = e i . (1.1) If d H ( v i )> 2, choose fi, f3 E EVifl ( E ( H )- E(T,)). For j = 2,3, form H,,j by splitting away the edges f, and fj. Define H = H,,, if H I , , is connected, and H = H,,, otherwise. If d,(vi) = 2, H

remains unchanged. (1.2) Write ei+l = vivi+l for the edge not in

H . Define Ti+, = Ti,ei+,, For a path p , v

Tiand incident with ui in

T J ; + ~(possibly

vi+, = vi).

E v(P)is called inner vertez if d p ( v ) = 2.

X.2

X. Algorithms: Eulerian Trails, Cycle Decompositions, Mazes

(1.3) Set i = i

Step 2. If i Step 3.

+ 1.

# q go to Step 1; otherwise, go to Step 3.

Tnis an eulerian trail of G.

As we shall see later this algorithm is not the fastest. Its advantage, however, lies in the fact that it can serve as the basis for a whole series of algorithms, all of which are good in the sense that their running time is polynomially bounded. In fact, the following algorithm can be viewed as a direct consequence of Algorithm X.l. It is one of the oldest algorithms for eulerian trails (see [LUCA94a, p.134-1351 for an early description). Algorithm X.2.Fleury 's Algorithm. Step 0. Let G,uo and To be as in Step 0 of the Splitting Algorithm. Step 1. Suppose the trail

G;

Ti= q,,el, vl,. . . ,e i , v i has been chosen. Set

= G - E(T;).

(1.1) Let ei+l E EVin E(Gi)be chosen such that ei+l is not a bridge of G; unless it is an end-edge of Gi.

(1.2) Write ei+, = 'U;'U~+~ and define Ti+l = Ti, ei+l, u ; + ~ (1.3) Set i = a

Step 2. If i

+ 1.

# q, go to Step 1; otherwise go to Step 3.

Step 3. Tn is an eulerian trail of G. The obvious difference between Algorithm X.l and Algorithm X.2 lies in the fact that Algorithm X.l successively produces graphs H with an increasing number of 2-valent vertices, while in Algorithm X.2 the trail Tiis stored separately so that Gi is strictly decreasing in size. Thus, Fleury's algorithm seems more suitable from a practical point of view when compared with the Splitting Algorithm. However, the choice of ei+l in Step 1 (1.1) of Algorithm X.2 corresponds precisely to the application of the Splitting Lemma, provided ei+l is not an end-edge of Gi.For, if we view the graph H of Algorithm X.l as the edge-disjoint union of Tiand G iof Algorithm X.2, then H I j , j E {2,3}, is disconnected if and only if f j is a bridge of G ibut not an end-edge. On the other hand, the Splitting Lemma guarantees that Hl,3 is connected if is not. Thus the choice of ej+l in Step 1 (1.1) of Algorithm X.2 is always possible, and therefore the Splitting Lemma ensures that both the

X.l. Algorithms for Eulerian Trails

x.3

Splitting Algorithm and Fleuy ’s Algorithm work. Incidentally) the trails Ti constructed in Algorithm X.2 are frequently called isthmus-avoiding because of the choice of ei+l (see, e.g. [TUTT66b, p.40-411). Despite its practical shortcomings, the Splitting Algorithm can easily be modified so as to obtain an algorithm for P(G)-compatible eulerian trails (and thus for eulerian trails in digraphs as well), or to obtain an algorithm for non-intersecting eulerian trails of a graph G embedded in some surface (and thus for A-trails if A(G) 5 4). All it takes is to restrict appropriately the choice of f2, f3 in Step 1 (1.1) of the Splitting Algorithm. It is therefore left as an exercise to derive such algorithms. However, developing an algorithm for X (D)-compatible eulerian trails in an eulerian digraph D requires the application of the Double Splitting Lemma (see Corollary VI.15 and its proof). As for developing an algorithm for eulerian trails in a mixed graph H , one can first produce an eulerian orientation D, of H (which can be done in polynomial time - see the outline of the proof of Theorem IV.ll based on flow theory), and then find an eulerian trail in D,. We also leave it as an exercise to develop algorithms for finding X(D)-compatible eulerian trails and eulerian trails in mixed graphs.

As for the complexity of the Splitting Algorithm (and thus of Fleury’s Algorithm as well - see the relation between these two algorithms, discussed above), its running time is at most O(p - q). For, deciding the connectedness of a graph can be done in O @ ) time. This and the Splitting Lemma ensure that the connectedness of H € {Hl 2 , Hl,3} can be decided in O(p) time. Since the splitting operation i i applied if and only if d(v) > 2, and since the splitting operation at u decreases d(v) by 2, it follows that deciding connectedness has to be performed at most CVEV(G) i(d(v) - 2) = q - p times. This gives the above bound O ( p . q). Consequently, the running time of the more special algorithms derived from the Splitting Algorithm is also no worse than O(p - q). However, employing parallel processing one can improve this upper bound (see below). As for X(D)-compatible eulerian trails and eulerian trails in mixed graphs, the complexity study of the corresponding algorithms is part of the Exercises X.2 and X.3. We note an interesting feature of Fleury’s algorithm when adapted for eulerian digraphs, which relates Fleury’s algorithm to Theorem VI.33. Theorem X.3. Let D be a weakly connected eulerian digraph, let Fleury’s algorithm be adapted for D (with bi and Di in place of ei and

X.4

X. Algorithms: Eulerian Trails, Cycle Decompositions, Mazes

G;, respectively), and let vo,Tibe defined as in Fleury’s algorithm. The following is true.

1) If b j E A(D,)nAui is a bridge (even if it is an end-arc), then bi E A:, and bj is uniquely determined. 2 ) For every i, if we mark bi E A ( D i ) n A,; provided

bi is a bridge of

Di, then the marked arcs of D induce an in-tree B rooted at uo (note that the same arc may be marked more than once). Moreover, the arcs of B appear as the ‘last arcs’ in the eulerian trail constructed by Fleury’s algorithm in the course of which the arcs have been marked. The proof of Theorem X.3 is left as an exercise. We note, however, that Theorems X.3 and VI.33 offer structural insight into Fleury’s algorithm and yield another explanation why this algorithm works. On the other hand, Theorem VI.33 can serve as the basis for an algorithm for eulerian trails in a digraph D : in the first step one has to construct a spanning in-tree Do rooted at a chosen uo E V ( D ) ,while in the second step the construction of a ( D - Do)-favoring eulerian trail takes place. The following algorithm is based on Hierholzer’s original paper (see Chapter 11). In fact, this paper can be viewed as an outline of the algorithm. This is also why I speak of Hierholzer’s Algorithm.’) Since it works faster than any of the preceding two algorithms, two formulations of this algorithm are presented here: the first formulation follows the style of the formulation used in preceding algorithms, the second is more formalized so as to estimate better the algorithm’s complexity.

Algorithm X.4. Hierhoher’s Algorithm. Step 0. Given a connected eulerian graph G, choose a starting point uo E V ( G ) .

Step 1. Produce a closed trail To starting at uo by traversing at each step any edge not yet traversed. To ends at uo with EVo& E ( T o )since G is eulerian. Set i = 0. 2, Possibly I have been the first t o do so, [FLEI83b], but I am definitely not the only one; see, e.g., [JUNG87a, p.39-411. In contrast, R.C. Read (to whom I owe many thanks for his elegant formulation of the more formalized version of Hierholzer’s algorithm) once attributed this algorithm t o Euler, admitting that he had no knowledge of Euler’s original paper at that time, [READ62b, p.511. However, this erroneous belief seems to have been upheld well into the 1980’s, among most graph theorists outside the German speaking countries and Hungary.

X.l.Algorithms for Eulerian Trails

x.5

Step 2. If E(Ti)= E ( G ) ,go to Step 4. If E(Ti)f E(G), choose v ; + ~ E V(Ti)such that E,i+l - E(TJ # 0 (this zli+l exists since G is connected). Produce a closed trail Ti’ as in Step 1 starting at ui+l in the component of G - E(T;)which contains w ; + ~ (T;’ exists since G - E(T;)is eulerian). Step 3. Produce a closed trail Ti+l such that E(Ti+,) = E(Ti)U E(T,I) by starting at w,,, going in Tito branching off into T;’,and running through 2”;’; after reaching ui+l in T;’ for the last time, finish the run through Ti.3) Set i = i 1 and go to Step 2.

+

Step 4. 2’; is an eulerian trail of G. The formulation of Algorithm X.4, however, is not specific enough to recognize that this algorithm is better, for example, than the Splitting Algorithm. The reason lies in the second part of Step 2 of Algorithm X.4 where no indication is given of how to find T J ; + ~for which E,;+l -E(Ti) # 8. 111 fact, it all depends on how the graph has been stored and on the use of pointers (a common tool in searching efficiently for given Let G be a simple connected eulerian graph.5) Assume “the graph to be stored as a number of adjacency lists - one for each vertex - each listing the adjacent vertices.” Denote these lists by L ( z ) , z E V ( G ) . The passing of an edge e = uv from 21 to w by one of the subtrails 7’; (see Algorithm X.4) amounts to deleting in L(u) the entry u and in L(u) the entry u. For this deletion procedure it is “sufficient to overwrite the entry [uin L ( u ) , u in L ( v ) respectively] with the last vertex in the list [ L ( u )L, ( v ) respectively], and then decrease by one the length of the list, that is, decrease by one the degree of the vertex”. As for the simultaneous deletion of u in L ( v ) ,it is “necessary to have a pointer to the u-entry in 3, That is, Ti+,results from a &absorption at w ~ + involving ~ the first transition of Tiand the last transition of T;’. 4, For the following I rely on a letter by R.C. Read in which he developed the following “. . .‘pidgin-algol’ listing of [Hierholzer’s] algorithm. . .” (this and the other phrasings put in quotation marks are taken from R.C. Read’s letter). For another specification of Hierholzer’s algorithm, see [EVEN79a, p.7-81.

5 , This assumption implies no real restriction since S(S(G)) is a simple graph for any graph G.

X.6

X. Algorithms: Eulerian Trails, Cycle Decompositions, Mazes

the v-list, and vice versa.” With these data structures given we present R.C. Read’s ‘pidgin-algol’ formulation of Hierholzer’s algorithm.

Algorithm X.4’. u is any vertex of G. HEAD and TAIL are stacks. Initially HEAD = ( u } , TAIL = 8. While HEAD# 0 do begin while degree of top vertex, u, of stack is > 0, do begin Let v be a vertex adjacent to u. Add v to HEAD. (v becomes new u). Delete edge uv from G. Decrease degrees of u and v by 1. end. while HEAD# 0 and top vertex u of HEAD has degree 0, do begin remove u from HEAD. add u to TAIL. end. end. Euler tour is in TAIL. To see that the time and space comp,dxity of t is algorithm is O(q), observe first that the search for v E N ( u ) is easy - just take the first entry in L(u). For, at any stage the first entry of L(u) is a vertex v for which uv has not been passed yet. Moreover, the continuation of the subtrail Tifrom v does not require any search procedure since v is on top of the stack HEAD (thus HEAD contains at most q objects at any given time), and L ( v ) is immediately accessible. Second, if the construction of Tiis completed, i.e., if Tiends at its initial vertex u after having used all entries of L(u),this vertex u is removed from HEAD as the top vertex of this stack and put on top of the stack TAIL. Then Tiis traced backwards by successively stacking the top vertex of HEAD on top of the stack TAIL until a top vertex w of HEAD is reached such that d(w) > 0. This is the only part of the algorithm which could absorb time unnecessarily. For, in this backtracking procedure one has to check at every top vertex z of HEAD whether d ( z ) = 0 or d ( z ) > 0, and this could cost time if one has t o search for d ( z ) first ( d ( z ) = IL(z)I since G is simple). However,

X.l. Algorithms for Eulerian Trails

x.7

given an appropriate storage of z and L ( z ) respectively, one reaches d ( z ) instantaneously so as to perform the comparison d ( z ) = 0 or d ( z ) # 0. At starts and ends. Then the above w for which d ( w ) > 0 the tracing of T;’ w is put on top of TAIL, and so forth. Thus, at any given time TAIL also contains at most q objects. Finally, an eulerian trail T has been read into TAIL in reverse order (‘reverse’ with respect to the orientation of the edges induced the first time an edge is used by the algorithm). Thus, reading T in TAIL from top to bottom this eulerian trail appears as a sequence in accordance with the orientation given by passing the edges in the construction of the corresponding Ti. This fact can be used to modify Algorithm X.4’ for digraphs. We leave it as an exercise to do this. Summarizing the preceding considerations on the working of Algorithm X.4’ we can say that every edge e is considered no more than three times: the first time in producing the corresponding subtrail Ticontaining e; the second time in the backtracking procedure searching for w with d(w) > 0 (if such exists); and finally in the reading of T in TAIL from top to bottom. Moreover, Step 3 of Hierholzer’s AZgorithm, i.e., the applicadoes not require any tion of a rc-absorption to Tiand T;’t o obtain Ti+,, particular effort inasmuch as this rc-absorption is implicitly contained in the backtracking procedure and in the way sections/segments of T are stacked in TAIL. In fact, this analysis shows that Algorithm X.4’ is linear. Moreover, a glance at the Edmonds-Johnson paper [EDM073a] reveals that the Nest-Node Algorithm combined with the Muze-Search Algorithm, as presented there, are - in content - basically the same as Hierholzer’s algorithm. The same holds true for the End-Pairing Algorithm established in that paper. We note that a generalization of the Maze-Search Algorithm will be studied in the next section. However, Algorithm X.4’ has been formulated for simple graphs only (which, by the preceding footnote, implies no loss of generality). Nevertheless, if one wants to avoid this restriction without using a subdivision graph, then one can modify Algorithm X.4’ by using incidence lists instead of adjacency lists. We leave it as an exercise to do this. We note in passing that Hoang Thuy’s algorithm as presented in [SYSL77a, p.121 can be viewed as a special case of Hierholzer’s algorithm inasmuch as it first constructs cycles Ciinstead of subtrails Ti, and then successively applies tc-absorptions so as to form an eulerian trail. Clearly, because it uses cycles instead of arbitrary subtrails in general, this algorithm cannot be as fast as Algorithm X.4’.

X.8

X. Algorithms: Eulerian Trails,Cycle Decompositions, Mazes

Another algorithm for constructing eulerian trails has been developed by A. Tucker, [TUCK76a]. It can be viewed as a combination of the Splitting Algorithm and Hierholzer's Algorithm (or - put differently - as lying between the End-Pairing Algorithm of Edmonds and Johnson and Hoang Thuy's algorithm). The idea is first to produce a trail decomposition S = {T,, . . .,T,; k 2 1) for the connected eulerian graph G by pairing arbitrarily the elements of E: for every ZI E V ( G ) ,and then to produce an eulerian trail by a sequence of tc-absorptions applied to pairs of connected subgraphs of G having a vertex in common and being induced by subsets of

s.

Algorithm X.5. Tucker's Algorithm. Step 1. Given the connected eulerian graph G form a 2-regular detachment G, from G by splitting away adjacent pairs of half-edges as long as possible. Label the vertices of G, with the same symbols attached to the corresponding vertices in G. Set i = 1 and let c, denote the number of components of Gi. Step 2. If ci = 1, set Ti= Gi and go to Step 4. If c, # 1, find two components Tiand T;' of G, such that vi+, E V ( T i )n V(T,')exists. Form a closed trail Ti+, by a tc-absorption at vi+, applied to Tiand 7';'. Step 3. Viewing Ti+, as a graph define G,+, = (Gi - (2'; U T;'))U Ti+l. Set i = i 1, and go to Step 2.

+

Step 4. Tiis an eulerian trail of G. It should be noted that once c, has been determined, the determination of ci, i > 1, requires no effort. For, owing to Step 2 and the definition of G,+l in Step 3 it follows that ci+l = ci - 1. Also, the construction of G, can be done instantaneously; and in the case ci+, # 1 one can simply remain with Ti+1constructed in Step 3, and f k d Ti+,such that vik2 exists. However, a search for Z I ~ +can ~ be avoided by using pointers again; they can be installed easily in the course of the construction of G,. Thus Algorithm X.5 is also linear. It is left as an exercise to work out the details of this argument and to produce a more formalized version of Algorithm X.5 analogous to the deduction of Algorithm X.4' from Algorithm X.4. We note in passing that [ABRA67a] contains a special version of Hoang Thuy's algorithm; it constructs eulerian trails in K p ,where p is a prime,

X.l. Algorithms for Eulerian Trails

x.9

9

starting from a decomposition of K p into hamiltonian cycles. However, starting with Hierholzer's algorithm, all these algorithms have in common the following basic idea on constructing eulerian trails. Namely,

given a trail decomposition S of the connected eulerian graph G, and considering a spanning tree B in the intersection graph I(S), an eulerian trail of G can be constructed from S by a set of &-absorptions, each of which corresponds to an edge of B and vice versa. (TE) We note in passing that statement (TE), being the essence of [AWER84a, Main Theorem], serves as the basis for a parallel algorithm developed in this paper. This algorithm requires q processors, uses a parallel connectivity algorithm and has a running time of O(Eog(q)).In presenting this algorithm and in commenting on it, we follow the cited paper.

Algorithm X.6. Let D be a weakly connected simple eulerian digraph. Step 1. Produce a trail decomposition S of D by choosing for every a- E A; some a+ E A:, 'u E V ( D ) ,such that a: # a: if a; # a;. Step 2. Describe the elements of S by telling each a E A ( D ) to which TiE S it belongs. Step 3. Determine a spanning tree B of the graph G, defined as follows: V ( G , ) = S, and for every a, b E A:, v E V ( D ) ,introduce an edge of the form TiTi f E(G,),provided a E V(T,),b E V ( T j ) ,where Ti,"' E S; possibly Ti =.''2 Step 4. For every e E E ( B ) perform a K-absorption corresponding to e .

In the implementation of Algorithm X.6 D is given by listing AS and A; for each v E V ( D ) .Moreover, q = qD processors are used, one processor assigned to each a E A ( D ) , carrying out the operations in which a is involved. Let L$ = ( a t , . . . ,a t ) and L; = (a;, . . . ,a i ) denote the lists corresponding to A$ and A , respectively, where k = od(v) = i d ( v ) . In carrying out Step 1 one defines t i ( v )= { a t , a;}, i = 1,. - . , k. This can be done simultaneously for all elements of A$ and all v E V ( D ) ,since we have q processors at our disposal. Thus Step 1 can be executed in constant time. Thus, one obtains a system of transitions X, of the trail decomposition S defined by the above t j ( v ) . In executing Step 2 one first produces an auxiliary graph G by defining

V ( G )= A ( D ) , a1a2E E(G)+,{ a 1 , a 2 }E X,

.

X.10

X. Algorithms: Eulerian Trails, Cycle Decompositions, Mazes

It follows that the components of G correspond bijectively to the elements of S (Exercise X.8). Thus, after applying a parallel connectivity algorithm of O(Zog(q)) running time one knows for every a E A ( D ) the corresponding '2'; E S for which a E 2';. However, in executing Step 3 one could run into difficulties if one deals with the whole of G, since 0 ( q G K )can be as large as O(&). On the other hand, since we need just one spanning tree of G, it suffices to deal with a connected spanning subgraph G C G, such that i j :=I E(G)I is comparatively small. We first observe that for the transitions ti(v),t j ( v ) E X,(v), the definition of a new set of transitions at v, X * ( v ) ,given by

x*(v>= (x,(v>- {ti(v),tj(v)})

u { { a t , a j - } ,{af,af}}

corresponds to a tc-absorption if and only if for Tl _> t i ( v ) and T, 2 tj(v), T,,T,,, E S, it follows that I # rn. In other words, the loops of G, are precisely those edges which correspond to tc-detachments. Now define

G=(

U

E E ( G , ) / t i ( u )G ~

j ti+l(v> ,

c'1

vEV(G)

forsome

ZE{I,

...,od(v)-l}}

We leave it as an exercise to show that G is a connected spanning subgraph of G, (the connectedness of G, itself follows from the strong connectedness of D ) . Now ij = Cv,v(Dj(od(v)- 1) = qD - p D . With the help of a parallel connectivity algorithm, one can then find the spanning tree B C G & G , in O(Zog(ij)) time. Finally, Step 4 can also be done in constant time since Step 4 can be executed such that every arc of D is involved in at most two changes of transitions. Namely, if ti,(v),. . . , ti,(v), 1 5 i, < . . . < i, 5 od(v), are the transitions involved in defining the edges of B, then t i j ( . ) = 1 5 j 5 T (putting T 1 = 1) are transitions corresponding to a sequence of tc-absorptions at v (the other transitions at v remain unchanged). Thus Step 4 can be performed simultaneously at every v E V ( D ) . It is left as an exercise to work out the details of the above argument and to show that this way of executing Step 4 yields an eulerian trail of D.

+

{~:,UG+~},

We note in passing that Algorithm X.6 can be extended to graphs G; this can be done by first finding an eulerian orientation DG of G in logarithmic

X.l. Algorithms for Eulerian 'Ikails

x.ll

parallel time and then apply Algorithm X.6 to D , (see [AWER84a, 4.0 and 4.11). In order to obtain an algorithm which produces all eulerian trails of a connected eulerian graph, we employ the Splitting Algorithm. Of course, owing to the exponential growth of the number N E := II(G)I of eulerian trails (see Proposition IX.45), we cannot expect to obtain a fast algorithm. Anyway, recall that for a graph H , H l , j , j = 2,.. . ,d( v ) , denotes the graph obtained from H by splitting away el and e j at v. Algorithm X.7. All-Duils Algorithm. Let G be a loopless connected eulerian graph. Step 1. Set 7 E = {G}. Step 2. If no H E IEwith A ( H ) # 2 exists, go to Step 6. Otherwise, choose v in H with d(v) # 2. Let e l , . . . ,e2k be the edges incident with 2). Form Hl,j for j = 2,. . . , 2 k = d(v). Step 3. Let j, be the least j , if any, for which Hl . is disconnected. If no such j exists define M ( H ) = {Hl,j/j = 2,. . . , 2 i j and go to Step 5. Step 4. Define M ( H ) = { H l , j / j = 2,. Step 5. Set 7 E = 7

E

. . , 2 k ; j # j,}.

u M ( H ) - { H } , go to Step 2.

Step 6. 7Eis the set of all eulerian trails of G.

In order to make sure that one does not need a search procedure in Step 2 one only has to observe that if one is a cycle, then all admissible Hl,j (namely two) are (in this casej, as defined in Step 3, exists). In this case, put these two at the end of the list representing IE(see Step 5). The is a cycle is easily performed by associating with H testing of whether the parameter a ( H ) = C ( d H ( v ) - 2 ) = 2 ( q H - p H ) where the summation can be taken over all v f V ( H ) - V2(H);c ( H ~ , = ~ )c ( H ) - 2 follows. Moreover, if H = Hl,j is not a cycle, the choice of v E V ( H )- V2(H)can be made instantly with the help of pointers in a way similar to Algorithm 4'. Thus either the first element of 7 E has a vertex which is not 2-valent, or else IEis the set of all eulerian trails of G (Step 6). Consequently, the only time consuming procedure is Step 3. For, although the Splitting Lemma applied to eulerian graphs tells us that Hl . is connected for $3. j # j , if j , exists, we do not know for which j Hl,j is disconnected, if such Hl,j exists at all. However, with the help of parallel processing the

X.12

X. Algorithms: Eulerian Trails, Cycle Decompositions, Mazes

construction of the graphs and the testing of being connected or not can be performed simultaneously, so Step 2 and Step 3 can be executed in O(log(q))time (see the discussion of Step 3 of Algorithm X.6). Thus the only decisive factor concerning the running time of the algorithm is the size of := 7- - { H E I E / A ( H )= 2). Next we discuss the set 'TE. Actually, the find set 7 E is (strictly speaking) not a set of eulerian trails; rather, it is the set of all connected 2-regular detachments of G. What , induced transition is important, however, is that for H , H' E 7 Ethe systems X H and xH,of G satisfy x, # XH!. That is, not only does a run through such 2-regular H describe an eulerian trail of G but we also have I IEI=] I ( G ) I owing to Definition VII.l and the construction of the final set 7E. Whence Algorithm X.7 does what it promises if we perform a run through each of the two 2-regular Hl,j E M ( H ) and add in Step 5 the corresponding sequences to 7 E instead of M ( H ) as defined in Step 4. The construction of these sequences can be done while Step 2 is repeated, since 2-regular H,,j are put at the end of the list of 'TE(see above). To cut down the size of IEone can introduce another set S, to which one adds the 2-regular elements of 7Estep by step as they are I while I S, I increases. created. This eventually leads to a decrease of I 7E However, to avoid an increase of I S, I one can use S, as the place where the eulerian trails corresponding to the 2-regular Hl,j E 'TE are actually described and then printed out. But even with this modification 'TE can grow exponentially depending on the choice of various vertices 2, E V ( H ) - V,(H) at which the splitting operation is performed (see Step 2). However, a further analysis shows that the ever-growing set 'TE (aslong as the second part of Step 2 can be performed) can be interpreted - at any stage - as a partition of the set I ( G ) of eulerian trails of G. Namely, if we denote by X" the set of transitions stemming from the various splitting operations performed in Step 2 and yielding H from G, then H corresponds to the set 7 ( G , X H )C_ I ( G ) of all eulerian trails T satisfying X" X,. Consequently, M ( H ) corresponds to a partition of 7 ( G ,X"). Altogether, the sequence of sets 'TE generated by Algorithm X.7 then corresponds to a partial order 0 by inclusion of subsets of I ( G ) with minimal element 7 ( G )and maximal elements {T},T E I ( G ) , and where 7 ( G ,X") is immediately preceding 7 ( G ,X " ' ) if and only if X H c XHr and lXHr- X" I= 1. With this structural insight one can modify the choices of 21 E V ( H ) V,(H) in Step 2 in order to cut down the growth of 7;. Namely, choose an initial vertex u0 and construct Ti+l as in the Splitting Algorithm

X.l. Algorithms for Eulerian Trails

X.13

or Fleury’s Algorithm, for example, but also construct the additional Hl,j at every vertex reached, according to Step 2 of Algorithm X.7, and perform also Steps 3,4,5 of this algorithm. When the construction of Tj+l stops, i.e., T,,, = T E 7 ( G ) ,then the corresponding set 7; in Step 5 of Algorithm X.7 contains the 2-regular graph T and all those H’ which are immediately consecutive to any H arising in the construction of T , in accordance with Steps 2 and 3 of this algorithm. In terms of the partial order 0 discussed above, this means that the partial order 0’ corresponding to 7; is induced by the above partial order 0 corresponding to (the final) 7E.Observing that the structure of 0 implies that the Hasse diagram D ( 0 ) is an out-tree with root 7 ( G ) it follows that D ( 0 ’ ) C D ( 0 ) is also an out-tree with root 7 ( G ) ;and one of the (two) sinks of D ( 0 ) belonging to D ( 0 ’ ) is { T } . Moreover, the path P := P ( I ( G ) , { T } )C D(0’) joining I ( G ) to { T } is a dominating path in D(O’), and if 7 ( G , X H ‘ )E V ( D ( 0 ) )is adjacent to some 7 ( G , X H )E V ( P ) in D ( 0 ) , then 7 ( G , X H ‘ )E V ( D ( 0 ’ ) ) . This completely characterizes D ( 0 ’ ) and O‘, respectively, and shows that for 2ki = d ( v i ) =: d i , 0; E V(G), i = 1,.. . , p , P

1qq I C ( ( d ; - 2 ) + ( d ; - 4) +. . . + 2)

+1

i= I P

i=l P

That is, 7; is of polynomial size. Observing that the other sink {T}’ of D ( 0 ) belonging to D ( 0 ‘ ) also represents an eulerian trail of G, we can now backtrack on P , branching off at 7 ( G ,X H ” ) € V ( P ) ,where d ( l ( G ,X H ” ) {, T } ) = 2 (with d denoting the distance function in D ( 0 ) ) . This amounts to forming first 7; = 7; - {T,T’, Ho}, where ( I ( G ,X H o ){, T } )E A ( P ) , then considering a path P’ = P ( 7 ( G ) 7, ( G ,X H 1 ) )2 P ( I ( G ) 7 , ( G ,X H “ ) )of minimal length in D ( 0 ’ ) - { { T } ,{T’), 7 ( G ,X H o ) } ,and extending P‘ by one of the two possible arcs in D ( 0 ) (note that I V ( H , ) - V2(H,) 1 and A ( H l ) = 4,and that, in general, d ( 7 ( G ) {, T l } )= d ( 7 ( G ) ,{T,)) = q - p for any T I ,T2 E I ( G ) ) .Thus 7; is extended first by H , and then by two

I=

X.14

X. Algorithms: Eulerian Trails, Cycle Decompositions, Mazes

elements of 7 ( G )to obtain 7F;whence 17;’ I=[ 7’ I. This backtracking procedure is repeated until all those elements of 7 ( G ) have been generated which contain the first transition t , of the originally constructed 2’; this transition corresponds to the first arc of P(T(G),{T}). Now one only has to continue on D(O - 0’) by constructing some eulerian trail T* starting at vo as well and with first transition t; # t,; a.s.0. Whence we may conclude from (*) that one has to consider a set of at most C:=,ke - q 1 graphs, at any stage. Moreover, parallel processing can be used in two ways: first in producing the various eulerian trails T together with the partial orders 0,and second in the backtracking procedure leading to the various sets 7F (see above).

+

We Ieave it as an exercise to produce new all-trails algorithms based on Algorithm X.7 and the discussion following it.

X.2. Algorithms for Cycle Decompositions Observing that every eulerian graph G has a nontrivial cycle C if q # 0 and that G - C is also eulerian, one obtains the following algorithm on cycle decompositions for arbitrary simple eulerian graphs. Algorithm X.8. Let G be a simple eulerian graph. Set S = 0 and H = G - Vo(G). Assume that G is given as a set of adjacency lists L ( u ) ,u E V ( G ) ,and that pointers are used as in Algorithm 4’.

Step 1. If q H = 0, go to Step 3. Otherwise, let vo E V ( H ) be chosen arbitrarily and construct a trail To starting at vo and ending at a vertex w as soon as w is reached for the second time; To = wo, . . . , w , T,, w (possibly vo = w). Set C = w,T,,w. Step 2. Set S = S U { C } , H = H - C,qH = q H - Z(C). Go to Step 1.

Step 3. S is a cycle decomposition of G.

Of course, on the whole the algorithm’s running time depends on the length of the paths vo, . . . , w in To and on the length of the cycles constructed in Step 1. Moreover, C has been written as an alternating sequence of vertices and edges instead of as an edge sequence solely for the purpose of simplifying the formulation of the algorithm. To rewrite Algorithm X.8 for digraphs one only has to adjust the adjacency lists and pointers. As for adapting this algorithm for mixed graphs

X.2. Algorithms for Cycle Decompositions

X.15

H one can produce an eulerian orientation D, of H (if such DH exists) and apply the adjusted version of Algorithm X.8 to D H . Another algorithm can be derived by using the same basic idea employed in Algorithm X.5. That is, starting from an arbitrary trail decomposition S of the eulerian graph G one performs %-detachmentsuntil every element of (the new trail decomposition) S contains different vertices only. It is left as an exercise to work out the details of that algorithm and its modified versions. Next we develop an algorithm which can be viewed as a specialization of Algorithm X.8 and which is based on Theorem IV.l. The starting point is the construction of an eulerian trail T of the connected eulerian graph G . By running through T one can ‘peel off’ one cycle C after another by deleting certain segments of T and such that the remainder of T forms an eulerian trail of G - C. Algorithm X.9.. Trail-Conforming Algorithm. Step 0 . Let T be an eulerian trail of the connected eulerian graph G, starting and ending at vo E V(G).Set S = (0). Step 1. If A(G) = 2, go to Step 3. Otherwise, starting a run through T at vo, let w be the first vertex such that w is reached for the second time while any other vertex appears at most once in this segment C starting and ending at w. Step 2. Set G = G - C,S = S U { C } , T = T - C (in the sense that T - C = vo,. . .,ew,l,zu,ezu,2,.. . ,vo for T = vo,. . . ,e,,l,C,e,,z,. . . ,vo). Go to Step 1. Step 3. S U { T } is a cycle decomposition of G. We observe that C as constructed in Step 1 is found with the help of pointers in at most q steps (one step being the passing of an edge). Since T - C is an eulerian trail of G - C and since the construction of T requires at most O ( q )time (see above), the running time of Algorithm X.9 is at most O(q2).However, one can combine the construction of T with the ‘peeling off’ procedure to improve the running time of the algorithm (Exercise X.13). The faster algorithm obtained by simply combining the ‘peeling off’ procedure with the construction of a trail decomposition based on traversing edges rather than on a given system of transitions X , can be viewed as a modification of Algorithm X.8. We also note that

X.16

X. Algorithms: Eulerian Trails, Cycle Decompositions, Mazes

while Algorithm X.9 produces a unique S once 2ro has been chosen as the initial vertex for a run through T,choosing another initial vertex 21; may yield a cycle decomposition 5” # S. Also, the choice of w as described in Step 1 of Algorithm X.9 is not essential to finding a segment C of T such that C is a cycle and T - C is an eulerian trail of G - C. For, we can take C t o be any segment of T which contains no proper subsegment that is also a segment of T . That is, ‘peeling off’ a cycle C such that T - C is an eulerian trail of G - C can generally be performed in many ways. Hence this ‘peeling off’ procedure, viewed as a general method for constructing cycle decompositions, associates many cycle decompositions with one eulerian trail. This observation applied to Sabidussi’s Compatibility Conjecture (which states that given an eulerian trail T in the connected eulerian graph G without 2-valent vertices, a cycle decomposition S exists such that X , n X, = 0) yields an approach to this conjecture. Namely, by ‘peeling off’ cycles from some eulerian trail T‘ which is compatible with T , one obtains a cycle decomposition S‘ which is compatible with T (but not with TI, of course). That such T’ exists if the above conjecture is true will be shown in the treatment of the Compatiblity Problem. Having secured the existence of eulerian trails T‘ compatible with T (Corollary VI.5), we note, however, that not every T’ compatible with T will yield S’ compatible with T (where S’ is obtained as described above). We shall also show by example that while a cycle decomposition S compatible with T may exist, no subset So of totally disjoint cycles will have the property that the system of transitions induced by T in G - UCESoE(C)defines an eulerian trail of this graph. Thus the ‘detour’ via TI compatible with T is (almost) unavoidable unless one generalizes Sabidussi’s conjecture. We finish this section with a short description of an algorithm for the construction of the set S(G)of all cycle decompostions of an eulerian graph G, [LING82a]. First, assume the edges of G labeled el,.. . ,ep. Choose an arbitrary e E E(G), e = e 11 . - A first subroutine produces in lexicographic order (with respect to the edge labeling) the set C(e) of all cycles containing e . A second subroutine produces for CiE C(e) the set Siof all cycle decompositions of G which have Ci in common. This is achieved by determining S(G - Ci)and forming

X.3. Mazes

X.17

Noting that Sj t l Sj= 8 for i # j, 1 5 i, j 51C(e)

I= ye it follows that

7 c

S(G)= U S i

.

i=l

Unfortunately, [LING82a] does not contain a complexity study, the main problem apparently being that no estimates on I S(G) I analogous to Proposition IX.45 seem to be known. Note that we have I S(G)I= 1 for a connected graph G each of whose blocks is a cycle, while I I ( G ) I can be arbitrarily large. On the other hand, the above algorithm, by its very structure, is a good example where parallel processing can essentially shorten the running time. It may thus be a worthy research problem to estimate the number min{y(e)/e E E ( G ) } for a 2-connected, 2lc-regular, loopless/simple graph, for example.

X.3. Mazes The history of mazes and labyrinths respectively and the development of the first escape algorithms are at least as old as Greek mythology itself. For, as the story goes, Theseus‘) used a thread given to him by Ariadne to track down the ‘maneating’ Minotaur in a labyrinth and - after killing the creature - to find his way out again. 7, Of course, mazes (labyrinths) can be defined in different ways; e.g., as a system of catacombs (in ‘real life’), or (mathematically) as a set of unit squares in the euclidean plane (see [SEND72a, DOPP7la]), or as some other geometrical configuration. However, for our purposes, a maze is a graph G (digraph D,mixed graph H ) which admits a covering walk W . Thus, G ( D , H ) must be (strongly) connected. In fact, the maze search problem can be formulated in the following way (unless stated otherwise, we restrict ourselves in the sequel to graphs): 1‘ one of the early supermen. 7, For an early account of various mazes, see [LUCA82a, p.41-551 which also contains TrCmaux’s algorithm (see Algorithm X . l l below). Other references to the history of mazes can be found in [ROSS82a, p.7891 which discusses in detail the labyrinth of Saint-Quentin, France. [KONI36a, p.351 contains several references which treat mazes as part of recreational mathematics.

X.18

X. Algorithms: Eulerian Trails, Cycle Decompositions, Mazes

Describe a general algorithm which constructs a closed covering walk W in a connected graph G such that, in the course of constructing W, this algorithm can only handle local information available at any vertex reached by W. W P ) That is to say that if we double the edges of G to construct an eulerian trail T in the new graph G, so that T corresponds to a double tracing W in G, then neither the Splitting Algorithm nor F l e w ’ s Algorithm applied to G would be a suitable algorithm, since each of them needs t o decide whether certain derived graphs are connected (which is global information available for G only). However, as we shall see below, Hierholzer’s Algorithm can indeed be viewed as one of the keys to the solution of (MSP); another key is Theorem VI.33. There are three classical algorithms attributable to Wiener, Tr6maux and Tarry respectively. They vary according to the available local information. In presenting them we follow [KONI36a]. However) there these algorithms are presented in a more descriptive way. Algorithm X.10. Wiener’s Algorithm.8) This algorithm is based on Theseus’ walk in which Ariadne’s thread is used. Thus one operates with the following Hypothesis. At any vertex v E V ( G ) reached in the course of the walk W, the set E, n E ( W ) is known (W denotes algorithmically the walk performed upon reaching v). Moreover, it is assumed that W-’ can be performed (this corresponds to the use of Ariadne’s thread).g) Step 0. Set i = 0 and choose

2r0

E

V ( G ) . Set W = vo.

According t o D. Konig, Chr. Wiener was the first t o treat mazes from a mathematical standpoint. For the historical record we also observe that

Wiener’s article [WIEN73a] immediately precedes the printing of Hierholzer’s work which, based on a talk by Hierholzer, was formulated and submitted by Wiener after Hierholzer’s death (cf. [FLEISOd, p.II.20]).

To perform W-l one may just label the half-edges in W as follows: if e E E, f l E, is the i-th edge passed in W and if it is passed from x t o y , then label e ( x ) with and e(y) with i’, thus assigning implicitly an orientation t o e (and this is, t o some extent, the way Wiener originally phrased his algorithm; namely, “One therefore marks the path passed, including the direction in which it is passed”). However, this use of labels for half-edges yields more local information than Ariadne’s thread; this becomes clear if one looks at the local information used in Trkmaux’s and Tarry’s algorithms below. ’)

X.3. Mazes

x.19

Step 1. Beginning at vUiE V(G) walk along an arbitrary ei E E,; E(w).~') Set = W,e i , vi+, (eiE E , ~ + ). ~Set i = i 1.

w

+

Step 2. Suppose W = v,, e,, . . . ,eUi-,,vi has been constructed. If v i is not an end-vertex and v h # vi for 0 5 h < i ,lo)go to Step 1. Otherwise, go to Step 3. Step 3. Consider the maximal j < i satisfying E,.- E ( W ) # O.ll) If j does not exist, set W = W,W-' and go to Step 5. Step 4. Define ei+k-l ._ .- ei-k and vi+! := vi-k, k = 1,.. . ,i - j. Set W = W , e j , .. . , e 2 j - j - l , v 2 i - j . Set i = 22 - j . Go to Step 1. Step 5. W is a closed covering walk in G.

To see that the algorithm terminates with a closed covering walk of G we consider G, := ( E ( W ) ) ,, where W is the last walk constructed by the algorithm. If E(G,) # E(G), then the connectedness of G implies that for some i, EVi - E ( W ) # 8. Choosing the maximal j = i with this property, it follows from Step 2 and Step 3 that Step 1 can be performed again (either directly, or indirectly via Step 4, depending on the position of vj in W and whether it has been reached before by W). Thus G, = G must hold; i.e., W is a covering walk in G. Assuming now that W is not a closed walk it follows that 21; # v, has been reached by W after having passed all edges of G at least once. That is, backtracking W in Step 3 shows that E, - E ( W ) = 8 for every v E V(G). But then the definition W = W,W-l in Step 3 yields a closed covering walk of G. Thus we conclude that the algorithm does what is said in Step 5. However, Wiener's algorithm is not very efficient in that the final W may use edges even more than twice.12) In fact, the other two algorithms (attributable to Tr6maux and Tarry respectively, and presented next) work faster than Wiener's algorithm. lo)

This can be decided given the first part of the Hypothesis.

The existence of j can be decided in the light of the second part and then in the light of the first part of the Hypothesis. 12) It is apparent that Wiener did not make use of Hierholzer's work in the most economical way. Although Wiener does not mention that his work was influenced by Hierholzer's, it is hard to believe that it was not.

X. Algorithms: Eulerian Trails,Cycle Decompositions, Mazes

X.20

Algorithm X.ll. Freinaux’s Algorithm. This algorithm operates with local information only; it is not assumed that W-’ is known for whatever W has been constructed algorithmically. Hypothesis. A t any vertex u E V ( G ) reached an the course of the construction of the walk W, the number Aw(e) is known f o r every e € E,. The direction in which an edge is being passed need not be known.13) Step 0, Step 1 and Step 2 are the same as in Algorithm X.10.14) Step 3 (it follows that ui is an end-vertex or ui= u j for some j < i ) . If XW(ei-,) > 1, go to Step 4. If Xw(ei-,) = 1, define ei := ei-,, ui+l := ui-, and set W = W,e i , vi+,. Set i = i 1 and go to Step 4.

+

Step 4. If Aw(e) > 1 for every e E E V i go , to Step 5 . Otherwise, choose e E EVi such that Aw(e) is minimal. Define ei := e and wi+, := y, where y = wi if e is a loop, e E Ey n EVi otherwise. Set W = W , e i ,ui+, and i = i 1. Go to Step 2.

+

Step 5. W is a bidirectional double tracing in G.

To see that the final W in Trkmaux’s algorithm is a bidirectional double tracing we first observe that

no edge of G is used b y W more than twice;

(*> this follows directly from Step 4. On the other hand, the first time Step 3 is performed it follows that Xw(ei-,) = 1. Thus we conclude from (*) that the final W is of the form

where ui-,= wi+, =: z,e;-, = ei =: e; hence

W , = . . . ,e i - 2 , z , q + , , . . . is a walk in G, := G - e. It now follows from Step 2, Step 3 and Step 4 that W , is a final walk in G, constructed according to Trkmaux’s 13) This is precisely the local information inherent in the use of Ariadne’s thread. 14) The local information used in Step 1 and Step 2 can be decided from the hypothesis. The same is true concerning Step 3 and Step 4 below.

X.3. Mazes

x.21

algorithm (the case-by-case considerations of this argument are left as an exercise). Noting that this algorithm yields a bidirectional double tracing whenever G is a cycle or a path, and applying induction we conclude that W , is a bidirectional double tracing. By the above construction, W, is bidirectional if and only if W is also bidirectional. Whence we may conclude that TrCmaux’s algorithm yields a bidirectional double tracing of G in general. We observe that Trachtenbrot’s algorithm as presented in [TRAC63a] is basically the same as TrCmaux’s algorithm, the only exception being that it terminates as soon as the Minotaur has been found; i.e., it stops at a specified vertex 5 # so of which Theseus has no a priori knowledge. That is, if G is disconnected and if u,,x belong to different components C,and C, respectively, then Ttachtenbrot’s algorithm acts on C, I)(u,) in the same way as TrCmaux’s. We note that this affinity between the algorithms of Trkmaux and Trachtenbrot was already observed in [ROSS7la]. Algorithm X.12. Tarry’s Algorithm. This algorithm also works with local information only. Hypothesis. At any vertex u E V ( G ) reached in the course of the construction of the walk W, the set EZ,w C E, of edges already passed f r o m v is known. Moreover, the edge ein(u) b y which v has been reached f o r the first time i s known. FOT the initial vertex u, of W we set {ei,(vo)} = 8. Step 0. Set i = 0 and choose u, E V ( G ) . Set W = 2ro. Step 1. Beginning at u j E V ( G ) walk along an arbitrary ei E k,; := E~,-(E,Oi,,u{ei,(ua)}). Set = W , e i , u i + l(ei E Evifl).Set i = i+l.

w

Step 2. Suppose W = u,,e,, . . . ,ei-l, 2ri has been constructed. If R,; # 0, go to Step 1. Otherwise, go t o Step 3. Step 3. If {ein(ui)} C EZ,,w go to Step 4. Otherwise, set ei = ein(ui), W = W,e i , uj+l ( e i E E , ; + ~ )i, = i 1; go to Step 2.

+

Step 4. W is a bidirectional double tracing in G.

In order to see that the final W is a bidirectional double tracing, we apply Theorem VI.33 and its proof technique in the following ‘way. Let D , be an orientation of G such that for e E E ( W ) , a , E A: n A; c A(D,) if and only if e E E, n E, is passed by W the first time from x to y, whereas if e $ Ei ( W ) ,let a , be one of the two possible orientations of e (in both cases, if e is a loop let a , be one of the two orientations of e ,

X.22

X. Algorithms: Eulerian Trails, Cycle Decompositions, Mazes

where we assume the topological point of view concerning loops). Define the strongly connected eulerian digraph

and consider for the final W Din:= ( { a , E A(D,)/e = ejn(v),V E V ( W ) } ) Dw:= ( { a e , / e ; E E ( W ) } ) 7 D , := ({a: E A ( @ ) / u e E A ( D i n ) } )=

06

7

,

where the orientation of a,, is defined by the direction in which e; is passed by W , and IA(Dw)I = Z(W). Denote by Tw the covering trail of D& which corresponds to W-’ in the natural way. Because of Step 1 and Step 3 we have Aw(e) 5 2 for every e E E ( G ) . By the same token, Aw(e) = 2 implies for e = ei = e j , i # j , that { a e , , a,,} = { a e ,a:}. Thus, D , D = D R 2 D E . Moreover, it follows from Tarry’s algorithm that

W is a closed walk which uses every e E EUoonce in each direction. (*) W-hence D , and D& are weakly connected eulerian digraphs, and thus T , E q,(D;). We want to show that D , = D and that D, is a spanning in-tree of D R = D rooted at uo. By definition of Din and D, and because of Step 3 (in which k,,. = 8 because of the second part of Step 2), it follows that D, is the diiraph induced by the ‘last’ arcs of T , (see the proof of Theorem VI.33), and thus D, is a spanning in-tree of D& rooted at uo and therefore, Din is a spanning out-tree of D , rooted at uo .

(**I

Whence it suffices to show that D& = D. This will imply D , which in turn classifies W as a bidirectional double tracing in G.

= D

Suppose D& # D. Since D E c D and both D E and D are weakly connected eulerian digraphs, it follows that D := D - D& is eulerian and A ( D ) # 8. Moreover, because of (*) and (**) it follows for some u E V ( D w )- {wo} = V ( D $ ) - { u o } that # 8. Among all possible choices for this 21 let u* be such that dDl (v*, uo) is minimum. This choice implies that the paths P ( ~ I * , 2 ~ ID~, )and P(uo,u*) 2 Din belong to D&. This and # 8 means, however, that in G some edge e E

X.3. Mazes

X.23

E,,,- {ei,(v*)} has not been passed by the final W from v*, while W does so with respect to e = ein(v*). This violates Step 2 of Tarry’s algorithm (since # 8 upon reaching v* by W just before passing ein(v*) from v*). Whence we may conclude D = D g ;thus Tarry’s algorithm produces a bidirectional double tracing of G indeed. We note in passing that a similar argument can be applied in the case of TrCmaux’s algorithm. For if we delete those edges which are not endedges but at which a retracting by (the final) W occurs, and if we orient the remaining edges according to the direction in which they are passed for the first time by W , we obtain the same digraph Din as in the above discussion of Tarry’s algorithm. This is no surprise, however, since a careful analysis of TrCmaux’s algorithm shows the validity of the following result whose proof is left as an exercise (see also [KONI36a, p.43-441). Proposition X.13. If W is the final walk in G constructed by Tr6maux’s algorithm, then for any v E V ( G )- (v,,}, the last edge passed by W from v is the first edge by which W reaches v (i.e., the edge represented by an arc in D i n ) . That is, W can be considered to have been constructed by Tarry’s algorithm.

However, it is not true in general that if W has been constructed by Tarry’s algorithm, the same W can be obtained by TrCmaux’s algorithm. For, comparing Step 1 of Tarry’s algorithm with the case A,(ei-l) =1 in Step 3 of TT6maux’s algorithm shows that Tarry’s algorithm admits more freedom of movement than TrCmaux’s algorithm. We note that a modification of Wiener’s algorithm using two different threads (a red and a white thread, say) also yields a bidirectional double tracing. Namely, in the backtracking procedure (Step 3 and Step 4 of Wiener’s algorithm) replace the corresponding section of Ariadne’s (white) thread with a red thread covering precisely this section, and specify that red sections must not be passed again. Thus in laying out the red thread one passes edges in the direction opposite to the one used in laying out the white thread. Because of the choice of j in Step 3 it follows that no edge is missed. When all the white thread has been removed upon finishing the walk, it follows that a bidirectional double tracing has been performed. Applying this idea to eulerian graphs and further specifying that at any stage of the algorithm edges not yet used should be passed whenever possible, one obtains Hierholzer’s algorithm. The above idea of using two types of thread can be found in [LIEB70a, 2.3.2, 2.3.31. We can do without threads, though, if we operate with orientations in accordance

X.24

X. Algorithms: Eulerian Trails,Cycle Decompositions, Mazes

with Wiener's original idea (see the footnote related to the hypothesis of Wiener's algorithm). The following algorithm uses this idea and can be viewed as a combination of the three preceding algorithms. Algorithm X.14. This algorithm assigns an orientation to every edge according to the first traversal of e by the walk W . Hypothesis. We operate with the hypotheses of Tr6maux's and Tarry's algorithms simultaneously. Consequently, for

Ew,; := { e E E,,/Xw(e) = i} defined upon reaching v in the course of constructing W , the sets EV",wn Ew,,, E , , and ECw := E , , - EV",w are known, and ECw is the set of edges passed towards v but not from v. Define Step 0 - Step 4 as in Tarry's algorithm with the only exception that we choose e in Step 1 with minimal Aw(e). The working of Algorithm X.14 is guaranteed by the working of Tarry's algorithm since the restriction concerning the choice of e in Step 1 has no negative impact; that is, the final W is a bidirectional double tracing in G anyway. The relevance of Algorithm X.14 lies in its application t o eulerian graphs which is demonstrated by the next result. On the grounds of Proposition X.13 and Algorithm X.14 we note that the rules of Tarry's algorithm or modifications of them cover all maze search algorithms which produce bidirectional double tracings (this was pointed out to me by P. Rosenstiehl) . Theorem X.15. Let G be a connected eulerian graph, and let W be a bidirectional double tracing in G arising from an application of Algorithm X.14. Let T = e j l , e i z ,...,eiq

be the subsequence of W obtained by listing the edges of G according to their second occurrence in W . It follows that T is an eulerian trail of G. Proof. The key to the theorem lies in the choice of e in Step 1 of Algorithm X.14, combined with an implicit application of Theorem VI.33.

We claim that if we denote by W ( i )the segments of (the final) W which are defined by the first traversal of the respective edges, then { W ( i ) / = i 1 , . . . , k} is a trail decomposition of G, and that the orientation D , of G defined by the first traversal of every e E E ( G ) is eulerian. In

X.3. Mazes

X.25

fact, starting at vo E V(G), let W(l) W be the segment ending at v, after having passed every element of Ev, precisely once. The choice of e in Step 1 of Algorithm X.14 guarantees not only that W(l) is well defined but also that is a closed trail (starting and ending at u,). Moreover, the orientation induced by W on E ( W ( ' ) )by the first passage of the respective elements is the orientation D,induced by W, := W ( l )in G, := (E(Wl)) and thus eulerian. Since A,(e) = 1 for every e E E,,, at this point of constructing W , it follows from Step 1 of Algorithm X.14 that W continues from v, by passing certain edges of G, in opposite direction (with respect to the orientation of 0,)until it reaches v(l) E V(G) for which Ew,o # 8, if such E , , exists. Suppose it does. Let Tl= T ( u , , d 1 ) ) denote this part of the walk W ; it is a trail since W is a bidirectional double tracing. It is again the choice of e E EUcr, in Step 1 which guarantees: (i) that W traces a closed trail W(,) starting and ending at dl); (ii) that every f E E ( W ( 2 ) )is an edge passed for the first time; and (iii) that the sequence W, := W ( 1 ) , T , , W ( 2C) W contains every e E E,,Cl, at least once. Now W traces in G, := (E(W,)) a trail ' T from d1)to a certain d 2 )for which E , , # 8 if such d2) exists. In any case we have E(T,) n E(T,) = 8 since W is a bidirectional double tracing and '"2 is also a sequence of edges used the second time. Observing that the final vertex of Tlis the initial vertex of T2 it follows that T,, T,is a trail starting at u, and ending at d 2 ) Generally . speaking, if W ( j )starting and ending at u(j-'), j 2 2, has been constructed then it is a closed trail containing edges used only for the first time such that W j := W ( l )T,, , W(,),T,,. . .,Tj-,, W ( j ) the , subsequence of W starting ~ ,least once. at uo and ending at u(j-'), uses every element of E , C ~ -at Thus { W ( l ) .,. . ,W ( j ) }is a trail decomposition of Gj := ( E ( W j ) )and , T,,T,,. . ., T j - l , is an open trail in Gj provided G j c G. However, if GjC G then Tj constructed in accordance with Step 1, Step 2 and Step 3 of Algorithm X.14 will end at u ( j ) for which Ew,,# 8 (if such dj)exists). The reason for this fact is Theorem VI.33 which is used implicitly in the definition of the algorithm. For ((ei,(u)/u E V(Gj)}) defines an out-tree B j rooted at uo in D j which is the orientation of (the connected graph) G j defined by . . . ,W ( j ) ,and which is eulerian since W ( t )is a closed trail, 1 5 t 5 j , and E(W("))n E ( W ( t ) = ) 8, 1 5 s < t 5 j , by construction (that is, by the choice of e in Step 1). Whence BF is an in-tree in Df rooted at uo, and TI,. . .,Tj-l corresponds to a

X.26

X. Algorithms: Eulerian Trails, Cycle Decompositions, Mazes

trail T ( j )in Df which can be extended to an eulerian trail of Df precisely because of Step 3 of the algorithm. For if T,, . . . ,Tj-,uses all edges of E, E ( G ) , x E V ( G j ) ,then the last edge used is precisely ei,(x); i.e., the last arc passed from z by T ( j )is the outgoing arc at x belonging to

BjR. It now follows that if G , := (E(W,)) = G for some k > 1 and T 2 , . . ,T,-l, W(’),then {W(’),. . . ,W ( k )). is a trail W , = W(l),T,, is a trail in G which can be extended decomposition of G and T, , . . . ,Tk-l to an eulerian trail of G by continuing the application of Algorithm X.14. That is, T as defined in the statement of the theorem is an eulerian trail of G. This had to be shown. Compare Algorithm X.14 with the Maze-Search Algorithm of [EDMO73a]: it follows that the latter is nothing but a special case of the former. For, the local information contained in the hypothesis of Algorithm X.14 can be extracted from the lists used in the Maze-Search Algorithm. Consequently, in their application to eulerian graphs, we observe that also the Maze-Search Algorithm satisfies Theorem X. 15. However, the eulerian trails obtained may be different even if the corresponding trail decompositions are the same. For, the Maze-Search Algorithm amounts to a backtracking procedure in constructing Tj.This is due to an implicit storage of the transitions of W ( j ) ,1 5 j 5 k, by producing at every ZI E V ( G )a list of the edges used for the first time and by which one enters ZI. Thus the Maze-Search Algorithm also yields implicitly ({ei,(v)/v E V ( G j ) } ) and the arborescenses B j and Bf as defined in the proof of the preceding theorem. Consequently, T obtained by this algorithm also leaves v for the last time along e i n ( v ) , v E V ( G ) . On the other hand, Algorithm X.14 if implemented, would not only list the elements of E:,, instead of the incoming edges (except for ein(v)), but also yield more freedom with which to choose the next out-going edge in producing T j , 1 5 j 5 k. This, however, is insignificant in relation to the complexity of these two algorithms; both operate in O ( q )time. The real difference between those two algorithms lies in their application to eulerian graphs: the Maze-Search Algorithm has Hierholzer’s algorithm as its point of departure, while it is Theorem VI.33 which serves as the basis for Algorithm X.14. Moreover, Theorem X.15 demonstrates the ailinity between Tr6maux’s and Tarry’s algorithm on the one hand, and the algorithmic construction of eulerian trails on the other. This ailinity seems to have passed unnoticed hitherto.

X.3. Mazes

X.27

Recall that Theorem VI.33 was already used to demonstrate the correctness of Tarry’s algorithm and can be used for the same purpose regarding Trkmaux’s algorithm (see the paragraph preceding Proposition X.13). Whence an affinity similar to the one just described is inherent in each of Trdmaux’s and Tarry’s algorithm; and it was this possibility of applying Theorem VI.33 which led to the formulation of Algorithm X.14 and Theorem X.15.However, this theorem no longer holds if we rely on only Trkmaux’s or Tarry’s algorithm. That is, in each of these algorithms if applied to an eulerian G, the sequence of edges listed according to their second occurrence in the final W need not define an eulerian trail of G . This is immediate for Trhmaux’s algorithm; just look at the second part of Step 3 of this algorithm. For the case of Tarry’s algorithm this fact is demonstrated in Figure X.l;the reason for this fact lies in Step 1 of Tarry’s algorithm where the edge e can be passed from zli = w during its second traversal by W , while f E E, has not been passed at all yet.

X

G Figure X . l . In Tarry’s algorithm, the sequence of edges listed according to their second occurrence can be of the form T = . . .,ei,(y),e,ei,(z), f,ein(w),ei,(zl), . . ., where ei,(z) is passed from z , z E { v , w , z , y } , e is passed from w and f is passed from zl. T is not an eulerian trail of G.

X. Algorithms: Eulerian Trails,Cycle Decompositions, Mazes

X.28

We note in passing that for the case of plane trees, no local information is needed to produce a bidirectional double tracing, but a simple rule called the “right-hand-on-the-wall” rule, suf€ices.15) Although each of the algorithms X . l l , X.12 and X.14 operates in O(q) time, basing itself on the existence of bidirectional double tracings in connected graphs, the question arises whether one can solve (MSP) such that every edge is passed twice at most. This problem is analogous to the CPP when compared with the construction of double tracings in known graphs. In fact, if, in addition to the local information of Algorithm X.14, one operates with a counter (e.g., in the form of pencil and paper), Algorithm X.14 can be improved. This has been done in [FRAE70a, FRAE71al.

Algorithm X.16. A.S. fiaenkel’s Algorithm. This algorithm also gives preference to untraversed edges as in Algorithm X.14. Hypothesis. In addition to the hypothesis of Algorithm X.14 we assume that a function (the counter) c : { W } t N U {0} is given, where { W } denotes the set of walks produced algorithmically, such that c(vo) = 1, and for W and W’ = W,e i , vi+l we have Ic ( W )- c(W’)I= 1.16) We also use the sets defined in the hypothesis of Algorithm X.14. This algorithm thus combines Algorithm X.14 and the counter c according to the following rules.

Rule 1. As long as c ( W ) > 0 proceed as in Algorithm X.14. Rule 2. The changing of c is defined in accordance with a certain property regarding vi+l in W’ = W,e i , u ; + ~ . a) If vi+l

# vj,

5 j 5 i, set c(W’) = c ( W ) + 1. for some j < i + 1 and if lE,i+l - E ( W )I> 0

b) If vi+l = v j 1, while lEw,,o12 1 for the same v i + l , set c(W’)= c(W)- 1. c ) If neither a) nor b) applies, set c(W’) = c ( W ) .

Rule 3. Suppose c(W) = 0. If Ew,o # 8 at vi, proceed as in Algorithm X.14. Otherwise, set ei = ei,(vi) if v; # w o , while the algorithm terminates for vi = vo. 15)

Left-handers are permitted to walk with their left hand on the wall.

16) A.S. Fraenkel defines c ( v o )= 0 because he views vo as an end-vertex attached to the original initial vertex, and begins the walk at v0.

X.3. Mazes

X.29

Outcome. The h a l W is a closed covering walk such that Aw(e) 5 2 for every e E E(G). If Aw(e) = 2 for some e E E(G)then W uses e in both directions. We now explain why the algorithm yields the required result. First of all, the counter c is a function as described in the Hypothesis. For c is increased by one if v E V ( G )is visited by (the final) W for the first time, while it is decreased by one at the same v if for the last time Ew,,# 0 for this v. Expressed differently, after the departure from v, all edges in E, are traversed at least once and c is not changed at v anymore. Whence at every vertex, c is increased by one and decreased by one precisely once. This increase and decrease of c is instantaneous if d(v) 5 2. It follows that c ( W ) is always a non-negative integer. Suppose now that c(W) = 0 for some W starting at vo and ending at v E V ( G ) ;possibly vo = v. If E w o # 8 at v, it follows from the above that no e E E , ,joins 21 to any z i V ( W ) ;otherwise, c is not decreased at z which in turn implies c ( W ) # 0 at v, a contradiction. Similarly, Go := ( E ( W ) )is an induced subgraph satisfying E, c E(G,) for every z E V(G,)-{v}. Thus (see Rule 1 and Rule 3) Algorithm X.16 deviates from Algorithm X.14 only after having passed all edges of G , at which point the algorithm stops if {ei,(v)} = 8 (i.e., v = v,), or else reaches vo via the unique path P( v,vo) satisfying E(P(v,vo)) E {e;,(v)/v E V ( G ) } . It follows that Algorithm X.16 constructs a closed covering walk such that the doubly traversed edges are used in both directions. Of course, Algorithm X.16 in general does not produce a solution of the CPP. This should not be surprising, however, since solving the CPP requires the construction of shortest paths which, in general terms, should be viewed as a matter of dealing with graphs globally. However, one may try to find faster algorithms by restricting the scope of graphs to which these algorithms apply. This approach was chosen, for example, in [RYTT82a] where the author deals with what he calls pyramidical Zabyrinths (these are, roughly speaking, graphs consisting of layers of plane graphs which decrease in size the higher they are placed, and only consecutive layers are connected by certain edges). By representing a labyrinth as a special set of squares from the plane integer grid, an upper bound on the length of a walk through such labyrinth was established in [SEND72a] (‘length’ refers basically to a walk in the dual graph of the

X.30

X. Algorithms: Eulerian Trails, Cycle Decompositions, Mazes

labyrinth”)). There is also an algebraic approach to handling mazes which we shall deal with in a more descriptive way. Namely, suppose the edges of a maze G are labelled with letters in such a way that the letter a, is assigned to e E E(G)together with an orientation of e , and the letter a‘, is assigned to the same e with the opposite orientation. Thus a walk W in G appears as a word w over the alphabet A := (a,,a’,/e E E(G)}.Let the set of walks in G be represented by the corresponding set L of words over A. The empty sequence stands for the trivial walks having just one vertex, while for two walks W,, W, where W, ends at the same vertex at which W, starts, WIW, is represented by wlwz E L.I8) Assume a, # a: and {ae,aI,}n ( a f , a ; }= 0 for every e , f E E(G), e # f . This way of handling mazes has been dealt with in [ROSS7la, ROSS73aI. In the first paper, P. Rosenstiehl develops what he calls Algorithm MINIREPLI and proves for connected eulerian G [ROSS7la, Thdorkme VIII] that the word formed by the letters corresponding to the second traversal of the edges of G corresponds to an eulerian trail of G. In other words, Rosenstiehl’s algorithm also produces a bidirectional double tracing which satisfies Theorem X.15.This is no surprise, though. For, although Rosenstiehl’s algorithm operates with words similar to the way in which Wiener’s algorithm operates with walks, a closer look at the former and the Mate-Search Algorithm of Edmonds and Johnson reveals that these two algorithms are equivalent. Namely, not only can the local information of the MazeSearch Algorithm be extracted from the words produced by Rosenstiehl’s algorithm and vice versa, but these two algorithms produce the same bidirectional double tracing in G and thus the same eulerian trail when applied to a connected eulerian graph. The following additional remarks are the result of a discussion I had with P. Rosenstiehl in the summer of 1990. Tremaux’s algorithm is the root of the Depth-First Search technique, one of the most basic techniques deployed in developing algorithms. The central rule of this technique is to retreat as soon as possible, i.e., when hitting a vertex already visited or 17) This dual graph is well-defined if we view the labyrinth as being defined by the vertices and edges of the integer grid which belong to at least one square of the set defining the labyrinth, and if we view precisely the squares of the set as faces. The dual graph of this labyrinth is, in fact, bipartite and plane. l a ) In a way we can view w E c as a multi-colored thread of Ariadne.

X.3. Mazes

X.31

when reaching an end-vertex (see Step 3 of Tremaux’s algorithm). This is in contrast to Algorithm X.14and its ‘derived’ algorithms by Edmonds and Johnson, and Rosenstiehl respectively, since there the governing rule is to retreat as late as possible which is the most elementary rule possible. Moreover, Algorithm X.14,when applied to eulerian graphs G enables one to produce all eulerian trails with respect to a given out-tree induced by the corresponding set of edges {ei,(v)/v E V(G)}(see Theorem VI.33 and the discussion following the proof of Theorem X.15). According to P. Rosenstiehl, about ten years ago R.E. Tarjan pointed out to him that this more general procedure was inherent in the Algorithm MINIREPLI. Finally, the trail decomposition {W(’),...,W ( ” } obtained in the proof of Theorem X.15 (and induced by the first traversal of edges) can be viewed as obtained by inductively contracting W ( i )onto a vertex v ( W ( ~to )) produce W(’+l) as starting and ending at v ( W ( ~ i) = ) , 1, . . . ,Ic - 1.

In another approach, [KOEG67a, THAL68a, SCHU72a1, a modified version of Wiener’s algorithm, applied to plane simple graphs, has been combined with creating binary sequences a,; the binary sequence corresponding to the final walk is called the Theseus word. One proceeds in the construction of W according to a right-hand rule. Having started at w, along a specified edge e,, suppose one has reached v along el (O+(v) = ( e i , . .. , e :,...,e’,(,,)). If v has been reached before, return to w defined by ei = vw; continue along e:+, otherwise. After retracing ei = fj (O+(w) = ,;(! . . . ,f;,. . . ,&,,)), if fj+l has been passed already continue retracing along ei-, = fj-, ; otherwise, continue along fj+,. At any stage, one associates with the walk W = e l , e,, . . ., . . ,ei-,,ei a binary sequence a, of length i such that the j-th digit of a, is 1 if e j is passed by W for the first time; otherwise, the j-th digit of a, is 0. It turns out that in general, the same Theseus word may correspond to non-isomorphic planar graphs. However, if G,, G, are plane and 2-connected and if they have the same Theseus word, then G, 11 G,, [KOEG67a]. The same can be said if G , and G, are trees, [THAL68a, Satz 11. Moreover, for a Theseus word corresponding to some plane G , there is a plane tree T having the same Theseus word, [SCHU72a, Satz 13, and G can be obtained from T by a certain identification process, [SCHU72a, Satz 21.

ej,.

Finally, if one is tired of walking or if the mouse goes on strike, one can try to let an automaton run through a maze. The problem one has to deal with in this case is the following.

X.32

X. Algorithms: Eulerian Trails,Cycle Decompositions, Mazes

Define a deterministic finite automaton capable of traversing an arbitrary maze. (AMSP)

K. Dopp deals with (AMSP) for mazes defined as a set of squares in the plane integer grid, [DOPP71a, DOPP7lbl. He assumes that an automaton has an initial state and that its memory capacity is defined exclusively by its (inner) state set; i.e, the various squares of the grid have no influence on the automaton’s memory. In fact, referring to other people’s attempts to solve (AMSP) K. Dopp and others conjectured that (AMSP) has no positive solution. While solving (AMSP) positively for special types of mazes and automata with a certain extended memory capacity, K. Dopp cannot solve (AMSP) in general. However, considering a maze to be a graph, H. Muller was able to prove that for every finite state automaton (in which input and output alphabets may be as large as the set of positive integers), a connected plane graph G exists which has 1- and 3-valent vertices only, such that the automaton does not reach every vertex of G, [MULL?la, Satz]. An analogous result for 2-regular digraphs has been proved in [RICD82a]. As for pebble automata searching mazes (where the pebble(s) is (are) used to mark vertices) we refer to [HOFF82b, HOFF84al which contain further references to this topic. Finally we quote [HEMM85a, RICD83a, ROSS72al which also deal with automata searching mazes.

Final Remark. The references to mazes are by no means complete. In fact, many important references have been omitted; they can be found in the references of the quoted papers. In general, whenever dealing with closed walks rather than eulerian trails (in Chapters V and VIII as well) I have not tried to gather all important papers. I have tried to reach such goal, however, whenever dealing with eulerian trails. I hope I have not missed too many significant papers pertaining to the topic of eulerian trails. Questions related to cycle decompositions, cycle coverings, enumeration (other than treated in Chapter IX), eulerian subgraphs, etc. will be treated in a separate volume.

x.4. Exercises Exercise X.l.Using the Splitting Algorithm a) derive an algorithm for P(G)-compatible eulerian trails provided P(G) satisfies Theorem VI.l.2);

X.4. Exercises

x.33

b) derive an algorithm for eulerian trails in digraphs; c) derive an algorithm for non-intersecting eulerian trails if G is embedded in some surface (or if (O+(v)/v E V(G)) is given). Exercise X.2. Suppose a weakly connected eulerian digraph D with a given system of transitions X ( D ) satisfies S(G) > 6. Using the Splitting Lemma and the Double Splitting Lemma, develop an algorithm for X ( D ) compatible eulerian trails whose running time is polynomially bounded. Exercise X.3. Develop an algorithm for eulerian trails in mixed graphs whose running time is polynomially bounded (hint: use flow theory to decide whether a given mixed graph has an eulerian orientation). Exercise X.4. 1) Adapt Fleury’s Algorithm to obtain an algorithm for eulerian trails in digraphs. 2) Prove Theorem X.3. Exercise X.5. Adapt Algorithm X.4’ for digraphs. Exercise X.6. Reformulate Algorithm X.4’ so as to embrace arbitrary connected eulerian graphs. Exercise X.7. Show that Algorithm X.5 can be formalized so as to become a linear algorithm. Exercise X.8. Let S be a trail decomposition of the simple graph (digraph, mixed graph) H . Define G by setting V(G) = E ( H ) U A ( H ) (where E ( H ) or A ( H ) may be empty), and z y E E ( G ) if and only if {z,y} E X,. Prove that there is a bijection ‘p between the components Ki of G and the trails Ti E S defined by

cp(Ki)= Ti* V(K,)

= E(Tj)U A(Tj)

.

Exercise X.9. Show that G is a connected spanning subgraph of G, (for the definition of G , and G, see Algorithm X.6 and the subsequent discussion of this algorithm). Exercise X.10. Show that the execution of Step 4 of Algorithm X.6 as described after the statement of this algorithm, can be done in constant time and yields an eulerian trail of D. Exercise X . l l . a) Derive a ‘pidgin-algol’ version of Algorithm X.7 similar to the manner in which Algorithm X.4’ has been derived from Algorithm X.4.

X.34

X. Algorithms: Euleriaa Trails, Cycle Decompositions, Mazes

b) Derive a new &-trails algorithm based on the construction of an eulerian trail by Algorithm X.l or Algorithm X.2, combined with the backtracking procedure developed in the discussion on the partial order defmed by the sets ‘TEas constructed in Step 5 of Algorithm X.7. c) Study the complexity of these algorithms for the case where parallel processing is (is not) employed.

Exercise X.12. Derive an algorithm for constructing a cycle decomposition of the eulerian graph G by starting from an arbitrary trail decomposition S of G and by applying &-detachmentsto elements of S which are not cycles. Determine the complexity of this algorithm under the additional assumption that parallel processing is (is not) employed. Exercise X.13. Improve Algorithm X.9 by combining the construction of an eulerian trail T with determining cycles C which are segments of T. Exercise X.14. Show that if W is a final walk constructed by T r k m a u ’ s Algorithm and having a retracting of ei-l = ei at vi, then W , = . . . ,eja2, 2ri-,(= vj+,),ei+,, . . . is a final walk in accordance with Trkmaux’s Algorithm applied to G, = G - e; . Exercise X.15. Prove Proposition X.13.

A. 1

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[STEI88a] Steinberg, R.; Younger, D.H.; Grotzsch’s Theorem for the Projective Plane, to appear with minor revision in Ars Combin. [SWAN63a] Swan, R.G.; An Application of Graph Theory t o Algebra, Proc. Amer. Math. SOC.14 (1963) 367-373. [SWANSSa] Swan, R.G.; A Correction to “An Application of Graph Theory to Algebra”, Proc. Amer. Math. SOC.21 (1969) 379-380. [SYSL77a] Syslo, M.M.; Skupieri, Z.; Applied Graph Theory 111: Eulerian and Hamiltonian Graphs. T h e 13.avelling Salesman Problem, Rocz. Pol. Tow. Mat., Ser. 111, Mat. Stosow. 10 (1977) 5-54. [SYSL83a] Syslo, M.M.; Deo, N.; Kowalik, J.S.; Discrete Optimization with PASCAL Programs (Prentice Hall Inc., New Jersey, 1983). [THALSSa] Thalwitzer, K.; Charakterasierung eines Labyrinthes durch ein Wort uber einem zweistelligen Alphabet, in: Beitrage zur Graphentheorie, Colloq., Manebach, 1967 (Teubner, Leipzig, 1968) 157-161. [THOM87a] Thomassen, C.; Bidirectional Retracting-Free Double Tracings and Upper Embeddability of Graphs, preprint. [TRAC63a] Trachtenbrot, B.A.; Algorithmes e t machines Ci calculer (Dunod, Paris, 1963) [TROYSSa] Troy, D.J.; O n Traversing Graphs, Amer. Math. Monthly 73 (1966) 497-499. [TUCK76a] Tucker, A.; A N e w Applicable Proof of the Euler Circuit Theorem, Amer. Math. Monthly 83 (1976) 638-640. [TUTT4la] Tutte, W.T.; Smith, C.A.B.; On Unicursal Paths in a Network of Degree 4, Amer. Math. Monthly 48 (1941) 233-237. [TUTT48a] Tutte, W.T.; The Dissection of Equilateral W a n g l e s i n t o Equilateral W a n g l e s , Proc. Cambridge Philos. SOC.44 (1948) 463-482.

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[wIEN73a] Wiener, C.; ober eine Aufgabe der Geometria situs, Math. Annalen VI (1873) 29-30. [WHIA8Oa] White, A.T.; A Note o n Conservative Graphs, J. Graph Theory, Vo1.4, No.4 (1980) 423-425. *[WILF87a] Wilf, H.S.; Yoshimura, N.A.; Ranking rooted trees, and a graceful application, Discrete algorithms and complexity (Kyoto, 1986), 341-349, Perspect. Comput. 15 (Academic Press, Boston, Mass., 1987). *[XIEL84a] Xie, L.T.; Liu, G.Z.; A Survey of the Problem of Graceful B e e s , (Chinese), Qufu Shiyuan Xuebao 1 (1984) 8-15. +[XION86a] Xiong, R.; On homomorphisms of the de Bruijn-Good graph, Kexue Tongbao (English Ed.) 31 (1986), no.7,439442. [XUON79a] Xuong, N.H.; How t o Detemnine the M a x i m u m Genus of a Graph, J. Combin. Theory Ser. B 26 (1979) 217-225. [YOUN83a] Younger, D.H.; Integer Flows, J. Graph Theory 7 (1983) 349-35 7. [ZALESSa] Zalesskii, A.E.; Assymmetric Analog of the Amitsur-Lewitzki Theorem, VestsiAkad. Navuk BSSR, no.2, Ser. FTz. Mat. 128 (1985) 110.16, 108-110. [ZEILSSa] Zeilberger, D.; A Combinatorial Approach t o Matrix Algebra, Discrete Math. 56 (1985), no.1, 61-72. [ZHAF86b] Zhang, F.J.; Xiaofeng, G.; Hamiltonian cycles in perfect matching graphs, (Chinese. English summary) J. Xinjiang Univ. Nat. Sci 3 (1986), no.4, l(r16. [ZHAF87a] Zhang, F.J.; Guo, X.F.; Hamilton cycles in directed Eder tour graphs, Discrete Math. 64 (1987), no.2-3, 289-298. +[ZHAF88a] Zhang, F.J.; Lin, G.N.; A note o n strong homomorphisms of de Bruijn-Good graphs, (Chinese. English summary) J. Xinjiang Univ. Natur. Sci. 5 (1988), no.4, 113-115. *[ZHA086a] Zhao, S.L.; Double isomorphic graceful trees are graceful stringing trees, (Chinese. English summary), Qufu Shifan Daxue Xuebao. Ziran Kexue Ban 1986, no.2, 54-55. [ZYKA88a] Zfka 0.; Nowhere-Zero 30-flow o n Bidirected Graphs, preprint .

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B.1

INDEX

Preliminary Remark. Instead of producing a separate index of symbols, I have included them in this index. In those few instances, where certain symbols play a role only in a smaller part of the book, they have been omitted. Also, most of the concepts developed in Part 1, Volume 1, but used in this Volume 2, are not included in this index. Finally, standard concepts from other branches of mathematics but not defined in this Volume 2, can be found in the corresponding literature.

B.2

Index

VIII.72 bipartition VIII. 31 (A;)+ VIII.72 bisecting double-star VIII.78 ( A 3VIII.72 Blossom Algorithm VIII.92 A:,* VIII.92 ~111.72 Blossom Inequalities A;,* IX. 73 bond-conforming VI r I.24 A(D) 1x.73 bond-faithful VIII.26 A*(D) 1 x 3 4 Borchardt IX.76 A**(D) IX.73 IX. 73 Bott A;,, ( D ) IX.80 Bouchet VIII.43 van Aardenne-Ehrenfest VIII. 134 Brucker H-absorbing IX.80 v111.127 de Bruin eulerian digraph IX.102 x . 5 de Bruijn graph x-absorption IX.102 IX.73 de Bruijn-Good graph adjacency matrix IX. 102 x.30 k-de Bruijn sequence Algorithm MINIREPLI x. 11 cancelling IX.20 All-Trails Algorithm Caplan IX. 76 All Minors vii, VIII.63 Matrix Tree Theorem IX.75 Catlin VIII.136 alphabet IX.99 (CB) VIII.56 Amitsur-Levitzki-Theorem IX.62 Celmins x.32 Chaiken IX .76 (AMSP) ~111.123 Chinese Postman Problem VIII.82 (AP) Ariadne x.17 chromatic trees IX. 106 ~111.127 Circular Flow Conjecture VIII.62 assigned euler network assignment problem (AP) VIII. 123 circular Patterns IX. 103 automaton ~111.124 (k,k)-cofactor of A * ( D ) IX.73 axis VIII.78 Colbourn vii balanced valuation VIII.64 X(D)-compatible eulerian trail X.3 basic orientation of G 1x.86 P(G)-compatible eulerian trail X.3 basic subgraph with root w i IX.73 concatenation IX.104 Bellman ~111.88 conservative flow VIII. 74 Berman IX.96 cost function VIII.82 BEST-Theorem IX. 80 counter X.28 bidirected graph ~111.72 CPP (Chinese ~111.72 Postman Problem) VIII.82 bidirection bidirectional double tracing VIII. 1 (cpp) VIII.83 bidirectional flow VIII.72 (cpp’) VIII.88 bidirectional (CPP” ) VIII. 9 1 (nowhere-zero) k-flow VIII.72 (cpp’”) VIII .91 bieulerian digraph IX.80 crossing trail IX.106

Index

B.3

IX.107 End-Pairing VI11.96 Algorithm VIII.92, X.7 equalizer (with respect to 21) IX.27 VIII.136 equalizing IX.20 IX.10 f-equalizing IX.27 1x.102 t-equalizing IX.45 IX.22 ~111.127 equivalence relation RE VIII.1 4,s IX.62 Eder Depth-First Search technique X.30 ederian conservative flow VIII.74 detachment X.8 eulerian H-augmentation VIII.134 diameter, diam(G) VIII.114 ederian orientation graph IX.98 (DIFF) ~111.115 eulerian permutation IX.2 VIII.25 EUL (Euler Tour Problem) VIII.89 diff (4 W J V ) VIII.28 even covering trail IX.3 Dijkstra's algorithm v111.89 even permutation IX.2 Diniz V I I I . ~(D ~ - Do)-favoringeulerian trail X.4 directed numbering VIII.74 Fleury's Algorithm x.2 Directed Postman flow VIII.19 Problem (DPP) VIII.114 k-flow VIII.19 directed postman's (mod k)-flow VIII. 19 tour (DPT) VIII.114 [a,b]-flOW VIII.64 X.13 Fouquet VIII. 74 dominating path domino problem IX.108 X.28 x.32 haenkel's Algorithm X.28 Dopp VIII.78 F'robenius IX.70 double-st ar Double Splitting Lemma x.3 G a e y VIII.17 ~111.1generating function IX.104 double tracing DPP (Directed Postman Gerards VIII.102 Problem) VIII.114 &&kn IX.106 VIII. 114 graceful graph VIII.75 (DPP) ~111.119 graceful labeling VIII.75 (DPP') Dynkin 1 x 7 0 Guan Meigu E circuits Ix.19 (The Chinese Postman) VIII.82 E paths IX.N Hadamad VIII.44 (elementwise) product IX.83 ~111.64 hamilt onian walk VIII. 109 IX.3, IX.104 Hamiltonian Walk Problem & VIII.90 (HWP) VIII.43, VIII.109 Edmonds VIII. 43 HarmY IX.73 Eight-Flow Theorem

crt(G) t-cut cycle-balancing cost function Di,j 0," (D+>

:2)

B.4

Index

X.13 Hasse diagram X.4 Hierholzer Hierholzer’s Algorithm X.4 IX.76 Hill X.7 Hoang Thuy vii Hosch Hu’s 2-commodity flow theorem VIII.108 Hut chinson IX.19 HWP VIII.109 identical permutation E IX.3 inner vertex X.l input alphabet VIII.124 integer flow VIII.19 integer linear programming problem (ILP) VIII.90 VIII.107 i-integer-valued integral MFP VIII.109 integral three commodity VIII.109 flow problem irreducible H-absorbing eulerian digraph VIII.130 VI11.96 Ising spin glasses X.3 isthmus-avoiding trail IX.106 Jackson, D.M. VIII.28 Jaeger Jaeger’s Four-Flow VIII.59 Theorem VIII.94 Jiirvinen VIII.17 Johnson, D.S. VIII. 90 Johnson, E.L. VIII.96 t-join VIII.6 Jungermann VIII.45 Kagno VIII.79 Kaluza IX.73 Kasteleyn VIII.74 Khelladi VIII.35 Kilpatrick IX.73 Kirchhoff

Kirchhoff matrix Klein group Kostant Kotzig Kronecker delta Kronrod Kundu’s Lemma Kwan Mei-Ko X(z,y)

Aw(e)

A, A(H) labyrinth Liebling Lillie Liskovec Liu Jiannong p ( G , cp)

IX.73 IX.14 IX.70 ‘I 76, IX.96 IX.62 VIII.90 VIII.59 VIII.82 IX.4 VI I I. 83 IX.24 IX.25 X. 17 VIII.94 vii IX.95 VIII.141 VIII. 27 IX.99, IX.107

Martin polynomial Matrix Tree Theorem for digraphs IX.73 Matrix Tree Theorem for graphs IX.76 maximum weight cycle packing problem (MCPP) VIII.95 maximum weight cycle packing problem in digraphs VIII.141 (DMCPP) IX.73 Mayberry X.17 maze Maze-Search Algorithm x.7 X.17 maze search problem VIII.96 MCCP McKay IX.95 VIII.95 MCPP IX.107 medial graph VIII .lo6 MFP VIII.135 Minieka minimum weight cycle covering

Index

B.5

problem (MCCP) VIII.96 oT ix.19 Minotaur X.17 odd covering trail ix.3 Mixed Postman odd permutation ix.2 VIII.124 one-way property Problem (MPP) VIII.80 mixed postman’s S-orbit VIII.4 VIII.124 orbit double cover tour(MPT) VIII.4 MMP (Minimum Weight Perfect Ore VIII.1 VIII.89 [a,b]-orientable Matching Problem) VIII.64 monoid of sequences IX.104 (mod (2k 1))Moon IX.76 orientation VIII.62 VIII.124 [a, b]-orientation VIII.64 (MPP) VIII.127 oriented cycle cover (MPP’) VIII. 23 VIII.128 Owen (MPP ’’) 1x.71 (MPP’”) VIII.133 PRE(v) 1x.22 VIII.124 ( 9 ,$) VIII.72 (MPT) X.18 k-packing of t-cuts VIII.103 (MW Miiller X.32 Palmer 1x.73 Multicommodity Flow parallel edges VIII.63 Problem (MFP) VIII.105 partial ((mod k)-)flow VIII.19 Next-Node Algorithm VIII.92, X.7 partition problem VIII.17 next state function VIII. 124 path-forest VIII.60 Nishizeki vii (PB) VIII. 137 non-backtracking pebble automata X.32 quasi-eulerian walk VIII. 1 perfect l-factorization c.3 nowhere-zero Petersen graph VIII.33 ((mod k)-) flow VIII.19 plane MFP VIII.109 null IX.20 Polesskii c.1 IX. 107 Polesskii-Kundu Lemma number of crossing trails 1 number of eulerian postman’s tour VIII.82 orientations IX.90 postman’s tour PT VII I. 83 number of spanning in-trees prime VIII. 130 of D rooted at v IX.79 PT (postman’s tour) VIII.82 VIII.83 number of spanning out-trees PTW of D rooted at v IX.73 Pulleyblank 1x.98 number of spanning trees IX.76 pyramidical labyrinth X.29 0 IX.74 Quenuille’s changeover 0 IX.60 designs IX.103 ?,(GI IX.90 RE 1x.22 6E(G) IX.20 Read x.4

+

c.

Index

B.6

(Eo;me, m)-restricted

Seymour

covering walk Eo-restricted double tracing retracing-free retracting retracting-free root of a covering trail rooted covering trail Rosa Rosenstiehl rotation equivalent Rowen Rural Postman Problem, RPP

W)

=,w

vii, VIII.22

VIII.14 Seymour’s Cut Condition Seymour’s Six-Flow Theorem S9n(4 s9n(T) sign of an edge sign-preserving sign-preserving bijection sign-reversing sign-reversing bijection signature signed graphic isthmus VIII. 140 Skilton IX.2 Smith

VIII.12 VIII.1 VIII.1 VIII. 1 IX.2 IX.2 VIII.76 vii, X.24 IX.22 IX.70

VIII. 103 x.5 IX.4 VIII.25, IX.4 VIII.28 VIII.l

Sabidussi Sabidussi ’s Compatibility Conjecture X.16 X.17 Saint-Quentin Schmidt VIII.24 Schiitzenberger IX. 17 Sebo vii, VIII.96 Serdjukov VIII.90 set of covering trails T in D starting at v IX.l set of eulerian orientations of G IX.20 set of independently prescribable words IX.102 set of trail decompositions IX.97 VIII.14 i-set i-set of the flow cp VIII.44

VIII.22 VIII.39 1x.2 1x.2 VIII.72 1x.25 1x.25 1x.27 1x.27 VIII. 72 VIII.72 V111.5 1x.80 VIII.56 1x.95

mark Sorokin x.1 Splitting Algorithm VIII.6 Splitting Lemma SPP (Shortest Path Problem) VIII .89 standard identity of degree m 1x.60 star-conforming VIII.24 VIII.26 star-faithful strictly upper embeddable VIII.4 strong double tracing VIII.1 swan 1x.4 Swan’s Theorem 1x.59 Swart VIII.56 symmetric digraph VIII.135 t(G) 1x.97 %f 1x.27 qf 1x.27 %(D) 1x.1 7v(f, 1) 1x.27 I,(G) := ?;(G, G) 1x.21 Tv(G,H ) IX.19

Index

IX.3 IX.27 IX.3 IX.27 IX.23 IX.76 IX.73 IX.79 IX.98 X.31 IX.108, X.18 X.21 IX.97 X.9 (k,n)-template digraph IX.102 Theseus X.17 Theseus word X.31 Thomassen VIII.3 Three-Flow Conjecture VIII.59 totally cancelling graph IX.66 Trachtenbrot x.21 Trail-Conforming Algorithm X. 15 transition-equalizing IX.45 transportation problem VIII.120 Traveling Salesman Problem (TSP) VIII.109 Tr’ emaux X.18 Trkmaux’s Algorithm x.20 triangle inequality VIII.110 Trivalent Four-Flow Conjecture VIII.59 VIII.3 Troy TSP VIII.109 VIII.109 (TSP) (TSP’) VIII.110 Tucker X.8 Tucker’s Algorithm X.8 Turgeon VIII.76

B.7

Tutte VIII.27, IX.80 Tutte’s 5-Flow Conjecture VIII.27 Tutte’s Four-Flow Conjecture VIII.57 Tutte polynomial 1x.99 type (i,j;k) 1x.28 /?-valuations VIII. 76 value-true cycle cover VIII.23 value-true walk VIII.14 vertex bipartition VIII. 21 Viches VIII.93 VLSI circuits VIII.96 Wagner VIII.14 Walton VIII.57 weak mark VIII. 56 Weak Three-Flow Conjecture VIII.59 weakly equalizing 1x.66 Weaver VIII .56 Welsh VIII.57 Wenger Erich 1x.4 Wiener X.18 Wiener’s Algorithm X. 18 windy postman’s tour VIII.136 (WPT) Windy Postman Problem VIII.135 (WPP) word 1x.99 VIII. 135 (WPP) (WPP’) VIII.135 WPT VIII.136 X, IX. 1 Xuong VIII.6 Younger VIII.43 Youngs VIII.45 zero matrix 0 1x.60 zero vector 0 1x.74 Zfka VIII.74

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c.1

APPENDIX: CORRECTIONS AND ADDENDA TO VOLUME 1

Since the publication of Part 1, Volume 1, of this monograph several colleagues discovered some errors related to grammar and concepts, but also to results. Moreover, as time goes by, preprints turn into publications. Hence I take this opportunity to correct these errors and to update the Bibliography of Volume 1. Part 2 of the monograph will (with probability tending to 1) contain an Appendix regarding errors and additional remarks related to the two volumes of Part 1. Let us correct errors first. Apart from the fact that I was taught Latin and not latin (p. II.l), the Shakespearian query (p. 111.73) should read “To P or to NPC ? . . .” (I forgot the letter C - a stupid mistake indeed). Another mistake in Chapter I11 relates to the question of priority regarding Kundu’s Lemma (Lemma 111.40, p. 111.39). Both Paul A. Catlin and Charles J. Colbourn drew my attention to V.P. Polesskii’s article [POLE7la] containing a reference to Lemma 111.40 published by Polesskii in Russian in 1970. Instead of trying to dig out the original reference I just quote [POLE7la], and shall call Lemma 111.40 the PolesskiiKundu Lemma’) from now on. Also, I forgot to define E* and A * , the respective sets of half-edges and half-arcs of a (mixed) (di)graph H (see, E,* , A* = UVEV(H) A:. e.g., p.III.41, line 16). Clearly, E* := UVEV(H)

‘1 C.J. Colbourn called this problem to my attention as early as November 1989, when we met at the University of Waterloo, but he didn’t recall Polesskii’s paper offhand. His letter of 1/2/90 did not reach me in time to be considered for Volume 1; in order to retain uniformity throughout Part 1 of this monograph I decided to wait with this change of names regarding Lemma 111.40 until Part 2. Finally, my thanks go to N.S. Merzlyakov, Moscow (USSR), who sent me the Russian version of [POLE71aJjust before I finished this volume.

c.2

Appendix

A more annoying mistake occurred on p. VI.146 where I (falsely) claimed [NISH83a] contains a proof that the hamiltonian walk problem can be solved in O(p2)time for triangulations of the plane. The truth is that this O ( p 2 )algorithm finds a V(G)-covering walk at most 3 ( p - 3)/2 in length, whereas the problem for such graphs is NP-complete. My thanks go to Taka0 Nishizeki who called my attention to this error; he also pointed out my confusion of first and last names in the reference [CHIB84a] on p. A.3: the authors of this article are - in fact - N. Chiba, T. Yamanouchi and T. Nishizeki. The next corrections relate to Chapter VII. These errors were pointed out by Mr. Emanuel Wenger. In fact, in the statement of Lemma VII.22 (p. VII.41) the equation K~ = K’K” should be replaced with K , = K I1 K I ; the index S in (*) on p. VII.42 should be replaced with s. Read ‘exist’ instead of ‘exists’ on p. VII.43, line 2, and on p. VII.44, last sentence, the ‘intested reader’ should become an ‘interested reader’. On p. VII.18, line 19, X$,,k should be replaced by XTI , and on p. VII.19, line 9, { e $ , e ; } 3?k should be replaced by {{eL,ei}}. Moreover, while it is true “that Ktransformations are not a sufficiently general tool to transform” eulerian trails in digraphs (see p. VII.45, first paragraph), it should be pointed out that segment reversals and therefore Ic-transformations cannot be defined for digraphs by analogy to the case of graphs. A similar problem of inaccuracy occurred earlier in the context of applying /+transformations to trail-decompositions of graphs with 2k > 2 odd vertices (p. VII.9, first paragraph). The last two questions were improperly phrased because I overlooked an elementary fact: If one applies a K-transformation to a trail T which is an element of a trail decomposition S of G, then the new trail TI necessarily satisfies E ( T ) = E(T’), and consequently, T and T’ join the same odd vertices of G. Therefore, if S* is obtained from S by a sequence of K;-transformations (or, for that matter, segment reversals), then S’ and S define the same partition of E(G)and the same pairing of the odd vertices of G. It follows that these last two questions have negative answers unless one specializes the range of graphs G for which one considers P(G,K ) . Finally, in the discussion following the statement of Theorem VII.28 (p. VII.52 and p. VII.53) I failed to state explicitly that for a plane eulerian G with A(G) = 4, G(TA(G), is edge-hamiltonian, provided G has at least three A-trails. This is in fact a direct consequence of Theorem VII.14 and Cummins’ result that the tree graph of a graph

Appendix

c.3

(having at least three spanning trees) is edge-hamiltonian. We note that an analogue to Theorem VII.28 holds for digraphs as well [ZHAF87a]; the underlying transformation is the .r-transformation (see Definition VII.25 and Theorem VII.27). I had already mentioned this on p. VII.52 but could not specify the transformation since at that time I did not have the article [ZHAF87a]. The same authors proved an analogue with respect to the set of all perfect matchings, where the transformation is defined by ‘rotating’ L-alternating cycles (ix. cycles in which certain edges of a 1-factor L alternate with edges not in L), [ZHAF86b]. We note, however, that this result had been proven already in [NADD8la, NADD84al (but by different methods). Finally, the article [LIXU88a] establishes a formula for the connectivity of G ( 7 , K). I would also like to cite some additional references on topics treated in Chapters IV--VI. First, it should be noted that Theorems IV.5 and IV.7 as well as some corollaries have been published separately in [FLEI89b]. - Theorem VI.35 has been proved in [FLEI89c] without relying on any of the Theorems VI.33 and VI.34 which can thus be derived easily from the more general theorem. - The problem of traversing certain edges in a specified order by an eulerian trail (see p. VI.34, first paragraph) has been considered in [POLJ9Oa] for the case of digraphs. - The theme of perfect 1-factorizations of K2,, (i.e., 1-factorizations such that any two 1-factors yield a hamiltonian cycle) which has been of relevance in the context of pairwise compatible eulerian trails (see p. VI.48), has also been treated in [DINI89a, IHRI87a, KOBA89a, SEAH89al. - An algorithm with O ( p ) running time regarding hamiltonian cycles in 4-connected planar graphs, has also been published in [CHIB89a]. This appendix ends with a full reference of certain articles which have been quoted as preprints in Volume 1. Thus, [FLEISSa] is now [FLEISOc], [GROP89a] is now [GROPSOa], whereas [HARBSga] remains that way; [READ86a], however, has become [READ87a].

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Eulerian Graphs and Related Topics: Part 1, Volume 2 (Annals of Discrete Mathematics, Volume 50)

Eulerian Graphs and Related Topics: Part 1, Volume 1 (Annals of Discrete Mathematics, Volume 45)

Eulerian Graphs and Related Topics: Part 1, Volume 1 (Annals of Discrete Mathematics, Volume 45)