Graph Colouring and Variations

GRAPH COLOURING AND VARIATIONS

ANNALS OF DISCRETE MATHEMATICS

General Editor: Peter L. HAMMER Rutgers University, New Brunswick, NJ, U.S.A.

Advisory Editors: C. BERGE, Universite de Paris, France M.A. HARRISON, University of California, Berkeley, CA, U.S.A. V KLEE, University of Washington, Seattle, WA, U.S.A. J. H. VAN LINT California Institute of Technology,Pasadena, CA, U.S.A. G. C. ROTA, Massachusetts Institute of Technology;Cambridge, MA, U.S.A.

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD *TOKYO

39

GRAPH COLOURING AND VARIATIONS

D. de WERRA A. HERTZ Departemen t de Ma thema tiques Ecole Polytechnique Federale de Lausanne Lausanne, Switzerland

1989

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD .TOKYO

Elsevier Science Publishers B.V., 1989

IC)

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Reprinted from the Journal Discrete Mathematics, Volume 74, Nos. 1-2, 1989

Library of Congress Cataloging-in-Publication Data Graph colouring and variations1 [edited by] D. de Werra. A. Hertz. cm. -- (Annals of discrete mathematics ; 39) p. Bibliography: p. ISBN 0-444-70533-3 1. Map-coloring problem. I. Werra, D.de. II. Hertz, A. Ill. Series. QA612.18.G73 1989 514'.3--dc 19

PRINTED IN NORTHERN IRELAND

88-28842 CIP

SPECIAL DOUBLE ISSUE GRAPH COLOURING AND VARIATIONS Guest Editors: D. de WERRA and A. HERTZ CONTENTS A. HERTZ and D. de WERRA, Foreword C. BERGE, Minimax relations for the partial q-colorings of a graph K. CAMERON, A min-max relation for the partial q-colourings of a graph. Part II: Box perfection P. DUCHET, On locally-perfect colorings M.O. ALBERTSON, R.E. JAMISON, S.T. HEDETNIEMI and S.C. LOCKE, The subchromatic number of a graph A. HERTZ and D. de WERRA, Connected sequential colorings A.J.W. HILTON, Two conjectures on edge-colouring B. BOLLOBAS and H.R. HIND, A new upper bound for the list chromatic number C.T. HOANG and N.V.R. MAHADEV, A note on perfect orders F. JAEGER, On the Penrose number of cubic diagrams R.E. JAMISON, On the edge achromatic numbers of complete graphs H.A. KIERSTEAD, Applications of edge coloring of multigraphs to vertex coloring of graphs M. KUBALE, Interval vertex-coloring of a graph with forbidden colors J. MAYER, Hadwiger's conjecture ( k = 6): Neighbour configurations of 6-vertices in contraction-critical graphs H. MEYNIEL, About colorings, stability and paths in directed graphs J. MITCHEM, On the harmonious chromatic number of a graph S. OLARIU, Weak bipolarizable graphs C.T. HOANG and B.A. REED, &comparability graphs H. SACHS and M. STIEBITZ, On constructive methods in the theory of colour-critical graphs E. SAMPATHKUMAR and C.V. VENKATACHALAM, Chromatic partitions of a graph M.M. SYStO, Sequential coloring versus Welsh-Powell bound S.K. TlPNlS and L.E. TROUER, Jr., A generalization of Robacker's theorem A.D. PETFORD and D.J.A. WELSH, A randomised 3-colouring algorithm

1 3

15 29

33 51 61 65 77 85 99 117 125 137 149 151 159 173 201 227 241 245 253

Discrete Mathematics 74 (1989) 1-2 North-Holland

1

FOREWORD Graph coloring has been a field of attraction for many years; a wide collection of papers has been dedicated to the study of chromatic properties of graphs. Initially such problems were just a kind of game for pure mathematicians; it was in particular the case of the famous four color problem. However, as people were getting used to applying the tools of graph theory for solving real-life organizational problems, chromatic models appeared as a quite natural way of tackling many situations. Among these are timetabling problems, or more generally scheduling with disjunctive constraints (pairwise incompatibility between jobs), clustering in statistics, automatic classification, group technology in production (partitioning a collection of parts into families of parts which are as similar as possible in their production process), VLSI design, etc. The theory of perfect graphs and particularly the perfect graph conjecture of Claude Berge provided a strong impetus for the development of the theory of coloring. Several papers in this volume are dealing with special classes of perfect graphs which are characterized by chromatic properties. A natural extension of coloring problems - motivated by a polyhedral formulation of optimization in perfect graphs - consists in expressing an integral vector in a polyhedron as a sum of integral vectors contained in a smaller polyhedron. This extension is considered in some contributions of the present volume. Besides this, colorcritical graphs have been a focusing point in many research works; such graphs, having some inherent structure, can hopefully be characterized by more and more properties. Many other variations and extensions of the basic node (or edge) coloring problem have been proposed: for instance a node or an edge may receive a set of “consecutive” colors, or a color chosen in a given set of admissible colors. A color class may be extended from a stable set of nodes to a union of disconnected cliques. Such types of variations are presented in the volume. Related optimization problems are also discussed; among those is the maximum q-coloring problem: find the largest number of nodes which can be colored with q colors in a given graph G (with chromatic number larger than q). Edge coloring problems form a special case of node coloring; deciding if there exists an edge coloring using A(G) colors in a simple graph G with maximum degree A(G) is an NP-complete problem. However, many results giving bounds of the edge chromatic number based on the degrees of the nodes and on some properties of the graphs can be obtained. This volume contains a few papers in this direction. Many algorithmic approaches have been developed and have provided large 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

2

D.de Werra, A. Hertz

classes of graphs for which coloring problems can be solved in polynomial time. Such results will certainly be extremely useful for applications. The papers presented in this text do not provide an exhaustive survey of the various fields of chromatic optimization; in particular they do not describe the numerous fields of applications of colorings. We hope however that they will bring to the reader a view of some facets of this active research area. We will have reached our aim if the contributions collected here will stimulate new investigations in the chromatic properties of graphs and hypergraphs. In such a wide field, it is difficult to unify notations. We have nevertheless encouraged the authors to use the definitions of Claude Berge, Graphs (North-Holland, 1985). Most of them followed these lines and the terms used in different ways are usually defined in the papers or given in appropriate references. Finally we would like to thank the authors who have been the active contributors to this volume. Our gratitude extends also to the many anonymous referees who have devoted much of their time to improve the quality and readability of the papers. We wish to acknowledge the enthusiastic support and the encouragements of Peter L. Hammer, Editor-in-chief of Discrete Mathematics. The help of North-Holland in preparing this volume is also gratefully acknowledged. Many thanks are due to Mrs. A.-L. Choulat for having handled the manuscripts with great care. Lausanne, June 15, 1988.

A. Hertz D. de Werra

3

Discrete Mathematics 74 (1989) 3-14 North-Holland

MINIMAX RELATIONS FOR THE PARTIAL 4COLORINGS OF A GRAPH C . BERGE C.M.S., E.R. 175 Combinatoire, 54 Bd Raspail, F75270, France A parfial q-coloring of a graph is a family of q disjoint stable sets, each one representing a “color”; the largest number of colored vertices in a partial q-coloring is a number aq(C), extension of the stability number a ( G )= a,(G). In this note, we investigate the possibilities, for 1 s 9 =sy ( G ) , to express a q ( C )by a minimax equality.

1. Optimal partial q-colorings Let G be a simple graph with a finite vertex-set X and with chromatic number y(G), and let q be an integer, 1 S q S y ( G ) . A partial q-coloring of G is a family 9,= (Sl, S,, . . . , S,) of q disjoint stable sets. If x E Sj, we shall say that the

vertex x is of color ( i ) . All the vertices need not have a color. The q-coloring 9, is optimal if the number of “colored vertices” lUirqSiI is as large as possible. The number of colored vertices in an optimal q-coloring will be denoted by a,(G), so that cul(G) = a ( G ) , the stability number of G. Consider a family V = (C, l j E J ) of cliques (complete subgraphs). The qcoloring Yqand the clique family ‘G: are associate if we have simultaneously: (A,) S j f l S j =0,

(A3) Sifl Cj # 0

CJl C j = O for i # j ;

for alli and all j .

Let G = ( X , E ) b e a g r a p h o n X = { x , , x , , . . . , x n } . Put Q = { 1 , 2 , . . . , q}, and let K , be the complete graph on Q. The Cartesian sum G K , is a graph on the Cartesian product X X Q, where ( x , i ) and ( y , j ) are joined if x = y and i # j , or if [ x , y ] E E and i = j . Every stable set So of G + K , defines a partial q-coloring as follows: color the vertex x of C with color i if and only if ( x , i ) E So. Conversely every partial q-coloring of G defines a stable set of G + K , . Thus, we have the following basic result:

+

Lemma 1. For every graph G and every integer q, there exists a one-to-one correspondence between the maximum stable sets of G K , and the optimal partial q-colorings of G , and

+

a,(G) = a(G

+ K,).

0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V.(North-Holland)

C. Berge

4

Lemma 2. A partial q-coloring Y, = (Sl, &, . . . , S,) is optimal if and only if for every partial q-coloring 9;= (Si,S;, . . . , S;), there exists an injective mapping u :US] +USi such that (1)

ux

E

{x} u r x

xES;

(2)

ux E S j

i#j

1

*ux = x

Clearly, a partial q-coloring Y, satisfying this condition is optimal, because if we consider for 9’; a partial q-coloring which is optimal, we obtain (since u is injective):

The converse follows immediately from Lemma 1 and the “maximum stable set Lemma” ([l], Chap. 13, 01): A stable set B is maximum if and only if every stable set S disjoint from B can be matched into B.

Lemma 3. Let G be a graph, and let q be an integer, 1s q s y ( G ) . Then every partial q-coloring which has an associate clique-family is optimal. S2, . . . , S,) be a partial q-coloring of G , and let So = Uirq(Six Let 9,= (Sl, {i}) be the stable set of C + K, defined as in Lemma 1. The existence of a clique-family % satisfying (Al), (A2),( A 3 )allows to construct a clique-partition of G + K , as follows: Take the cliques { x } x Q for all x E X - U j C j , and the cliques Cj x {i} for all i and j. Clearly, these cliques are pairwise disjoint and cover X; furthermore, by (A3),each clique of the partition contains one element of So, and only one. This shows that So is a maximum stable set. From Lemma 1, it follows that .4., is optimal. We denote by 8 ( G ) the least number of cliques which partition the vertex-set of G.

Theorem 1. Let G be a graph with chromatic number y ( G ) , and let q be a positive integer, q < y(G). Then the following conditions are equivalent: (1) a ( +~K,) = e(c + K ~ ) ; (2) every optimal q-coloring has an associate clique-family ; (3) some optimal q-coloring has an associate clique-family ; (4) if M denotes a clique-partition of G , then

Proof. (1) implies (2). Assume that a,(C + K , ) = 8 ( G + K , ) . Let Y q = (Sl, S,, . . . , S,) be an optimal q-coloring of G. By Lemma 1, the set So =

Minimax relations for partial q-colorings

5

+

(Si x { i } ) is a maximum stable set of G K q . Let %,, be a minimum clique partition of G K q , so that I %,I, = ISo/.Some cliques are of the type { x } x 0,with x E B (for some B c X) and the others are of the type Cj x { i }. We may replace each 0,by Q, and assume that the Ci satisfy:

+

(I)

cjn Cj. = 0

(2)

U Cj = X - B

for j # j';

i

for all i ;

Ci f l Sj # 0 for all i and all j . (3) From (2) and (3) we see that for the subgraph Gx-Binduced by X - B , each Sj - B is a maximum stable set, and each %'= (Cj 1 j) is a minimum cliquepartition; therefore any minimum clique-partition % = (Ci \ j) of G X p Bsatisfies (A,), (A2)and (A3).This shows that Yqhas an associate clique-partition %.

( 2 ) implies (3). Obvious. ( 3 ) implies (4). I f we consider a q-coloring 9,and a clique-partition M, we have: (i)

I u s/= c SEYq

c

I C ~ U S I ~min{lCI,q}.

CEM

CEM

Let 9,be some optimal q-coloring. Let % be the clique family associate with 9,. Let M be the clique-partition obtained from V by adding the singletons { x } for Then equality holds in every vertex x which is not covered by a member of Yq. (i). The property (4) follows immediately. (4) implies (1). Let pqand M be a q-coloring and a clique-partition of G for which (i) holds with equality. Since the inequality (i) is true for all 9,and all M , the q-coloring pqis optimal. Let % be a clique-partition of C K , obtained from

+

M

- b y replacing each CEM with [ C I > q by C x { l } , c x { 2 } , . . . , c x { q } , and - by replacing each vertex x which belongs to a E M with lcl s q by { x } x Q. Clearly, % is a clique-partition, and

c

Hence, a(G

+ K , ) = a ( C )= 1 % 1 =

e(G + K,).

0

Remark. In Theorem 1 we assume that q < y(G), because for q = y(G), the condition (1) is always true. Indeed, by Lemma 1 every graph G with chromatic number y(G) = y satisfies

a(G + K y ) = e(C + K y ) = n.

C. Berge

6

The complement of a graph C is a graph G with the same vertex-set, two vertices being joint in G if and only if they are not joined in G. We have:

+

Corollary. If a graph G satisfies a(G K 4 ) = B(G + K 4 )f o r some q < y(G), then its complement G satisfies a(G K r ) = B ( G K r ) f o r some r < y ( G ) .

+

+

+

+

Proof. a(G K 4 ) = B(G K 4 ) is equivalent to the existence of a partial q-coloring having an associate clique-partition. Consequently the result follows from the symmetric role played by the cliques and the stable sets in the conditions (AlL (A*),(Ad. 0 Examples. We study first the Haj6s graph H (a triangle inscribed in a hexagon) with two different optimal 2-colorings (Fig. 1 and Fig. 2). This graph is perfect, but we cannot prove the optimality of a 2-coloring by associate clique-families. No such associate family does exist: In both Fig. 1 and Fig. 2, the cliques C , and C2cover all the uncolored vertices, but are not disjoint; therefore (Cl, C,) is not an associate family. On the other hand a 2-coloring of the complement H of the Haj6s graph is represented on Fig. 3, and its optimality follows from the existence of the associate family (Cl). The main problem we shall study here is: for which classes of graphs does an optimal q-coloring have an associate clique-family? H H 0

c2

--

.

r

c - -

-.

\ c1

/

I

\

I

\

I

I

I

\

I

\

/

\

Fig. 2. cu,(H) = 4.

Fig. 1 . cu2(H) = 4.

-

-

_

-

/

H

Fig. 3. cu2(fi) = 5.

Minimax relations for partial q-colorings

7

2. Existence theorems due to the perfectness of G + K q Let G, denote the subgraph of G induced by a set A of vertices. Recall that a graph G is perfect if u(GA)= 6(GA) for every A c V ( G ) .A chain p is odd (resp. even) if the number of edges in p is odd (resp. even). A chain p is chordless if its vertices induce an elementary chain. A graph G is a parity graph if for x , y E V ( G ) , all the chordless chains joining the vertices x and y have the same parity. For an elementary cycle, we say that two chords [ x , y ] and [ z , t ] cross if the vertices x , z , y , t are encountered in this order on the cycle. It is easy to show: A graph is a parity graph if and only if every odd elementary cycle of length 2 5 has two crossing chords (Burlet and Uhry [ 5 ] ) . These parity graphs have been considered first by Sachs [15], who has shown that every parity graph is perfect. This follows also from a more general result of Meyniel [13]. Burlet and Uhry [5] gave a polynomial time recognition algorithm for a parity graph. Many graphs considered in the literature belong to this class.

Example 1. A bipartite graph, defined by two vertex-sets X and Y, is a parity graph, because every chain joining x E X and x ’ E X is even, and every chain joining x E X and y E Y is odd. Example 2. Let G be a graph such that each block (“maximal 2-connected subgraphs”) is a clique. For x , y E V ( G ) , there is only one chordless chain that joins the two vertices x and y , therefore C is a parity graph. If T is a tree, its line-graph L ( T ) has the following property: each vertex of L ( T ) which does not correspond to a pendant edge of T is an articulation vertex, and each block is a clique. So L ( T ) belongs to this class. Note that one can easily show: Each block of G is a clique if and only if G satisfies the two following properties: (i) each cycle of length 2 4 has a chord (ii) G does not contain as an induced subgraph the graph C ; (cycle of 4 vertices plus one chord). Let us mention a few graphs which, are obviously not parity graphs. For k 2 2 , denote by C2k+lthe odd elementary cycle of length 2k 1; denote by C ; k + l any graph obtained from C 2 k + l by adding one chord (Fig. 4). Denote by CIS‘the graph obtained from C5 by adding two non crossing chords (Fig. 5). We see on the figure that the two vertices a and b are joined by an odd chordless chain and by an even chordless chain. Hence a parity graph does not contain a C 2 k + l , nor a c ; k + 1 , nor a C;. In fact, Burlet and Uhry have shown in [ 5 ] : A graph is a parity graph if and only if it does not contain as an induced subgraph any one of the following configurations : C 2 k + l : cycle of length 2k + 13 5 c ; k + ,: cycle of length 2k + 12 5 plus one edge cycle of length 5 plus two non-crossing chords. Cl;:

+

C. Berge

8 a

a

ci C5 b

d

b

C

Fig. 4.

Fig. 5.

In order to show that a q-coloring of a graph G has an associate clique-family, we know from Theorem 1 that it suffices to show that a(G + K q ) = 8(G + K q ) , which can follow from the perfectness of the graph G + K,. Various necessary and sufficient conditions for the perfectness of the Cartesian sum G + H have been found independently by Ravindra and Parthasarathy [14] and by de Werra and Hertz [7]; for H = K q , the result is simpler:

Theorem 2. Let G be a graph, and let 2 G q S y(G). The Cartesian sum G perfect if and only if q = 2, and G is a parity graph ; or q 3, and G is a graph whose blocks are cliques.

+ K , is

Proof. The condition is necessary. Let q = 2 . Let G be a graph which is not a parity graph. Then G contains a C;, or a C2k+lwith k 3 2, or a C&+l (as shown by Burlet and Uhry [ 5 ] ) . In the first case, C;+K 2 contains an odd chordless cycle: a2, b2, b l , c l , d l , d2, e2, a2 represented by the arrows on Fig. 6. Similarly an odd chordless cycle exists in C2k+l+ K 2 and in c;k+,+ K 2 (remove in Fig. 6 the edges [a2, c2] and [ a l , cl], and then the edges [c2, e2] and [ c l , e l ] ) . In all cases, the graph G + K 2 contains an odd chordless cycle, and consequently cannot be perfect. I

a2

e2

I

I

Fig. 6 .

a

I

C’j+K:,

Minimax relations for partial q-colorings

9

C; + Kg

Fig. 7.

Let q 2 3, and let G be a graph where some blocks are not cliques. Then G contains a C; or a c k with k 2 4 (see note after Example 2). If G contains a c k with k odd 3 5 , we see as above that the Cartesian sum Ck K2 contains a C k + 2 . If G contains c k with k even 24, then C, K3 contains a C k + , (see the arrows in Fig. 7). If C contains a C;, we see that Ci + K3 contains a C7 (see Fig. 7). Thus, in every case, the graph C + K, contains an odd chordless cycle, and, consequently is not perfect. The condition is sufficient. Let q 3 3, and let G be a graph whose blocks are cliques. Then G does not contain a C;. The graph G + K, does not contain a C ; , because otherwise, at least two vertices of this induced C; would have the same vertical projection (but not the four of them), and every possibility leads to a contradiction. By the same argument, G + K , does not contain a C 2 k + l with k s 2. Then, the perfectness of G K, follows from a theorem of Ravindra and Parthasarathy which asserts that a graph which contains no C; and no C 2 k + l with k 3 2 is perfect (for a complete proof, see Tucker [16], Theorem 2). For q = 2, the proof is similar.

+

+

+

The line-graph of a tree satisfies all the requirements of Theorem 2, thus any optimal q-coloring of such a graph has an associate clique-family. However, this argument does not help to construct an associate clique-family, and we can prove more directly:

Theorem 3. Let G be the line-graph of a tree, and let q S y(G). Then, for every optimal q-coloring, an associate clique-family can be efficiently constructed. Furthermore, the graph G K , is perfect.

+

Proof. Let p, denote an optimal q-coloring of a line graph G = L ( T ) of a tree T . We have to show that p, has an associate clique family. If q = y(G), pq is a q-coloring of C and any maximum clique forms an associate clique family. So assume q < y(G). For a vertex a of the tree T, denote by w T ( a )the set of all edges incident to a : a set of edges 4 contained in o T ( a )is

C. Berge

10

called a star of center a . We shall construct a partial q-coloring (El, E2, . . . , E q ) for the edges of T , and a family of stars (4 I j EJ), such that: ( A ; ) 4 n Fk = 0 for j # k ; (A;) every edge e $ UEi s is contained in some 4; (A;) 16 f l Eil= 1 for all i, j . Clearly, ( A ; ) , (A;) and (A;) are equivalent to the axioms (A,), ( A2 ) ,(A3 )which characterize an associate clique-family in G = L ( T ) . Put

B = {x Ix

E

V ( T ) ,d T ( x )> 4).

The subgraph TB of T induced by B is a “forest” (union of disjoint trees). Let b , be a pendant vertex of some tree in TB. Label with a “0” the edge of T B incident to b , and d T ( b l )- q - 1 other edges of T which are incident to b , . Let T, be the forest defined by the unlabelled edges of T, we have d T , ( b l= ) q. Put Bi = I.{ x E V ( T ) ,d ~ , ( x ) > q ) . Then consider a pendant vertex b2 of a tree in the forest T B , , and define as before B2; and so on. The procedure terminates with a forest Tk such that d T k ( x s ) q for every vertex x ; since Tk is a bipartite graph, it follows from Konig’s Theorem that there exists a q-coloring of its edges. This defines a partial q-coloring, and every star F; = w T ( b i )contains all the q colors. Clearly, every uncolored edge of T is contained in some F;, and the E’s are disjoint. This shows that the E’s satisfy (A;), (A;), (AS) and achieves the proof. 0

3. Other classes of graphs In this section, we shall show that some known properties of balanced hypergraphs give easily some new classes of graphs for which the optimal q-colorings have an associate clique-family. An hypergaph H = ( E lE,2 , . . . , E m ) on X = {xl, x 2 , . . . , x , } is a collection of non-empty finite sets (called “edges”) whose union is X (the “vertex-set”); a cycle of fengfh k is a sequence ( x , , E l , x 2 , E2,. . . , Ek,xI)where the xi’s are different vertices, the Ej’sdifferent edges, and x i , x i + , E E, for all i . The hypergraph H is balanced if every cycle of odd length has an edge which contains at least three vertices of the cycle. Clearly, a bipartite graph is also a balanced hypergraph. The balanced hypergraphs have been introduced to generalize the main properties of the unimodufar hypergraphs (hypergraph whose incidence matrix edge-versus-vertex is totally unimodular). Let H be a balanced hypergraph with n vertices and m edges. Let A be its m X n incidence matrix, whose columns represent the edges, and whose rows represent the vertices. Let A * be its transpose. We denote by d = ( d l , dZ,. . . , d,) and y = ( y l , y,, . . . , Y , ~ )two m-dimensional vectors whose coordinates are non-negative integers; we denote

11

Minimax relations for partial q-colorings

by c = (cl, c,, . . . , c,) and t = (tl, t,, . . . , t n ) two n-dimensional vectors whose coordinates are non-negative integers.

Property 1 (Corollary of Berge-Las [12]). For every d E N”, max{x diyi/y E N“, Ay

G

Vergnas [4]; generalized by Lovasz

1) = ,in[

2 tilt E N“, A*t 2 d

1.

Property 2 (Theorem of Fulkerson-Hoffman-Oppenheim [S]). For every d E N“, max

Ic

tilt

E N”, A*t

sd

Property 3. A hypergraph H = ( E l , E,, . . . , En)with “rank” r ( H ) = max lEil is balanced if and only if every partial subhypergraph H’ (obtained from H by removing vertices and edges) is colorable with r ( H ’ ) colors so that no two vertices belonging to the same edge of H ’ have the same color. For the proofs of these results, see [3], Chapter 5.

Theorem 4. Let G be a graph with vertex-set X and let q < y ( G ) . Let H ( G ) be the hypergraph on X defined by the maximal cliques of G. If H ( G ) is balanced, then every optimal partial q-coloring of G has an associate clique-family. Proof. Let H ( G ) = ( E l ,E,, . . . , E m )be the hypergraph of the maximal cliques of G , with incidence matrix A , and assume it is balanced. Let fi = ( E l , . . . , Em, F , , . . . , F,) be the hypergraph obtained from H ( G ) by adding an edge 8 = { x i } for i = 1, 2, . . . , n , and let A be the incidence matrix of fi. Clearly, l? is balanced. Let q be a positive integer, and let ij = (q, q, . . . , q, 1, 1, . . . , 1) be an (rn + n)-dimensional vector with m coordinates equal to q and n coordinates equal to 1. Let t = (t,, t,, . . . , tn) be an integral vector satisfying (A)*Zs(5. and maximizing C t i . Clearly, every coordinate ti is equal either to 0 or to 1. Put S = {xj/l S j S n, ti = 1). Thus, S defines a maximum set of vertices of H with the property: the subhypergraph Hs induced by S has a rank r(Hs) = maxEEH,[El at most equal to q. Since Hs is balanced, S is then a maximum set of vertices of G which can be colored with q colors (by Property 3). Hence:

Let jj = (y,, y,,

. . . , y,,

z , , I,,

. . . , I,) be an (m + n)-dimensional integral vector

C. Berge

12

satisfying Ar

3 (1, 1,

. . . , 1 ) and minimizing the scalar product

m

n

(~l,~)=qC~i+Czj* ;=I

j=]

Since y defines a covering of H minimizing a linear function with positive coefficients, all its coordinates are G l . Put Zo = { i / lG i s m, y j = l }

Jn = { j / l ~j s n, zj = I } . Thus, we have:

Then, we can write (using Property 2), IS( G

C IS n E;I+

ielo

IS n 41s q jeJo

IZ,~ + lJnl= (4, r) =

ti = 1st.

Hence each of these inequalities holds with equality, and, consequently, IS n E;l = q

(i E 10);

(2)

ISnFil=l

(jEJn).

(3)

(S

n EJ n (S n E j . )= 0 (i, i' E Z,, i #i n ).

(4)

Let pq be a q-coloring of G,s.For i E 10 the clique E j contains q different colors, by (2). Every uncolored vertex is contained in at least one of E,'s by ( 1 ) and (3). We may assume that the Ej's are pairwise disjoint: otherwise, we may replace each Ej by I?; c Ei without violating ( l ) ,(2), (3), (4). Thus, (El, E2, . . . , Em)constitute an associate clique family. This achieves the proof. 0

Corollary 1. Let G be a graph whose maximal cliques constitute a unimodular hypergraph. Then every optimal partial q-coloring of G is characterized by the existence of an associate clique family. This follows from Theorem 4, since every unimodular hypergraph is balanced. Note that this statement was obtained first by other methods by Cameron [ 6 ] .

Corollary 2. Let G be the line-graph of a bipartite multigraph. Then every optimal partial q-coloring of G has an associate clique family. The result follows from the Corollary 1, since the hypergraph H ( G ) of the maximal cliques has no odd cycles, and therefore is unimodular.

Minimax relations for partial q-colorings

13

Corollary 3. Let G be an interval graph. Then every optimal partial q-coloring of G has an associate clique family. (Same argument).

Corollary 4. Let G be a comparability graph with no induced K5 minus two dkjoint edges. Then every optimal partial q-coloring of G has an associate clique family. It is easy to show that H ( G ) is balanced. Thus the result follows from Theorem 4. Corollary 4 is in fact a special case of a theorem of Greene and Kleitman, reformulated as follows:

Theorem of Greene and Kleitman. Every optimal partial q-coloring of a comparability graph has an associate clique family. In fact, the Greene-Kleitman theorem [lo] states that the directed graph G of a partial order satisfies:

where M denotes a path-partition. Since every path induces a clique (by the transitivity) and every clique is spanned by a path (by Redei’s Theorem), this is equivalent to the condition (4) in Theorem 1, and we have shown that (4) is equivalent to (1). The Greene theorem [9] can be reformulated as follows:

Theorem of Greene. In the complement of a comparability graph, every optimal partial q-coloring has an associate clique family. Note that all the graphs mentioned in these results are perfect. If G + K , is perfect, it is well known that the determination of a maximum stable set of G Kq is a polynomial problem. But if G is perfect, is the determination of an optimal q-coloring also a polynomial problem?

+

References [l] C. Berge, Graphs (North-Holland, New York, 1985). [2] C. Berge, Minimax relations for normal hypergraphs and balanced hypergraphs, Ann. Discr. Math. 21 (1984) 3-19. [3] C. Berge, Hypergraphs (North-Holland, 1988). French version: Hypergraphes, combinatoires des ensembles finis (Gauthier-Villars, Paris, 1987). [4] C . Berge and M.Las Vergnas, Sur un thCor&rnedu type Konig pour Hypergraphes, Ann. N.Y. Acad. Sc. 175 (1970) 32-40.

14

C. Berge

151 M. Burlet and J.P. Uhry, Parity graphs, Ann. Discrete Math. 21 (1984) 253-277. [61 K.B. Cameron, A minimax relation for q-colorings, Discrete Mathematics 73 (1988). [71 D. de Werra and A. Hertz, On perfectness of sums of graphs, Report ORWP 87/13, Swiss Federal Institute of Technology in Lausanne (1987). [8] D.R. Fulkerson, A.J. Hoffman and R. Oppenheim, On balanced matrices, Math. Programming Study l(1974) 120-132. (91 C. Greene, Some partitions associated with a partially ordered set, J. Combinat. Theory A 20 (1977) 669-680. [lo] C. Greene and D. Kleitman, The structure of Sperner k-families, J. Combinat. Theory A 20 (1976) 80-88. [Ill A.J. Hoffman and D.E. Schwartz, On partitions of a partially ordered set, J. Combinat. Theory B 23 (1977) 3-13. [12] L. Lovasz, Normal Hypergraphs, Discrete Math. 2 (1972) 253-267. [13] H. Meyniel, The graphs whose all cycles have at least two chords, Ann. Discrete Math. 21 (1984) 115-120. (141 G . Ravindra and K.R. Parthasarathy, Perfect product of graphs, Discrete Math. 20 (1977) 177- 186. [ 151 H. Sachs, On the Berge conjecture concerning perfect graphs, in: Combinatorial Structures (Gordon and Breach, New York,1970) 377-384. [I61 A. Tucker, Coloring K,- e free graphs, J. Combinat. Theory B 42 (1987) 313-318.

Discrete Mathematics 74 (1989) 15-27 North-Holland

15

A MIN-MAX RELATION FOR THE PARTIAL qCOLOURINGS OF A GRAPH. PART 11: BOX PERFECTION Kathie CAMERON* Department of Management Sciences, University of Waterloo, Waterloo, Ontario, Canada N2L 3G 1

This paper examines extensions of a min-rnax equality (stated in C. Berge, Part I) for the maximum number of nodes in a perfect graph which can be q-coloured. A system L of linear inequalities in the variables x is called TDI if for every linear function _a_ such that _c is all integers, the dual of the linear program: maximize {cx: ;x satisfies L } has an integer-valued optimum solution or no optimum solution. A system L is called box TDI if L together with any inequalities 1 c x 6 LC is TDI. It is a corollary of work of Fulkerson and LoQasz that: where A is a 0-1 matrix with no all-0 column and with the l-columns of any row not a proper subset of the l-columns of any other row, the system L ( G ) = (A& s 1, g 3 0) is TDI if and only if A is the matrix of maximal cliques (rows) versus nodes (columns) of a perfect graph. Here we will describe a class of graphs in a graph-theoretic way, and characterize them as the graphs G for which the system L ( G ) is box TDI. Thus we call these graphs box perfect. We also describe some classes of box perfect graphs.

1. Introduction Consider a graph H for which every induced subgraph G of H satisfies the following min-max equalities for every positive integer q (equivalently, for every positive integer q < w ( G ) , the maximum size of a clique in G).

I=

(1.1) maximum{lSI: S E V ( G ) ;V clique C in G , IS n CI s q } (1.2) minimum{q

1x1+ IV(G) - U XI:Xis a set of cliques in G } .

(1)

( V ( G ) denotes the node-set of G. In this paper, cliques need not be maximal.) Restricting q to be 1 in the above would say H is perfect. Also then, (using the Perfect Graph Theorem 123,241) a set S as in (1.1) is the same as a set which can be partitioned into no more than q stable sets; that is, a partial q-colouring. Thus: For a perfect graph G, (1.1)

[=

(1.3) maximum{lSI: S E V ( G ) ,S is a partial q-colouring in G}.

* Research supported by the Natural Sciences and Engineering Research Council of Canada 0012-365X/89/$3.50 01989, Elsevier Science Publishers B.V. (North-Holland)

K. Cameron

16

Fig. 1.

It is clear that: For any graph G,

l(1.4) minimum Note that the X of (1.2)and the V of (1.4)may be taken to be node-disjoint. Thus, for perfect graphs, the equality (1) is the same as (4)in ([3], this volume) which says (1.3)= (1.4)- Lov6sz [25] called a graph H q-perfect if for each induced subgraph G of H, (1.3) = (1.2). Greene [19]gave the graph of Fig. 1 to show that not all perfect graphs satisfy (1). It does not satisfy (1) for q = 2. The Dilworth-Greene-Kleitman min-max theorem ([8],[20])says that comparability graphs satisfy (1) for all q. Greene’s min-max theorem [191 and a more general theorem proved independently by Edmonds and Giles [9]say that cocomparability graphs satisfy (1) for all q. Lov6sz [24]proved that the substitution operation preserves perfection: if H and K are disjoint graphs and v is a node of H, then to substitute K for v in H, join each node of K to each neighbour of v, and delete v. Two important special cases of substitution are joined and unjoined duplication: to create m joined duplicates of node v, substitute a clique of m nodes for v ; to create m unjoined duplicates of v, substitute a stable set of m nodes for v. Note that creating 0 joined or unjoined duplicates of node v corresponds to deleting v, and thus taking an induced subgraph is a special case of duplication.

G2

Fig. 2.

17

Min-max relation for partial q-colourings V

Fig. 3.

Creating joined duplicates need not preserve min-max (1): The graphs GI and G2 in Fig. 2 satisfy (1) for all induced subgraphs and all q , but G3 does not satisfy (1) for q = 3. (See Section 6 for more examples and proofs.) Jean Fonlupt pointed out that creating unjoined duplicates need not preserve min-max (1). The graph in Fig. 3 satisfies (1) for all induced subgraphs and all q, but if node v is replaced by two unjoined duplicates, the new graph does not satisfy (1) for q = 2. Let us examine the effect of creating joined or unjoined duplicates in G on the min-max equality (1). For each v E V ( G ) ,let a, be a non-negative integer. Replace each v E V ( G )by a set of a, joined duplicates to get a new graph G’-that is, substitute a clique of size a,, for v. G’ satisfies (1) if and only if

I

c

(2.1) maximum(

I J EV

xu: v clique

c in G ,

xu 6 q ; ueC

(C)

Vv E V ( G ) ,0 =sxu =s a,, xu integer

-

I 1x1+ 2

(2.2) minimum q

veux

a,:

I

1

(2)

x is a set of cliques in c .

Since creating joined duplicates need not preserve min-max (l), equivalently if a graph satisfies (1) for all induced subgraphs and all q , it need not satisfy (2). The graph of Fig. 4 does not satisfy (2) for q = 3 and the aI,’sas shown. 2

2

Fig. 4.

K. Cameron

18

For each v E V ( G ) ,let w,,be a non-negative integer. Replace each v E V ( C )by a set of w,,unjoined duplicates to get a new graph G’-that is, substitute a stable set of size w,,for v. G’ satisfies (1) if and only if: ((3.1) maximum

1

I=

Lc

(3.2) minimum{q

w,,:S E V ( C ) ;V clique C in G, IS r l CJS q ]

2 yc+

cliques C

y , , : V v ~V ( G ) , weV(G)

yc+y,,3w,; U€C

I

V cliques C , y , 3 0, y , integer; Vv

E

V ( G ) ,y,, a 0, y,, integer

(3)

.

Where q = 1, Fulkerson [14] called a graph pluperfect if it satisfies (3) for every non-negative integer-valued w = (w,,: v E V(G)). We now look at a unification of (2) and (3). A graph G is called box perfect if vE for every positive integer q, and all non-negative integer-valued w = (w,,: V ( G ) )and a = (a,,: v E V ( G ) ) ,the following min-max equality holds:

c

((4.1) maximum(

w,,x,: V clique C in G,

V€V(C)

I=

Vv

4

2

cliques C

c

xu s q ;

WE,

E

V ( G ) ,0 S X , ~s a,,, x, integer

yC+

I

(4)

a,,y,: U€V(C;)

Vv

E

V(G),

c y c + y , a w,,;

we,

V cliques C , yc

3 0,

yc integer;

Vv E V ( G ) ,y,, 3 0 , y, integer

Where w,,= 1 V v E V ( G ) , (4) is (2). Where a, = 1 V v E V ( G ) , (4) is (3). Where w, = 1 and a, = 1, V v E V ( C ) ,(4) is (1). In Section 4, we will prove that box perfect graphs are precisely the graphs for which a certain system of linear inequalities is box totally dual integral. Note that if C is box perfect then so is any induced subgraph of G : choose a,, = 0 (or w, = 0) for v not in the induced subgraph.

2. Box perfection and joined and unjoined duplicates

For a fixed w = (w,,: v E V ( G ) ) and a = (a, : v E V ( G ) ) , let G,(w, a) be the graph obtained from G by substituting a stable set S,, of size w,, for each

Min-max relation for partial q-colourings

19

v E V ( G ) ,and then substituting a clique of size a, for each u E S,,. Then:

(4)holds for G e ( 1 ) holds for Gl(w, a)

(5)

If G is box perfect, it turns out that it is not necessary to substitute a clique of the same size for each node of S,, in order to conclude that (1) holds. In Section 5 we will prove:

Theorem 1. Creating unjoined duplicates preserves box perfection. Corollary 1. G is box perfect A n y graph obtained from G by first creating unjoined duplicates and then creating joined duplicates satisfies ( 1 ) . Alternatively, for a fixed w and a, let Gz(w, a) be the graph obtained from G by substituting a clique C, of size a, for each v E V ( G ) , and then substituting a stable set of size w,, for each u E C,,. In general, G,(w,a) # Gz(w, a). However:

(4)holds for G e (1) holds for Gz(w, a).

(6)

Similar to before, if G is box perfect, it turns out not to be necessary to substitute a stable set of the same size for each node of C,, in order to conclude that (1) holds. In Section 5 we will prove;

Theorem 2. Creating joined duplicates preserves box perfection. Corollary 2. G is box perfect. A n y graph obtained from C by first creating joined duplicates and then creating unjoined duplicates satisfies ( 1 ) . Corollary 3. G is box perfect. 9 Any graph obtained from G by creating a series of joined andlor unjoined duplicates satisfies (1).

3. Classes of box perfect graphs Theorem 3. The following classes of graphs are box perfect. (i) Comparability graphs [4]. (ii) Cocomparability graphs [9, 41. (iii) Graphs whose clique-node incidence matrix is totally unimodular. (vi) p-Comparability graphs [4],defined immediately below. A p-comparability graph is a graph which arises in the following way: start with a digraph C that has a set T E V ( G ) , IT1 S p , such that every edge of G is in a

20

K . Cameron

Fig. 5.

dicircuit, and every dicircuit of G intersects T exactly once; add the chords of every dicircuit, delete T, and make all edges undirected. l-comparability graphs are precisely comparability graphs [4]. It was proved in [4] that p-comparability graphs (and thus comparability graphs), and cocomparability graphs are box perfect. These proofs were based on our Coflow Polyhedron Theorem ([4,51) which gives strong min-max properties for any digraph. In particular, this provides new proofs of the Greene-Kleitman Theorem and Greene’s Theorem. It also follows from the Greene-Kleitman Theorem and Corollary 1 that comparability graphs are box perfect since creating joined or unjoined duplicates preserves being a comparability graph. Similarly, it follows from Greene’s Theorem and Corollary 1 that cocomparability graphs are box perfect. The Edmonds-Giles Theorem says cocomparability graphs are box perfect.

Proof of (iii). It is easy to see that if the clique-node incidence matrix A of graph G is totally unimodular, then G is box perfect: For a postive integer q, let q be a vector of all 4’s. Then it is well known [22] that for integer-valued vectorsb and a, the following linear program (7) and its dual have integer-valued optimum solutions. By the linear programming duality theorem, the optimum objective values of a linear program and its dual are equal. This is precisely (4). maximize

wx subject to

Ax sq

Qaxsa.

(7) 0

We comment that if G is box perfect, its complement need not be. The graph of Fig. 5 is the line-graph of a bipartite graph, and hence box perfect, but its complement, the graph of Fig. 1, does not satisfy (1) for q = 2.

4. Total dual integrality Let A be a matrix, d and _c vectors of constants, and x a vector of variables. A system, Ax a d, of linear inequalities with rational A and d is called totally dual

Min-max relation for partial q-colourings

21

integral (TDI) if the dual of the linear program: maximize {ex:Ax s d } has an integer-valued optimum solution for every integer-valued c such that it has an optimum solution [9]. TDI systems are interesting because of the following result.

The TDI Theorem (Edmonds and Giles, [9]). If Ax S d is a totally dual integral system with integer-valued d, then for any c such that maximum{cx:Ax s d } exists, there is an integer-valued optimum solution x*. Thus a TDI system with integer-valued 4 provides the following integer min-max equality for any integer-valued _c for which either the min or the max exists. maximum{cx: Ax S d , x integer-valued} minimum{yd: y- A = c, y

3 0,

y integer-valued} .

A system, Ax S d, of linear inequalities is called box totally dual integral (box TDI), if it together with any upper and lower bounds on the individual variables

u:::{f

is TDI; that is, if for any 1, u E (Q U {kt.})", the system

is TDI.

A x s d is called upper box TDI if it together with any upper bounds on the variables is TDI; that is, if for any u E (Q U { +t.})", the system

{,":Zd

is TDI.

Groflin [21] proved that Ax S d is box TDI if and only if for any subset J of the variables, and any values uj E Q for j E J, the system

is TDI. Also, if Ax S d is TDI, so is any system obtained by changing some of the inequalities to equations (for a proof, see [28], Theorem 22.2). It follows that if Ax s d is upper box TDI, it is also box TDI, and thus box TDI and upper box TDI are equivalent. We will consider the following system of clique inequalities and non-negativity constraints for graph G. (10.1)

v clique c in G , C xu s 1; "€C

(10.2) v v E V ( G ) ,X"

3 0.

For our discussion here it does not matter if we consider only maximal cliques in (10) or all cliques.

Theorem 4. G is perfect.

a The system (10) of

clique inequalities and non-negativity constraints is TDI.

K. Cameron

22

Proof. The “if” part of this theorem follows by the TDI Theorem. The “only if” part is immediate by Lovasz’s theorem that creating unjoined duplicates preserves perfection (and thus for perfect graphs, (3) holds for q = 1). Theorem 4 motivates us to study graphs for which the system (10) is box TDI, which we will now show are the box perfect graphs, defined earlier in Section 1.

Theorem 5. G is box perfect.

e The

system (10) of clique inequalities and non-negativity constraints is box

TDI.

Lemma. If y- is an optimum solution to the dual of the linear program maximizeicx: Ax s d } , and r is a positive rational then y is an optimum solution to the dual of the linear program maximize {cx:Ax S r d } . Thus if Ax s d is TDI, so is Ax s r d . Proof of Theorem 5. Suppose G is box perfect. We will show that (10) is upper box TDI, and hence box TDI. We must show that for an integer-valued _c, the dual of the linear program (11) below has an integer-valued optimum solution.

2

maximize

CJ,

subject to

usV(C)

v clique C , C xu G I ; usc

v v E V ( C ) ,o s x ,

SU,.

We may assume that c, 0 and 0 s u, < cc, Vv. Let q be a positive integer so that for each u,, qu, is an integer. Then for this q , for w,,= c,,, and a,, = qu,, (4) holds. By the lemma, the y of (4.2), which is integer-valued, is an optimum solution to the dual of (11). Now suppose that the system (10) is box TDI. We must show that for a positive integer q, and non-negative integer-valued a and w , (4) holds. The system:

v clique C , C xu G I; usc

Vv E V ( G ) ,O S X , s a , l q ,

is TDI. Thus by the lemma, so is the system:

v clique C ,

x,

=S q ;

usc

[VV

E

v(G), o ~ x ==au. ,

Then the min-max (8) where c = w and Ax

G

d is (12) is the same as (4).

Min-max relation for partial q-colourings

23

5. Proofs of Theorems 1 and 2 Theorem 1. Creating unjoined duplicates preserves box perfection. Proof. Assume G is box perfect. Consider the graph G’ obtained from G by creating an unjoined duplicate t of u E V ( G ) . We will show that G’ is box perfect by showing that (1) holds for G;(w,a ) for all non-negative integer-valued w and a. Fix w and a, and let H’ denote the graph obtained from G’ by substituting for each node u E V ( G ’ ) a stable set S, = { u , : i = 1, . . . , w,} of size w,. It suffices now to show that (2) holds for H’ with the upper bound a,, = a, for each uiE S,. If a, = a , , then (2) holds for H’ since (1) holds for Gl(w‘, a ’ ) where wh = w, + w,,w: = w, for E V ( G )- u, a: = a,, for u E V ( G ) . Thus without loss of generality, 0 < a, < a,. We will use the fact that (2) holds in each of the following instances: H = H’ - S, (i.e. H is obtained from G as H’ was from G’) with the same upper bounds a,, as H’.

(13)

H’ with the upper bounds on the nodes u, E S, lowered from a, to a,, but the other upper bounds unchanged.

(14)

If in some optimum x for (13), xu, 3 a, for some uiE S,, then setting x:, = a, for t, E S,, and x; = xp for p E V ( H ’ )- S,, and taking X ’ = any optimum X for (13), it is clear that x’ and X ’ satisfy (2) for H‘ with the given upper bounds. Thus we may assume that for every optimum x for (13), xu, < a,, for all uiE S,.

(15)

Note that every I feasible for (14) is feasible for If’with the given upper bounds. If for some optimum X for (14), at least w, members of S, U S, are in U X , then we can assume that all members of S, are in UX, and then this X and any optimum x for (14) satisfy (2) for H’ with the given upper bounds. Thus we may assume that for every optimum X for (14) fewer than w, members of S, U S, are in UX . Let X * be an optimum X for (14) such that no node of S, is in U X . Let x* be an optimum x for (14). By complementary slackness, since some node uiof S, is not in U X , x:,=a,. It is easily seen that x* restricted to H and X * are optimum for (13). But this contradicts (15). 0

Theorem 2. Creating joined duplicates preserves box perfection. We comment that Theorem 2 also follows from Edmonds and Giles’ Theorem ([9, 10, 111) that duplicating variables preserves box total dual integrality [4].

Proof of Theorem 2. This proof is similar to the proof of Theorem 1. Assume G is box perfect. Consider the graph G’ obtained from G by creating a joined duplicate t of u E V ( G ) . We will show that G‘ is box perfect by showing

K . Cameron

24

that (1) holds for Gi(w, a) for all non-negative integer-valued w and g. Fix w and g, and let H' denote the graph obtained from G' by substituting for each node v E V ( C ' )with a clique C, = {vi:i = 1, . . . , a,} of size a,. It suffices now to show that (3) holds for H' with the weights w",= w, for each v, E C,. If w , = w , , then (3) holds for H' since (1) holds for Gl(w', g') where a: = a , a,, a: = a , for v E V ( G )- u , wl= w, for v E V ( G ) .Thus without loss of generality, 0 < w,< w,. We will use the fact that (3) holds in each of the following instances:

+

H = H ' - C, (i.e. H is obtained from G as H' was from G') with the same weights w, as H'.

(16) H' with the weights on the nodes u, E C, lowered from w, to w,, but the other weights unchanged. (17)

If in some optimum y- for (16), y,, s w, - w,, for some u, E C,; that is, Cu,Ecyc3 w,; then setting y & = y , for cliques C in H ' - C, with u, E C ; y & = yc for cliques C in H' - C, with ui@ C ; y:, = 0 for t, E C,; and yj, = yp for p E V ( H ' )C,; and letting S' be some optimum S for (16), it is clear that S' and y ' satisfy (3) for H' with the given weights. Thus we may assume that: for every optimum -y for (16), y , > w, - w,> 0 , for all u, E C,.

(18)

If for some optimum S for (17), at least a, members of C, U C, are in S, we can assume all members of C, are in S, and then this S together with any optimum y for (17) with y , increased to y,, + (w, - w,)for each u, E C, satisfy (3) for H' with the given weights. Thus we may assume that for every optimum S for (17), fewer than a, members of C, U C, are in S. Let S* be an optimum S for (17) such that no node of C, is in S*. Let y * be an optimum y for (17). By complementary slackness, since some node u,-of C, is not in S * , y:, = 0. It is easily seen that S* and y- * restricted to H are optimum for (16). But this contradicts (18). 0

6. An infinite class of graphs which satisfy (1) for all induced subgraphs and all q , but which are not box perfect. An rn-trampoline (rn 2 3 ) is the graph C with 2rn nodes

V ( G )= { v i :0 s i srn - 1) U { u i :0 s i srn - l } and E ( G ) = { ( v i , u i ) : O s i s r n - l } U { ( v i~, ~ + , ( ~ ~ ~ , , , ) : O ~ i ~ r n - 1 } U {(ui,u j ) :0 s i < j

s rn - l}.

A trampoline is a graph which is an rn-trampoline for some rn. A trampoline is called odd or even according to whether rn is odd or even. The rn 3-cliques {ui, vi,u ~ + ~ ( 0~ G~i s~rn~-)1,} of , the rn-trampoline are called the outer

25

Min-ma* relation f o r pariial 9-colourings

triangles. The graph of Fig. 1 is the 3-trampoline. The graph G, of Fig. 2 is the 5-trampoline.

Proposition 1 [4]. All the trampolines except the 3-trampoline satisfy ( 1 ) for all induced subgraphs and all q. Proposition 2 [4]. None of the odd trampolines are box perfect: The (2q - 1)trampoline does not satisfy (4) where a , = 1, a,,, = q - 1, and wu,= w,,, = 1, Osism-1. Proof of Proposition 1. In [4], it was shown that any induced subgraph F of an rn-trampoline G, except G itself, is a p-comparability graph, and hence satisfies (1). Assuming rn 3 4, Table 1 give an S and X satisfying ( 1 ) for G, for all values of

q<m. 0 Table 1

Is

9

1 2 9 = 3, . . . ,m - 1

Uu, u1,.

. . , U,-l

uo, u1,. . . I

Urn-!,

uo, u , , . . . ,

uo, u2 uo, u , , . . . , u q - l

the outer triangles the m-clique the m-clique

Proof of Proposition 2. We will display an x feasible for (4.1) except that it is not integer-valued, and a y feasible for (4.2) except that it is not integer-valued, such that the objective values of this x and y are equal, but are not an integer. (=m-1)

Osis2q-2

y, = 4 for each of the 2q

- 1 outer

triangles

y, = 0 for any other clique Y , = 0I} Y,, = 2

o s i s 2q - 2

2 F 2 x u+ a i=O

c

2cl-2 i=O

xv, = (2q - l)(t)

+ (2q - l ) ( q - 1 ) = 2q2 - 2q + 4

=2+2q+;.

0

K. Cameron

26

7. Balanced graphs A 0-1 matrix is called balanced if it has no odd order submatrix with exactly two 1’s in each row and column. A graph is balanced if its incidence matrix of maximal cliques versus nodes is balanced. Fulkerson et al. [16] proved that if matrix A is balanced, and d and a are non-negative integer-valued vectors, then the following linear program (19) and its dual have integer-valued optimum solutions:

maximize 1 & subject to Ax sd OS&SQ

Where A is the incidence matrix of maximal cliques versus nodes of balanced graph G and d is a vector of all q’s, this implies that (2) holds for G. The graph of Fig. 3 is balanced but not box perfect.

Acknowledgement I would like to thank Les Trotter and Jack Edmonds for helpful comments,

References [l] [2] [3] [4]

C. Berge, Balanced matrices, Math. Programming 2 (1972) 19-31. C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973). C. Berge, A min-max relation for the partial q-colorings of a graph, Part 1, this volume. K.B. Cameron, Polyhedral and Algorithmic Ramifications of Antichains, Ph.D. thesis, University of Waterloo (1982). [5] K.B. Cameron and J. Edmonds, Coflow polyhedra, to appear. [6] V. Chvatal, On certain polytopes associated with graphs, J. Combin. Theory (B) 18 (1975) 138- 154. [7] G.B. Dantzig, Linear Programming and Extensions (Princeton University Press, Princeton, New Jersey, 1963). [8] R.P. Dilworth, A decomposition theorem for partially ordered sets, Ann. Math. Ser. (2) 51 (1950) 161-166. [9] J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs, Ann. Discrete Math. l(1977) 185-204. [lo] J. Edmonds and R. Giles, Box TDI polyhedra, unpublished manuscript. [ l l ] J. Edmonds and R. Giles, Total dual integrality of linear inequality systems, in Progress in Combinatorial Optimization, W.R. Pulleyblank, ed. (Academic Press, Toronto, Ontario, 1984) 117- 129. [12] M.R. Farber, Application of L.P. Duality to Problems Involving Independence and Domination, Ph.D. thesis, Rutgers University, New Brunswick, New Jersey (1981). [13] M.R. Farber, Characterizations of strongly chordal graphs, Discrete Mathematics 43 (1983) 173-189. [14] D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Math. Programming 1 (1971) 168-174.

Min-max relation for partial q-colourings

27

[15] D.R. Fulkerson, Anti-blocking polyhedra, J. Combin. Theory (B) 12 (1972) 50-71. [16] D.R. Fulkerson, A.J. Hoffman and R. Oppenheim, On balanced matrices, Mathematical Programming Study 1 (1974) 120-132. [17] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980). [18] C. Greene, Sperner families and partitions of a partially ordered set, M.I.T. Math. Center Tracts 56 (1974) 91-106. [19J C. Greene, Some partitions associated with a partially ordered set, J. Comb. Theory (A) 20 (1976) 69-79. [20] C. Greene and D.J. Kleitman, The structure of Sperner k-families, J . Combin. Theory (A), 20 (1976) 41-68. [21] H. Groflin, On switching paths polyhedra, Combinatorica 7(2) (1987) 193-204. (221 A.J. Hoffman and J.B. Kruskal, Integral boundary points of convex polyhedra, in H.W. Kuhn and A.W. Tucker (eds.), Linear Inequalities and Related Systems, Annals of Mathematics Study No. 38 (Princeton University Press, Princeton, New Jersey, 1956) 233-246. [23] L. LovBsz, A characterization of perfect graphs, J. Combin. Theory (B) 13 (1972) 95-98. [24] L. LovBsz, Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972) 253-257. [25] L. LovBsz, Perfect graphs, in Selected Topics in Graph Theory 2, L.W. Beineke and R.J. Wilson, eds. (Academic Press, London, 1983) 55-87. [26] A. Lubiw, r-free matrices, Masters’ thesis, University of Waterloo (1982). (271 M.W. Padberg, On the facial structure of set packing polyhedra, Mat. Programming 5 (1973) 199-215. [28] A. Schrijver, Theory of Linear and Integer Programming (Wiley and Sons, Chichester, 1986).

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Discrete Mathematics 74 (1989) 29-32 North-Holland

29

ON LOCALLY-PERFECT COLORINGS Pierre DUCHET C.N . R.S., Uniuersite'Paris 6, Park, France

Received April 1988 A coloring of a graph is locally-perfecf if for every vertex u, the closed neighborhood of u contains no more than o ( u ) colors, where w ( u ) is the order of a largest clique containing u. Here is constructed, for any 9 a 3, a 9 + 1-chromatic graph, with clique number q, that admits a locally-perfect coloring. This answers a problem of Preissmann [3].

A (proper) coloring of a finite simple graph (G) is perfect if it uses exactly

o(G)colors, where o ( C )denotes the order of a largest clique in G. A coloring is locally-perfect [3] if it induces on the neighborhood of every vertex v a perfect coloring of this neighborhood. A graph G is perfect (resp. locally-perfect) if every induced subgraph admits a perfect (resp. locally-perfect) coloring. Preissmann proved that locally-perfect graphs form a proper subclass of the class of perfect graphs, and she asked for the following:

Problem 1 [3]. Is it true that if a graph C has a locally-perfect coloring then it has a locally perfect coloring in w ( v ) colors? The answer is affirmative for o(C)= 2 since a triangle-free graph that admits a locally-perfect coloring is necessarily bipartite. We show below that the answer is negative when o(G)L 3. In fact we exhibit a graph G admitting a locally-perfect coloring but which is not even o ( C ) colorable. A ( p , q)-coloring [ l ] of a graph is a coloring with p colors such that at most q appear in the neighborhood of every vertex.

Proposition 2. If a graph G admits a ( q + 1, q)-coloring, it is q-colorable. Proof. Let Q, : V ( G ) + (1, . . . , q + l} be (q + 1, q)-coloring of G. A q-coloring Q,' can be defined as follows, If q ( v )# q + 1, then Q,'(v) = Q,(v). If ~ ( v =) q + 1; then Q,'(v)is the smallest color not appearing in the neighborhood of v. 0 A stronger form of (2) is proved in [l]. More generally, [l] gives rather good estimates of the smallest integer f ( s , q) such that there exists a graph admitting a ( f (s, q), q)-coloring but which is not s-colorable. The next theorem yields f ( q , q) = q 2 and answers negatively to Problem 1.

+

0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

30

P . Duchet

+

Theorem 3. For any integer q 2 3, there exists a q 1-chromatic graph, of clique number q, which admits a locally-perfect coloring into q + 2 colors. Clearly, it suffices to construct a non-q-colorable vertex-transitive graph' Gq, with clique number q , which admits a (locally-perfect) ( q 2, q)-coloring. Notice that G9will be q + 1-chromatic, by Proposition 2. Let X be a q 2-element set. We define G, = (V, E) by:

+

+

Vertices ( x , A ) and ( y , B) are joined in G9 whenever x $ B, y $ A and x Z y . Obviously, G, is vertex-transitive and has order

(z)

the respective coordinate mappings. The We denote by cp : V +X and $J : V + clique number of G, is at most q since if {(xl, A , ) , . . . , (x,,,, A,)} is a clique of G,, we necessarily have xi + x i (for 1=z i , j S w ) and { x , , . . . ,x,} nA , = 0. For x E X and A E (f),set:

v, = v n c p - ' ( ~ ) W, = V n $J-'(A)

z,

= {x

E

v; x E $J(tJ)}.

The Vx's are stable sets of order (9 g I ) which partition V. The W,'S are cliques of order q which partition V. Observe also that for any distinct elements A, p, Y of X,the set

v, u (2, n Vp) u { Y , { A , P}} of order (" ;I ) + q + 1= (, ;*).

SLp.v:=

is a stable set It follows that cp is a ( q + 2, 4)coloring of G, and that G, has clique number q and stability number (7' ;):'. It remains to prove that G9 is not q-colorable. We first establish the following lemma:

Lemma 4. In G,, every stable set of order (" g2) is some Slmn. Proof. Let S be a stable set of order (" 12).Put S,:=S observation is immediate:

n V,. The following

If S, and S, are not empty and if x Z y , then we have x all v E S, or y E ~ ( vfor ) all v E S,.

E

~ ( v for ) (1)

'In a vertex-transitive graph, all vertices play the same role, i.e. the automorphism group acts transitively on vertices.

31

Locally -perfect colorings

Because of (l), it is convenient to consider the digraph D whose vertices are the non-empty sets of the form S,, with an arc (S,, S,) each time we have x E v ( v ) for all v E S,. By the definitions, the in-degrees d-(S,) in D are not greater than 2. Moreover, we have: If d-(S,) = 1 then IS,l s q. If d-(S,) = 2 then IS,l = 1. By (l), D is complete; since q 2 3, it follows, by (2) and (3) that if D has order 24 then IS1 < ( 9 t’). The same inequality holds if D has order 1 or 2 or if D contains a directed 3-cycle. Hence D contains the transitive tournament T3 with vertices, say SA,S,, S,,. Henceforth, by (2) and (3), we have S = SAU S, U S,, with

Since IS( = ( 9 t 2 ) ,we have S = SAW,,.0

Proof of theorem (3) (continued). If G9 is q-colorable, it admits by Lemma (4) partition into q sets S ( i ) = SA,,,,,,, 1zs i S q. Necessarily the Aj’s are all distinct hence at least two pi’s coincide, say p , = p2. But this is a contradiction, since the vertex ( p , , {A,, A?}) = ( p 2 , {A2, A,}) belongs to both S(1) and S ( 2 ) . 0 Graph G9 defined above looks like a Kneser graph [ 2 ] . Define, more generally a graph G,,,,, = ( V , E ) by:

v := [( R , T ) E);(

[

x

(f)R

E : = { ( R , T ) , ( R ’ , T ’ ) }E

;

(r)

T)

:R

I

n T’ =R’ nT =0 .

Here 0 S r 6 t S n and X is an n element set. Kneser’s graph is G , , , , and G9 corresponds to G,,2,9+2.

Problem 5. What is the chromatic number of G,,,,,? Remark 6. Graph G9, as defined above, it not perfect. Problem (1) remains open for perfect or locally-perfect graphs [3].

Acknowledgement Our sincere thanks are due to M. Preissmann who kindly pointed [ l ] to our attention.

32

P. Duchet

References [l] P. Erd6s, Z. Furedi, A. Hajnal, P. KomjBth, V. Rod1 and A . Seress, Coloring graphs with locally few colors, Disc. Math. 59 (1986) 21-34. [2] L. Loviisz, Kneser’s conjecture, chromatic number and homotopy, J. Comb. Th. 25 (1978) 319-324. [3] M. Preissmann, Locally-perfect graphs, preprint IMAG, Grenoble, France (1986) submitted.

33

Discrete Mathematics 74 (1989) 33-49 North-Holland

THE SUBCHROMATIC NUMBER OF A GRAPH M.O. ALBERTSON Smith College, U.S.A.

R.E. JAMISON and S.T. HEDETNIEMI* Clemson Uniuersify, U.S.A.

S.C. LOCKE Florida Atlantic University, U.S.A.

The subchromatic number X,5(G)of a graph G = (V, E) is the smallest order k of a partition { V , , V,, . , . , V,} of the vertices V ( G ) such that the subgraph induced by each subset V, consists of a disjoint union of complete subgraphs. By definition, &(G) s X ( G ) , the chromatic number of G . This paper develops properties of this lower bound for the chromatic number.

(v)

1. Introduction An n-coloring of a graph G = ( V , E) is a function f from V onto N = { 1, 2, . . . , n } such that whenever vertices u and 21 are adjacent, then f ( u ) #f(v). An equivalent definition, which provides much of the motivation for this paper, is that an n-coloring of C is a partition {Vl, V2,. . . , V,} of the vertices V ( G )into color classes, such that for every i = 1, 2, . . . , n, the subgraph (V,) induced by V, is totally disconnected, or equivalently, is the disjoint union of K,’s (complete graphs with one vertex). A partition {V,, V,, . . . , V,} of V ( G )is called complete if for every i , j , 1 G i <j S n , there exists a vertex u in V, and a vertex v in V, such that u and 21 are adjacent. The chromatic number X ( G ) and the achromatic number Y ( G ) are the smallest and largest integers n , respectively, for which G has a complete n-coloring. Notice that in the definition of the chromatic number the completeness of the partition is not required but follows easily from the definition, whereas the completeness of the partition used in defining the achromatic number is essential. The chromatic number, of course, is a very well studied parameter, whose history dates back to the famous Four Color Problem and the early work of Kempe in 1879 [25] and Heawood in 1890 [22]. The achromatic number was first studied as a parameter by Harary, Hedetniemi and Prins in 1967 [20], and later named and studied by Harary and Hedetniemi in 1970 [19]. This paper was motivated in part by an interest in developing interesting lower * Research supported in part by Office of Naval Research Contract N00014-86-K-0693. 0012-365X/89/$3.50 01989, Elsevier Science Publishers B.V. (North-Holland)

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M.O. Alberrson et al.

bounds for the chromatic number, ones that would involve partitions {Vl, V2,. . . , V,} of the vertices of a graph, with certain conditions imposed on the induced subgraphs ( V , ) . In general, let P be an arbitrary property of a graph. We say that a property P is (induced) hereditary if whenever a graph G has P, then so does every (induced) subgraph G’ of G. A subset S of V ( G )is a P-set of C if (S), the subgraph induced by S, has property P. A (generalized) P-coloring of G is an assignment of colors to the vertices of G such that for any given color, the set of vertices having this color is a P-set. A (P, m)-coloring of G is a partition n = {Vl, V2,. . . , V,} of V ( G )such that each subset V, is a P-set. The P-chromatic number, denoted XJG), is the smallest integer m for which G has a (P, m)-coloring. The notion of generalized P-colorings of graphs, or of P-partitions of the vertices of a graph, is certainly not a new one. It seems to have been ‘independently discovered’ by several authors around 1968 and again by several authors around 1985. The definitions involving P-sets and P-colorings given here are those given by Hedetniemi in 1968 [23], whose studied the non-hereditary property PI: that a graph be disconnected or the trivial graph, and the hereditary property P2: that a graph be acyclic, in which case the P2-chromatic number is called the point arboriciq of a graph. Independently in 1968, Chartrand, Kronk and Wall also defined and studied the point arboricity of a graph [lo]. In 1968, Chartrand, Geller and Hedetniemi [8] studied the hereditary property P,(k): that a graph contain no path of length k, for some fixed value of k. Also in 1968, Sachs and Schauble [37] studied the hereditary property P4(k): that a graph contain no complete subgraph on k vertices, for some fixed k. P4(3)-colorings were also studied by Harary and Kainen in 1977 [21], where they showed that the vertices of any planar graph can be partitioned into two sets V, and V2such that neither ( V , ) nor ( V2) contains a KJ. In 1969 Kramer and Kramer [27], [28] and [26] studied colorings in which no two vertices in (V,) whose distance apart in C is 4 receive the same color; this of course is equivalent to studying (normal) colorings of the kth power of a graph. In 1970, Lick and White [32] introduced k-degenerate graphs, which are defined in terms of the hereditary property P,(k), called k-degenerate: that no induced subgraph of a graph have minimum degree exceeding k, for some fixed value of k. K-degenerate graphs have since been studied by Cook [13] and Borowiewcki [4]. Also in 1970, Hedetniemi [24] continued to study disconnected colorings of graphs, and offered this comment: “In spite of the unnaturalness of the concept of disconnected colorings, more than two dozen results have been constructed for D-colorings and X, which are virtually identical to corresponding results that have been established for the traditional colorings and chromatic number X ( G ) of graphs. These new results indicate that the established results reveal much less about properties of colorings than they do about concepts which are much more general; they also reveal that most of the established results on coloring can be proved using little more than purely set theoretic arguments”. Again in 1970, Folkman [16] proved a result

Subchromatic number of a graph

35

which in effect said that for any, nontrivial hereditary property P of a graph, there are graphs whose P-chromatic number is arbitrarily large. Three other studies of generalized P-colorings were offered by Cockayne [ l l ] , and Cockayne and Miller [12], who proved that for any hereditary property P, if a graph G has a complete (P, k)-coloring and a complete (P, n)-coloring (k < n), then for every m, k s m < n, G must also have a complete (P, m)-coloring. In 1977, three other papers appeared along these lines: Lesniak-Foster and Straight [30] studied the hereditary (co-chromatic) property P6: that a graph be either a complete graph or an empty graph; Lesniak-Foster and Roberts [29] studied, among other things, the property P3(2), i.e. ( K ) has no paths of length 2; and Sampathkumar, Neeralagi and Venkatachalam [38] studied the hereditary property P,(k): that no two vertices in ( K ) lie at distance k from one another in G, for some fixed value of k. Around 1985 the subject of generalized colorings of graphs seems to have experienced a “rebirth”. In [18] Harary referred to generalized colorings as conditional colorings and considered, in particular, the general classes of properties PF, that a particular graph F is forbidden to be induced by any of the vertices in a set K. Also Andrews and Jacobson [l], [2], studied the hereditary property P,(k): that the maximum degree of a vertex is less than or equal to k, for some fixed value of k. Mynhardt and Broere [35], [5], also studied the general property PF: that a graph contain no induced subgraph isomorphic to a given graph F; Brown and Corneil [7] studied general P-chromatic numbers for hereditary properties, vertex P k-critical graphs and uniquely P k-colorable graphs; and Domke, Laskar, Hedetniemi and Peters [14] studied the hereditary property Pg: that a graph is isomorphic to a complete r-partite graph. We have not attempted here to complete a comprehensive survey of generalized colorings or point partition numbers of graphs, rather we have attempted to indicate something of the wide variety of properties which have been studied. Table 1 summarizes some of the these properties. Table 1. Some P-colorings which have been studied. Let for:

n=[V,, V,, . . . , V,,,) be a partition of V ( G ) ,where for every i, 1

i G m, (V,) has property P

PI: ( V , ) is disconnected or trivial (231, [24] P,: (V,) is acyclic [lo], [23] P,(k): (V,.) has no path of length k , for some fixed k [8] P,(k): (V,) has no complete subgraph of size k , for some fixed k [37]

P&): (V,) has no induced subgraph whose minimum degree is bk for some fixed k [32] P6: ( V , ) is either a complete graph or a graph without edges [30] P,(k): (V,) has no two vertices at distance k in G, for some fixed k [38] P,(k): (V,) has maximum degree G k , for some fixed k [l], [2] Ps: (V,.) is a complete r-partite graph, for any r [14] PF: (V,) has no induced subgraph isomorphic to F [5], (351, [7]

PF: (V,) has no induced subgraph isomorphic to any graph F i n F [7] P,”: (V,) is a disjoint union of complete subgraphs [35], [this paper] P,,(k): ( V , ) contsins no induced K , , k [13]

36

M.O. Albertson et al.

In this paper we study another generalized coloring of a graph, one which provides both an interesting lower bound for the chromatic number and which has interesting properties in its own right. A subcoloring of order n of a graph G is a partition {Vl, V,, . . . , V,} such that for every i = 1, 2, . . . , n, the subgraph (F)is a disjoint union of complete subgraphs (of various sizes), i.e. = UKj’s. The subchromatic number X,(G) is the smallest n for which G has a complete subcoloring of order n ; while the subachromatic number Y,(G) is the largest integer n for which G has a complete subcoloring of order n. The following chain of inequalities follows immediately from these definitions:

(v)

For any graph G, X,d X d Y d Y,.

(1)

It also follows from the definition of a subcoloring that if 17 = { V,, V,, . . . , V,} is a subcoloring of minimum order, i.e. X,(G) = n, then I7 is in fact a complete partition of V ( G ) . Although we have newly named the subchromatic number here, it was originally studied in a different framework by Mynhardt and Broere, [35] and [5], who referred to X,(G) as the F-chromatic number, F(G), for the graph F = K1,2. That is, if a graph has no induced subgraph isomorphic to the path on three vertices, also denoted K1,*, then it must be a disjoint union of complete subgraphs. Subcolorings are also related to the co-colorings of Lesniak-Foster and Straight [30], in which the induced subgraphs must either be complete graphs or empty graphs. If we define the co-chromatic number X c ( G ) to equal the minimum number of colors in a co-coloring of the vertices of G , then it follows that, in fact X,S X c S X , since every co-coloring is a subcoloring. The remainder of this paper reviews results which have either been proved or can readily be inferred from previous results in the literature, and develops new properties and bounds for the subchromatic number of a graph.

(v)

2. Results from previous studies Because the property Plo: (K) is a disjoint union of complete subgraphs is hereditary, we can immediately infer a number of results about the subchromatic number, which are corollaries of earlier studies. [7] For any induced subgraph H of a graph G, X , ( H ) S X , ( G ) .

(2)

[16, 35, 71 For every k 3 1, there is a graph Gk with x,(Gk) = k.

(3)

[35] For every k 5 1, there is a K3-free graph Gk with X,(C,) = k.

(4) (5)

[30] For any graph G with p

6 8 vertices,

X,(G) S X , ( C ) d 3.

[30] for any graph G with p vertices, minimum degree 6 and maximum degree A, X,(G) S X c ( G )d min{l + A, p - S } . (6) ,...,P n , [30, 351 For the complete multi-partite graph, G = KplPz whereplSp2d.-.~pn,X,(G)=Xc(G)=minl,j,n{n, n-j+pj}. (7)

Subchromatic number of a graph

31

A graph G is vertex k-subcritical if X,(G) = k but X,(G - v ) < k for all v in V ( G ) .

[7] Any k-subchromatic graph contains a vertex I-subcritical sub(8) graph, for all 1 S k. [7] If G is a vertex 2-subcritical graph, V ( G )= [vo, v,, . . . , v,} and H,, H2, . . . , H, are k-subchromatic graphs (k 2 l), then the graph F which results from successively substituting Hifor vi (every vertex in Hi is joined to every vertex adjacent to vi) is not k-subcolorable.

(9)

Note in (9) above, that if Hi= H for all i = 1, 2, . . . , n and IV(H)I = rn then IV(F)(= rnn + 1. Note also that K l , 2is the only vertex 2-subcritical graph.

[35] If X,(G) = k and H = K1,2[G], then X , ( H ) 3 k + 1, where V(G,[G2]) = C(CJ x V(G2)and ( u , , u 2 ) is adjacent to (v,,v2) if and only if either u l v l is in E l or u 1= v1 and u2v2is in E2. (10) [5] For every outerplanar graph G, X,(G)S3, and this bound is tight, i.e. there exists an outerplanar graph H for which X , ( H ) = 3. (11) [5] For every planar graph G, X , ( C ) S 4, and this bound is tight. (12) Next, let X J G ) , the partite chromatic number, equal the minimum order of a partition V,, V2,. . . , V, such that for every i, 1 < i s m , (V;:) is a complete r-partite graph, for some (variable) r.

[14] For any graph G, X,(G)=X,(G'), where G' denotes the complement of G. (13) [14] For any graph G with n vertices, (14) (i) 1<X,(G) *x,(G') s n2/4; (ii) 2 ~ X , ( G ) + X , ( G ' ) s X ( G ) + X ( G ' ) s n+ 1; (iii) X,(G) +X,(G') S 2X(G). Finally, in (111 Cockayne generalized a classic result of Szekeres and Wilf [39], that X ( G )S 1 + maxC,
M.O. Alber&on et al.

38

&(G) = minvinVd,(v).And last, let P , ( G ) equal the maximum number of edges in a subset F of E, no two of which have a vertex in common. [ll] for any graph G, X,,(G) =s1 + maxG.
3. Graphs with small subchromatic numbers Clearly, the subchromatic number of a graph G equals one if and only if G is a disjoint union of complete graphs, and in particular, the subchromatic number of any complete graph K,,, or the complement of a complete graph, is one, i.e. X,(K,,) = X,(K;) = 1. Unfortunately, no straightforward characterization seems to exist for graphs whose subchromatic number is two. The next several results give an indication of the difficulties involved in characterizing the class of 2subchromatic graphs.

Proposition 1. (i) For any bipartite graph G , which is not a disjoint union of complete graphs, X , ( C ) = 2; (ii) for any graph G, which is not a disjoint union of complete graphs and is the complement of a bipartite graph, X , ( G ) = 2. ; (iii) for any split graph G = ( V , E ) (i.e. a graph whose vertex set can be partitioned into two sets ZJ and W , such that ( Z J ) is totally disconnected and ( W ) is a complete graph), which is not a disjoint union of complete graphs, X,(G) = 2; (iv) for any cycle C,, with n 3 4 vertices, X,(C,,) = 2; (v) except for the three graphs in Fig. 1, f o r which X , ( G ) = 3, every graph G with G6 vertices satisfies X,(G) S 2; (vi) X,(K,, - X ) = 2, for any non-empty set X of pairwise disjoint edges of K,,.

w 5

K,+ K Fig. 1. The smallest three 3-subchromatic graphs.

2,3

Subchromatic number of a graph

39

A block of a graph G is a maximal, nonseparable (connected, nontrivial and has no cutvertices) subgraph of G. The block graph B ( G ) of G is the graph whose vertices correspond to the blocks of G, and two vertices are adjacent if the blocks to which they correspond have a cutvertex in common. As characterized by Harary [17], a graph H is the block graph of some graph is and only if every block of H is a complete graph. Block graphs are another family of 2-subchromatic graphs.

Theorem 2. For any block graph H , X , ( H ) S 2. Proof. We may assume w.1.o.g. that the block graphs considered are connected. We proceed by induction on the number of blocks b of H. Clearly, if b = 1 then H is a complete graph and X , ( H ) = 1. Assume therefore that for every graph H with b s k blocks, X , ( H ) s 2. Let H be a block graph with b = k 1 blocks. By the definition of a block graph, H must have at least one block B which contains exactly one cutvertex, say u. If we delete every vertex in B , except v , from H then the remaining graph G’ will be a block graph having k blocks. therefore, by our inductive hypothesis, X,(G’)s2. Let the vertices of G’ be subcolored with 2 colors, and assume without the loss of generality that vertex u is colored 1. Then all the other vertices of B can be colored 2. Since they form a complete subgraph, the resulting coloring is a 2-subcoloring of H. 0

+

A cactus is a connected graph each of whose blocks is either a K 2 or a chordless cycle.

Theorem 3. For any cactus C with n 2 4 vertices, X , ( C ) = 2. Proof. We again proceed by induction on the number b of blocks in C. Clearly, if b = 1 then C is a cycle of length 2 4 , and X,?(C)= 2. Assume therefore that X , ( C ) = 2 for all cacti C with more than three vertices and with b 6 k blocks for some fixed k , and let C be a cactus with b = k + 1 blocks. Then C has at least one block B having one cutvertex, say v . Consider the graph C ’ which is obtained by removing from C all of the vertices of B except the cutvertex u. Clearly, C ’ is a cactus with S k blocks. Case 1. C’ has at least 4 vertices. Therefore, by our inductive hypothesis, X , ( C ’ ) = 2. Let the vertices of C‘ be subcolored with 2 colors. Assume, without loss of generality, that vertex v is colored 1, and consider the color class containing v. Subcase la. Vertex v is not adjacent to any other vertex colored 1. In this case we can combine any 2-subcoloring of B in which vertex v is also colored 1 with the given subcoloring of C‘ to produce a 2-subcoloring of C.

40

M.O. Albertson

et

al.

Subcase lb. Vertex v is adjacent to at least one other vertex colored 1. In this case we can combine the given subcoloring of C' with any 2-subcoloring of B in which 21 is colored 1 but is not adjacent to any other vertex colored 1, to produce a 2-subcoloring of C. Since B is, by definition, either a K 2 or a cycle, we know that for any vertex v in B we can always construct a 2-subcoloring in such a way that IJ is colored 1 and is not adjacent to any other vertex colored 1. Case 2. C' hasfewer than 4 vertices. In this case we know that C' = K 3 , K,,z o r K z ; and in each of these cases we can assume that a 2-subcoloring of C' exists, in

which vertex v is colored 1 and is not adjacent to any other vertex colored 1. We can then refer back to Case la. A corollary of the next theorem provides another family of 2-subchromatic graphs.

Theorem 4. Let G be an interval graph which contains no induced K,,,+,. Then X,(G) s n. Proof. Represent G by a family F of intervals. Let I be a set of mutually disjoint intervals of maximum cardinality m. Without loss of generality we may assume that I consists of minimal intervals, since if an interval I in I contains a smaller interval J , we could replace I by J and still have a set of disjoint intervals of the same size. Let the intervals in I be: I, = [ai,b , ] , i = 1, 2, . . . , m, listed from left to right. Let us further assume that among all such sets of intervals satisfying the above conditions, I is chosen to minimize C a,. Claim. If an interval A in F meets I, for some i and no contains b,.

4,

for j # i , then A

Proof of claim. Let A = [a, b] and assume that A does not contain b,. Then since 4 is minimal, a
Subchromatic number of a graph

41

A

B

c

+ dn

c+ en Fig. 2.

clique, since by the claim, any interval A in C,+, which meets only Ic+dn contains bc+dn.If A meets two or more intervals Zi, the first is since 0 (A)= c + dn, and since A meets a later interval, it must contain bc+dn. We now show that if A is in c,+d,,and B is in C,,,, with e > d , then A and B cannot meet. Indeed, suppose they did meet. Note that A meets tc+dn, B meets and neither A nor B meets an earlier interval (cf. Fig. 2). Thus, the left endpoint of B is larger than bc+,,-,. Since A intersect B is not empty, it follows that A contains the left endpoint of B. Thus A meets B and the n intervals Zi for i = c + dn, . . . , c + ( d + 1)n - 1, and the set of vertices of G corresponding to this set of n + 2 intervals fcims an induced K1,,+l, which contradicts our assumption that G contains no induced K I , , + l . Thus, no A in C C + , can meet any B in C,,,, if e > d , and we produce a legitimate n-subcoloring of C .

Corollary 5 . For any indifference graph G , X,v(G)S 2. An indifference graph is a graph G = (V, E) for which a real-valued vertex function f exists satisfying: (u,v ) in E if and only if I f ( u ) -f(v)l< 1. In [36] Roberts showed that G is an indifference graph if and only if C is an interval graph containing no induced K1,3. We close this section by noting the following about 3-subchromatic graphs.

Proposition 6. (i) For any wheel W, ( = C , + K l ) , f o r n 2 5 , XJW,) = 3; (ii) [35]for any graph G with n S 8 vertices, X,(G) s 3, and this bound is tight; (iii) [5] for any maximal outerplanar graph G , X , ( C ) S 3 , and this bound is tight. We suspect that the upper bound in Proposition 6(ii) can be considerably improved. Although we do not know the smallest value of n for which there exists a 4-subchromatic graph on n vertices, a construction presented in the next section can be used to produce a 4-subchromatic graph with 13 vertices. Proposition 6(iii) is a bit disappointing since the chromatic number of any maximal outerplanar graph with n 2 3 vertices is 3; it is easy, however, to construct maximal outerplanar graphs whose subchromatic number also equals 3.

M.O. Albertson

42

et al.

4. General results It is easy to construct graphs whose chromatic number is arbitrarily large. For example, the process of adding a new vertex and joining it to every vertex in a given graph G produces a graph (denoted K, G) whose chromatic number is one greater than the chromatic number of G, i.e. X ( K I G) = 1 + X ( C ) . However, this is not always true for the subchromatic number.

+

+

Proposition 7. For any graph G, X,(K, + G) = 1 + X , ( G ) if and only if there does not exist a X,-subcoloring of C in which one color class V , induces a complete graph As previously mentioned in Section 2, several authors [16], [35] and [7] have established results which are sufficient to show that there are graphs whose subchromatic number is arbitrarily large. The following is a simple proof of this fact.

Theorem 8. [35] For any positive integer n, there exists a K,-free graph G with X J G ) 3 n. Proof. It is well known that there exist K3-free graphs having arbitrarily large chromatic numbers (cf. [34]). Therefore, let G be a K,-free graph for which X ( G )= 2n. Let X J G ) = m and let {V,, V2,. . . , V,} be a subcoloring of G. Since G is K,-free, each is a disjoint union of K,'s and K,'s, and therefore the vertices in each V , can be 2-colored in the normal sense. Therefore, X ( G ) = 2n G 2m = 2Xs(G),or A',((?) 3 n. 0

(v)

Corollary 9. If C denotes the size of a largest clique in a graph G, then

X(G ) 6 C * X,(G ), or X / C6 X,. Theorem 8 establishes the existence of graphs with arbitrarily large subchromatic numbers. Theorems 10 and 13, which follow, provide methods for constructing such graphs. Let C and H be graphs and let C v H denote the graph which results from adding a new vertex v to the disjoint union of G and H and joining v to all vertices in G and H.

Theorem 10. For any graphs G and H with X , ( G ) > k and X , ( H ) > k , X,(G v H ) 3 k + 1. Proof. Assume that X,(G v H) 6 k and let n = { V , , V2,. . . , V,} be a subcoloring of G v H of order k. Then k colors must be used to color the vertices of G and k colors must be used to color the vertices of H. Consider the color assigned to vertex v , call it 'red'. It follows that there must be at least one vertex u in G

Subchromatic number of a graph

x S =1

x

x S= 2

x

S

43

=3 S

=4

Fig. 3. Small k-subchromatic graphs.

and at least one vertex w in H which are also colored ‘red’. That is, there is a u - v - w red path. But since u and w are not adjacent in G v H, this cannot be a legitimate subcoloring of C v H. Therefore, X,(G v H) k + 1. 17 Fig. 3 illustrates the smallest k-subchromatic graphs constructible by the G v H-process, for k = 1,2, 3 and 4. Even though S, is not drawn with a planar embedding, it is easy to see that S, is a planar graph. Thus, as is pointed out in [5],X,(G) = 4 is possible for planar graphs. Notice that these graphs have 2k - 1 vertices and contain the full binary trees with 2k - 1 vertices as spanning subgraphs. These graphs, furthermore, can be seen to be interval graphs and they are also chordal graphs (graphs in which every cycle of length strictly greater than three possesses a chord, i.e. an edge joining two nonconsecutive vertices of the cycle), for which the size of the largest clique, the subchromatic number and the chromatic number are all equal. The next two results illustrate properties of the G v H construction. A perfect elimination ordering of a chordal graph G = (V, E) is an ordering of the vertices in V, say

44

M . 0. Albertson et al.

vl, v2, . . . , v,, such that each vertex ui is simplicia1 (i.e. ( N ( v , ) )is a complete subgraph) in the subgraph induced by the vertices vi, vi+,,. . . , v,. A Hamiltonian elimination ordering (HEO) is a perfect elimination ordering in which ui is adjacent to viClfor 1 6 i 6 n - 1. Lemma 11. If G and H are both chordal (resp. Interval) graphs, then so is G v H , and if G and H have Hamiltonian elimination orders, then so does G v H . Proof. The fact that G v H is chordal is obvious. If C and H are interval graphs, represent G by negative intervals and H by positive intervals, and then represent the vertex v by an interval covering all the others. For the H E 0 for G v H take the H E 0 for G , next take vertex v, and then take the H E 0 for H. 0 Theorem 12. For every integer k > 0, there is a chordal graph G with clique number k and subchromatic number k. Moreover, there is such a graph that is an interval graph with an HEO. Proof. We proceed by induction. Let GI be the graph with one vertex. Clearly GI has all of the required properties for k = 1. Inductively define G,,, = G,, v G,. The result follows from Lemma 11 and the fact that a largest clique in G,,,, consists of a largest clique in G, together with vertex x. 0 The following result, which is a corollary of (6) in Section 2 [35], describes another family of graphs for which the size of the largest clique, the subchromatic number and the chromatic number are all equal.

Theorem l3 [35]. For any positive integer m , X,( Km,m . _ _, m . ) = X ( Km,m,., , m ) =m , is the complete m-partite graph containing m classes of m vertices where Km,m,...,m each. ,

Proof. Trivially, X,(K,,, , _ . m _ ,) S m since X ( K m , ,,_..,m ) = m and X,(G) 6 X ( G )for any graph G. Therefore, assume that Xs(Km,m,...,m) = k < m , and let II= {V,, V,, . . . , V,} be a k-subcoloring of Km,m ,__., m. It follows that at least one color class, say V,, must have more than m vertices. Therefore, (V,.) cannot be a totally disconnected graph and must contain at least one pair, say ( u , v ) , of adjacent vertices. But every other vertex in is adjacent to either u or v. Therefore ( V , ) is a connected graph. But clearly ( K ) cannot be a complete graph since it has more than m vertices. Therefore I7 is not a legitimate subcoloring and x s ( K m , m ,...,m ) 3 m* 0

(v)

In 1941 Brooks [6] proved his classical theorem: for any connected graph G with maximum degree A, X ( G )< A 1, where equality holds if and only if G is a complete graph or an odd cycle. For the subchromatic number we can derive a stronger result.

+

Subchromatic number of a graph

45

Theorem 14. Let G be a k-regular graph, k 3. Then C contains a ( [ k / 2 ]+ 1)colorable, spanning subgraph H such that d H ( v )2 k - 1, for every v in V ( G ) , where d H ( v )denotes the degree of vertex v in subgraph H.

+

Proof. Let H be a ( [ k / 2 ] 1)-colorable, spanning subgraph of G having a maximum number of edges, and let the vertices of H be colored with [ k / 2 ] 1 colors. Suppose d H ( v )< k - 1 for some v in V ( G ) . Let NH(v)and N,(v) denote the set of vertices adjacent to v in H and G, respectively, and let N,[v] = {v} U N,(v). Consider the colors assigned to the vertices in NG(v).Since d H ( v ) < k - 1, there must be at least two vertices, say w1and w2, which are adjacent to v in G but not in H. Let c ( v ) denote the color of v. If c ( v ) # c ( w 1 ) (or (c(v)#c(w2))then we can add the edge between v and w1 (or w2) to H, contradicting the maximality of E ( H ) . Therefore, c(v) = c(w,)= c(w2). Now consider the colors assigned to the vertices in N H [ v ] .If fewer than [ k / 2 ]+ 1 colors are used to color these vertices, then we can recolor v with one of the unused colors and add edges (v, w l ) and (v2,w2)to H , again contradicting the maximality of E ( H ) . Therefore, we assume that [ k / 2 ]+ 1 colors are used to color the vertices in N H [ v ] .It follows that at least one color is used at most once to color the vertices in NH(v)(i.e. [ k / 2 ]colors must be used to color N,(v); if each of these colors were used at least twice, we would have at least 2 [ k / 2 ]2 k - 1 vertices in N,(v), but by assumption INH(v)l< k - 1). Let vertex u be the only vertex in N,(v) assigned this color. Then if we delete edge (v, u ) and recolor v the same as u, we can add edges (v, w l ) and (v2, w2) to H, contradicting the maximality of E ( H ) . Therefore, every vertex v in H must have d H ( v ) b k - 1. 0

+

Corollary 15. For any k-regular graph G , k 0, X,(G) s [ k / 2 ]+ 1. Furthermore, there is a subcoloring of G with [ k / 2 ]+ 1 colors in which every color class is a disjoint union of only KI’s and K2’s. Proof. A 0-regular graph G is simply a disjoint union of K,’s; hence X,(G) = 1= [0/2] + 1. A 1-regular graph G is a disjoint union of K2’s; hence X , ( G ) = 1 = [ 1 / 2 ]+ 1. A 2-regular graph G is a disjoint union of cycles. From Proposition 1 it follows that X,(G) S 2 = [ 2 / 2 ]+ 1. For k 2 3 we apply the proof of Theorem 14 as follows. Let H be a ( [ k / 2 ] 1)-colorable, spanning subgraph of C having a maximum number of edges. If H = G , we are done. Suppose H Z G . By the proof of Theorem 14, d H ( v )b k - 1 for every vertex v of G. Let the vertices of H be colored with [ k / 2 ]+ 1 colors, let v be a vertex with d H ( v ) = k - 1 and let w be the vertex adjacent to v in G but not in H. Then, as in the proof of Theorem 14, it follows that c(v) = c(w). Thus, if we add the edge (v, w ) to H we will produce a subcoloring of a spanning subgraph H‘ in which the set of vertices assigned color c(v) is a disjoint union of only K l ’ s and K 2 k . If H’ = G , we are done; if

+

46

M.0.Albertson et al.

not, we repeat the above argument for some vertex v' of G with d H . ( u ' )= k - 1, and so on. Eventually we obtain a subcoloring of G with [ k / 2 ] 1 colors, in which every color class is a disjoint union of only K,'s and K,'s. 0

+

Corollary 15 enables us to produce another large class of 2-subchromatic graphs.

Corollary 16. For every cubic, i.e. 3-reguIar, graph G , X , ( G ) s 2. Corollary 17. For every graph G with maximum degree A, X,(G) S [A/2]

+ 1.

We conclude this section with the following results which establish another upper bound for the subchromatic number. Let f ( n ) = max{X,(G): IV(G)l = n}.

Lemma 18. f ( n )

i1+f(n - [log,

n]).

Proof. First observe that the Ramsey number r ( k , k ) < 4k, which follows from the facts that r(k, 1) = r(1, k ) = 1, and r(k, 1) < r ( k , I - 1) < r(k, 1 - 1). By induction one can show that r(k, I) < 2k+' (cf. Bondy and Murty, p. 104 [3]). Let G be a graph on n vertices, 4k S n < 4k+1.Then G has an independent set of size k or a clique of size k and k = [log, n ] . Then f ( n ) 6 1 + f ( n - [log, n]). 0 Using Lemma 18 one can prove the following: we omit the details.

Theorem 19.

[fi] S f ( n ) G n/(log,

n

- 1)

+ O(n/(log4 n)').

5. Further directions, open problems An interesting refinement of the concept of a subcoloring is the following. Given a graph G, a partition of the vertices into t sets V , , V,, . . . , V, is called an (I,, I*, . . . , r,)-subcoloring if the induced subgraphs ( V , ) consist of disjoint

Fig. 4. Graphs with no (1,2)-subcoloring.

Subchromatic number of a graph

47

I

Fig. 5 . XJK, X K6) 5.

complete subgraphs, each component of which has cardinality no more than ri. A standard coloring is then a (1, 1, . . . , 1)-subcoloring. It follows from Corollary 15 that every k-regular graph has a ( 2 , 2 , . . . ,2)-subcoloring with [k/2] + 1 colors. Thus, for example, every cubic graph has a (2,2)-subcoloring. This raises the question: which cubic graphs have ( 1 , 2 ) -subcolorings? For example, the Petersen graph has a (1,2)-subcoloring, but the two graphs in Fig. 4 do not. One can show without too much difficulty that if G is toroidal, then X , ( G ) 6 6. However, we believe that X,(G) G 5 holds for all toroidal graphs; in fact, we have not been able to construct a toroidal graph that cannot be 4-subcolored. A number of results can be obtained about the subchromatic number of various products of graphs. For example, for the join of two graphs one can see that max{X,(G), X , ( H ) } GX,(G + H) G X , ( C ) + X , ( H ) . For the Cartesian product G X H, where V(G X H) = V ( G )x V ( H ) and E(G x H) = { ( ( g , , hl), ( g 2 , h 2 ) ) : g l= g 2 and h l h 2 in E ( H ) or h , = h2 and glg2 in E ( G ) } , one can show that: X,(G x H) 6 min{max{X,(G), X ( H ) } , max{X(G) X , ( H ) } 1. The subcoloring in Fig. 5, however, shows that even for complete graphs this bound can be inexact. Finally, we ask, can one bound the subchromatic number of (i) a minimally 2-connected graph? (ii) a chordal graph? or (iii) a perfect graph? Acknowledgements The authors would like to thank the referees for their extremely detailed comments and excellent suggestions which greatly improved the presentation of this paper. References [l] J.A. Andrews and M.S. Jacobson, O n a generalization of chromatic number, Congr. Numer. 47 (1985) 33-48.

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[2] J.A. Andrews and M.S. Jacobson, On a generalization of chromatic number and two kinds of Ramsey numbers, Ars Combin. 23 (1987) 97-102. [3] J.A. Bondy and U.S.R. Murty, Graph theory with Applications (American Elsevier, New York, 1976). [4] M. Borowiecki, Point partition numbers and generalized Nordhaus-Gaddum problems, Colloq. Math. 47 (1984) 279-285. [S] I. Broere and C.M. Mynhardt, Generalized colorings of outerplanar and planar graphs, Graph Theory with Applications to Algorithms and Computer Science (Kalamazoo, Mich., 1984) 151-161 (Wiley, New York, 1985). [6] R.L. Brooks. On colouring the nodes of a network, Proc. Cambridge Philos. SOC. 37 (1941) 194- 197. (71 J.L. Brown and D.G. Corned, On generalized graph colorings, J. Graph Theory 11 (1987) no. 1, 87-99. [8] G. Chartrand, D. Geller and S. Hedetniemi, A generalization of the chromatic number, Proc. Cambridge Philos. SOC.64 (1968) 265-271. [9] G . Chartrand and H. Kronk, The point arboricity of planar graphs, J . London Math. SOC.44 (1969) 612-616. [lo] G. Chartrand, H. Kronk and C. Wall, The point arboricity of a graph, Israel J. Math. 6 (1968) 169- 179. [11] E.J. Cockayne, Colour classes for r-graphs, Canad. Math. Bull. 15 (1972) 349-354. [12] E.J. Cockayne and G.G. Miller, An interpolation theorem for partitions which are complete with respect to hereditary properties, J. Combin. Theory Ser. B 13 (1972) 290-296. [13] R.J. Cook, Point partition numbers and girth, Proc. Amer. Math. SOC.49 (1975) 510-514. [14] G.S. Domke, R. Laskar, S.T. Hedetniemi and K. Peters, The partite-chromatic number of a graph, Congr. Numer. 53 (1986) 235-246. [15] P. Erdos and A. Hajnal, On chromatic number of graphs and set-systems, Acta. Math. Acad. Sci. Hungar. 17 (1966) 61-99. [16] J. Folkman, Graphs with monochromatic complete subgraphs in every edge-colouring, SIAM J. Appl. Math. 18 (1970) 19-24. [17] F. Harary, Graph theory (Addison-Wesley, Reading, Mass., 1969). [18] F. Harary, Conditional colorability in graphs, Graphs and Applications, Proc. First Col. Symp. Graph Theory (Boulder, Colo., 1982) 127-136 (Wiley-Intersci. Publ., Wiley, New York, 1985). [19] F. Harary and S. Hedetniemi, The achromatic number of a graph, J. Combin. Theory 8 (1970) 154-161. [20] F. Harary, S. Hedetniemi and G. Prins, An interpolation theorem for graphical homomorphisms, Portugal. Math. 26 (1967) 453-462. [21] F. Harary and P.C. Kainen, On triangular colorings of a planar graph, Bull. Calcutta Math. SOC. 69 (1977) 393-395. [22] P.J. Heawood, Map colour theorems, Quart. J. Math. 24 (1890) 332-338. [23] S. Hedetniemi, On partitioning planar graphs, Canad. Math. Bull. 11 (1968) no. 2, 203-211. [24] S. Hedetniemi, Disconnected colorings of graphs. Combinatorial Structures and Their Applications, (Proc. Calgary Internat. Conf., Calgary, Aka., 1969) 163-167 (Gordon and Breach, New York, 1970). [25] A.B. Kempe, On the geographical problem of four colors, Amer. J . Math. 2 (1879) 193-204. [26] F. Kramer, Sur le nombre chromatique K ( 9 , G) des graphes, Rev. Francaise Automat. Informat. Recherche Operationnelle 6 (1972) Ser. R-1, 67-70. [27] F. Kramer and H. Kramer, Un probleme de coloration des sommets d’un graphe, C.R. Acad. Sci. Paris Ser. A-B 268 (1969) A46-A48. [28] F. Kramer and H. Kramer, Ein Farbungsproblem der Knotenpunkte eines Graphen bezuglich der Distanz p, Rev. Roumaine Math. Pures Appl. 14 (1969) 1031-1038. [29] L. Lesniak-Foster and J . Roberts, On Ramsey numbers and graphical parameters, Pacific J . Math. 68 (1977) no. 1, 105-114. [30] L. Lesniak-Foster and J . Straight, The cochromatic number of a graph, Ars Combin. 3 (1977) 39-45. [31] D.R. Lick, A class of point partition numbers, Recent Trends in Graph Theory, (Proc. Conf. New York, 1970). Lecture Notes in Mathematics, Vol. 186, 185-190 (Springer, Berlin, 1971).

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[32] D.R. Lick and A.R.White, k-degenerate graphs, Canad. J. Math. 22 (1970) no. 5, 1082-1096. [33] L. LovBsz, On chromatic number of finite set systems, Acta Math. Acad. Sci. Hungar. 19 (1968) 59-67. [34] J. Mycielski, Sur le coloriage des graphes, Coll. Math. 3 (1965) 161-162. [35] C.M. Mynhardt and I. Broere, Generalized colorings of graphs, Graph Theory with Applications to Algorithms and Computer Science, (Kalamazoo, Mich., 1984) 583-594 (Wiley-Intersci. Publ., Wiley, New York, 1985). [36] F.S. Roberts, Indifference graphs, Proof Techniques in Graph Theory, F. Harary, Ed., 139-146 (Academic Press, New York, 1969). [37] H. Sachs and M. Schauble, Konstruktion von Graphen mit gewissen Farbungseigenschaften, Beitrage zur Graphentheorie, (Kolloquium, Manebach, 1967) 131-136 (Teubner, Leipzig, 1968). 1381 E. Sampathkumar, Prabha S. Neeralagi and C.V. Venkatachalam, A generalization of the chromatic and line chromatic numbers of a graph, J. Karnatak Univ.: Sci. 22 (1977) 44-49. [39] G. Szekeres and H.S. Wilf, An inequality for the chromatic number of a graph, J. Combin. Theory 4 (1968) no. 1, 1-3.

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Discrete Mathematics 74 (1989) 51-59 North-Holland

51

CONNECTED SEQUENTIAL COLORINGS A. HERTZ and D. de WERRA Ecole Polytechnique Fidirale de Lausanne, Switzerland

Let us consider a graph G = (V, E). A k-coloring (Sl, . . . , S,) of its nodes is called canonical if any node u E V of any color i is contained in a clique K of size i such that K f l S, # 0 for l s j s i . A connected order on a connected graph G = (V, E ) is any order u I<. . . < up such that { u , , . . . , u i } induces a connected graph for any i 1 Cis IVI = p . We prove that any sequential node coloring based on any connected order gives a canonical coloring of any connected subgraph G' of G if and only if G is a parity graph without Fish (a Fish is a forbidden graph on 6 nodes).

1. Introduction Given an order of the nodes of a graph G, a sequential node coloring is an algorithm which scans the nodes in this order and gives to each one the smallest available color. A connected order in a connected graph G is any order v I <. . < up such that { v l , . , . , v i } induces a connected graph for any i 1 S i 6 IVI = p . For a non connected graph G it will simply be the concatenation of connected orders of the connected components of G. Gi will denote the subgraph induced by { v l t . . . v i } . The algorithm consisting of a Sequential node coloring based on any Connected ORdEr will be called SCORE. In other words, we choose at the beginning any node of the graph. Then, at each step, we choose any node which is adjacent to at least one already colored node and give it the smallest available color. When there is no such node, we start with a node in another connected component (provided all nodes are not colored yet). Our purpose is to study a class of graphs for which SCORE always produces optimal colorings for the graph itself and for a collection of subgraphs. This class will be closely related to parity graphs. These were introduced by Olaru and Sachs [4] as graphs characterized by the following property: every odd cycle of length 2 5 has two crossing chords.

Later they were called parity graphs because they can also be characterized by the 0012-365X/89/33.50 @ 1989, Elsevier Science Publishers B.V. (North-Holland)

52

A. Hertz, D. de Werra

equivalent property: for every pair x , y of nodes, all chordless chains connecting x and y have the same parity.

Burlet and Uhry [2] gave a good algorithm to recognize a parity graph. In the next section we shall give the main result; its proof will be developed in Section 3. Relations with perfectly orderable graphs [3] will be discussed in Section 4. All graph theoretical terms not defined here can be found in Berge [l]. All “induced” subgraphs will simply be called subgraphs according to [ 11.

2. The main result A k-coloring (Sl, . . . , S,) is called canonical [5] if for any node x of any colour i, there exists a clique K 3 x of size i such that K n S j # O for j = i , i - 1 , . . . , 1. Canonical colorings are interesting since it was proved in [5] that a graph G is perfect if and only if every subgraph G’ of G has a canonical coloring. A k-coloring (&, . . . , S,) is called strongly canonical [5] if for any clique K there exists a clique K’ =I K such that K ’ n Sj # 0 for j = 1, 2, . . . , min{r/S, n K # 0}. Clearly a strongly canonical coloring is canonical. A graph G = (V, E ) will be called SCORE-perfect (resp. strongly SCOREperfect) if for any k (1 S k S IVl), SCORE provides a canonical (resp. strongly canonical) coloring on G,. It is equivalent to say that a graph G is (strongly) SCORE-perfect if any connected order on any connected subgraph G’ of G gives a (strongly) canonical coloring of G’. A Fish is a graph G = (V, E ) such that V = (1, 2, 3, 4,5, 6) and E = {[l, 21, [ l , 31, [2, 31, [3,4], [3, 61, [4, 51, [4, 61, [ 5 , 61) (see Fig. 1). A parity graph G containing no Fish will be called a Fish-Free Parity graph (or shortly FFP graph). It would be interesting to know when SCORE provides an optimal coloring. Instead of answering this general question, we will prove the following theorem:

Theorem. For a graph G , the following statements are equivalent: (1) G is an FFP graph (2) G is SCORE-perfect (3) G is strongly SCORE-perfect. A

Fig. I .

Connected sequential colorings

53

3. The proof Let G = ( V , E ) be a graph, < an order on its nodes and x a node in V. We shall denote by:

4)

the color of x. si the set of nodes with color i. N(x1 the set of all nodes adjacent to x . ax) the subgraph induced by { y E V / y s x } [xo, . * . xnl a chain with edges [xi, xi+J (0 s i s n - 1). [xo, . . . , x,, xo] a cycle with edges [ x i , x i + J (0 S i s n - 1) and [x,, xo].

In order to show by induction that SCORE gives a strongly canonical coloring of the first k nodes of an FFP graph, we shall need the following results.

Lemma 1. Let

- G = ( V , E ) be a connected parity graph colored by SCORE (with I V I 2 3 ) - x E V be the last node of the order; assume c ( x ) > 1. - y E V be any node adjacent to x such that c( y ) > 1. Then there exists some node v E S, such that the clique K = { x , y } can be extended to a clique K ‘ = {x, y , v } with K ’ n S1# 0.

Proof. If IVJ= 3 the result is clear. So let us suppose that the result is true for any parity graph with at most r - 1 nodes and suppose that I VI = r. Let v and y be two nodes adjacent to x such that c ( y ) > c ( v ) = 1. If [ y , v ] E E the lemma is true; so let us suppose that [ y , v ] $ E. Since G - { x } is a connected parity graph, there must be some chordless chain of even length (same parity as [ y , x , v ] ) joining v to y in C - { x } . Let C = [xo = v, xl, . . . , x, = y ] be such a chain with a maximum number of nodes of color 1. Since C is even, there is some index i (0 < i < n ) such that xi and x i + l have colors >1. By the induction hypothesis applied to G(xi) or G(xi+,) (whichever is the larger), there is some node v ’ E S1 such that [v’,x i ] E E and [v’,xi+J E E. We show now that [v’,x i - l ]$ E and [v’,xi+2]$ E if i < n - 1. Let us suppose we have one of these edges, [ V ’ , X ; - ~ ] for example (the other case is similar). Then [v’,xi+*]$ E (if i < n - 1) since otherwise [v’,x i - l , . . . , x i + 2 , v ’ ] would be an odd cycle with no two crossing chords. Similarly [v’,xi-2]$ E (if i > 1). Thus, in order to avoid an odd cycle with at most one chord, we must have [v’,xi] 4 E for any j # i - 1, i, i 1. But now [xo = v, xl, . . . ,x i - , , v’,xi+,, . . . ,x, = y ] is a chordless chain joining v to y with more nodes of color 1 than C . This contradicts the maximality of C. So [v’,x i - l ] 4 E and [v’,xi+2]$ E if i < n - 1 and now [v’,xi] $ E for any j # i, i 1 since otherwise we would have an odd cycle with at most one chord.

+

+

A. Hertz,

54

D. de Werra

. . . , x,, x ] has length at least five and all The odd cycle [x, x o , . . . , x i , v’, possible chords except [ x i , x i + l ] must be incident to x . Thus, in order to have two crossing chords, the edge [ x , v ’ ]must exist. If i is even, [ x , xg, . . . ,x i , v’,x ] is an odd cycle of length at least five with no two crossing chords. Therefore i is odd. It follows that [x, v’,x i + l , . . . ,x,, x ] is an odd cycle with no two crossing chords. This is allowed only if i 1= n, and thus v ’ is the desired common neighbor of x and y of color 1. 0

+

Lemma 2. Let - G = ( V , E ) be a connected FFP graph colored by SCORE (with I VI 2 3) - x E V be the last node of the order; assume c ( x ) > 1 - y E V be any node adjacent to x such that c ( y ) > 1. Then for any color k < min{c(x), c ( y ) } there exists some node z E S, such that the clique K = { x , y } can be extended to a clique K ’ = { x , y , z } with K ‘ n s k # 0. Proof. If IVI = 3 the result is trivial. So we shall make a proof by induction on IVI. Let us suppose that the lemma does not hold; we take a node y violating the lemma and having the smallest possible color. Let us consider a color k < min{c(x), c ( y ) } such that s k n N ( x ) n N ( y ) = 0. By Lemma 1 we know that k > 1. Let z be any node of S k n N ( x ) . We have [ y , z] r$ E. Claim. There is‘no node w E N ( x ) n N ( y ) n N ( z ) with c( w )< k. To see this, we can observe that since k < c ( y ) , there exists some node Z‘ E Sk r l N ( y ) . We have [z’, x ] r$ E since S k n N ( x ) n N ( y ) = 0. Now [ z ’ , w ]r$ E since otherwise [z, x , y , z’, w ,21 would be an odd cycle with no two crossing chords. Since c ( w ) < min{c(z‘), c ( y ) } = k , we know, by induction hypothesis applied to G ( y ) (which contains z ’ ) , that there exists some node w’ E Sccw,n N ( z ’ )f l N ( y ) ) . We have:

E since otherwise [ w , x , w ’ ,z’, y , w ] would be an odd cycle with no two crossing chords. - [w’, z ] @ E since otherwise [ x , y , z’, w’, z, x ] would be an odd cycle with one chord.

- [ w ’ ,x ] r$

Now, the subgraph induced by { x , y , z, z ‘ , w , w ’ } is a Fish. This ends the proof of the claim. 0 There is some even chordless chain joining z to y in G - { x } . Let [z = xo, . . . , x, = y ] be such a chain with maximum number of nodes with color 1. If have colors > 1, then with the same there exists some index i such that xi and proof as in Lemma 1 we can prove that i = n - 1 or i = 0.

Connected sequential colorings

55

Now two cases are possible: (a) c(x,)=l for all even indices i such that 2 6 i 6 n - 2 ; x1 and x,-~ have colors > 1 (b) c(xJ = 1 for all odd indices i such that 1s i s n - 1. Case ( a ) . By Lemma 1 applied to either G(xl) or G ( z ) , we know that x1 and z have a common neighbor v of color 1. As in Lemma 1, it follows that [v,x i ] $ E for 2 s i s n and that [v,x ] E E. Similarly, there is some node v‘ E S, such that [v’,y ] E E , [v’,x] E E , [v‘,x , - ~ ]E E and [v’,x i ] $ E for any i < n - 1. Nodes v and V ’ are distinct since [v,z ] E E and [v’,21 @ E. We have n = 2 since otherwise the subgraph induced by {x, xl, z , y , v, v’} would be a Fish. Since k < c ( y ) , let Z’ be any node of S, n N ( y ) . We have [z’, x ] $ E since s k n N ( x ) n N ( y ) = 0. We also have [ z ,x l ] $ E since otherwise [z’, y , x, z , xl, 2’1 would be an odd cycle with no two crossing chords. So s k n N ( y ) n N ( x l ) = O . Now c ( x l ) < k since otherwise color k would be smaller than min{c(xl), c ( y ) } and by the induction hypothesis applied to either G ( y ) or G(x,) s k n N ( y ) f l N ( x l ) would not be empty. Now, since x1E N ( y ) n N ( z ) , we know by the claim that [x, x l ] 4 E. Since c ( z ) < c ( y ) and c(xI)< min{c(z), c(x)} = k, we know by minimality of c ( y ) that there is some node xI,E n N ( z ) r l N ( x ) . By the claim we have [xl., y ] 4 E and now [xl.,x, y , xl, z , x I . ]is an odd cycle with one chord. Case ( 6 ) . If [x, x i ] E E for some odd i then n = 2 and i = 1 (otherwise we have an odd cycle with no two crossing chords) and by the claim this is impossible. So [ x , xl] r$ E. By Lemma 1 there is some node v E S1n N ( x ) n N ( z ) . Let j be the greatest index such that 0 s j s n and [v,xi] E E . By the claim we have j < n. Now let i be the smallest index such that j < i 6 n and [x, xi] E E ; [v,xi, . . . , xi,x , v ] is an odd cycle with at most one chord. 0

Lemma 3. Let - G = ( V , E ) be a connected FFP graph colored b y SCORE (with I V I 2 3) - x E V be the last node of the order; assume c(x) > 1 - K = {xl, . . . , x,, x } be a clique where c ( x i ) < c(xj) if 1 S i < j S rn. Then for any color k < min{c(x), c(xl)} there exists some clique K’ 3 K such that

~‘ns,#ti Proof. If m = 1, this result is true by Lemma 2. So let us suppose that m > 1 and that the lemma is true for any clique with at most m - 1 neighbors of x. We know that there exists cliques K’ = { y , x 2 , . . . , x,, x } and K“= { y ’ , xI,. . . , xrnv1,x } such that c ( y ) = c ( y ’ )= k. So we must have [ y , x I ]E E or/and [ y ’ ,x,] E E since

A . Hertz, D. de Werra

56

otherwise [ y ’ , x l , x , , y , x , y ’ ] would be an odd cycle with no two crossing chords. 0

Proof of the theorem (1).$(3) It is sufficient to show this implication for a connected FFP graph G. Let us consider any clique K = { y l , . . . , y r } of G and let s > 1 be the smallest color in K (if s = 1 there is nothing to prove). Using Lemma 3 (x = last node in K) with index k decreasing from s - 1 to 1, it is easy to see that there exists cliques K’ = {xkr. . . , xsPl,y , , . . . ,y k } with .(xi) = j for any j (k S j S s - 1). For k = 1, this proves that the coloring in G is strongly canonical.

* *

(3) (2) This implication is obvious. (2) (1) If G is not an FFP graph, it must contain as induced subgraph at least one of the following: (a) a Fish (b) the odd cycle [x,, . . . x 5 ] with two non crossing chords [x2, x4] and [xz, x s ] (c) a chordless odd cycle [xl, . . . ,x Z k + , ](k > 1) (d) an odd cycle [xl, . . . , x 2 k + l ][k > 1) with one chord [ x 2 k + l 4. , Now we construct the connected order on G, beginning by this subgraph. In order to see that (2) is not true, it is sufficient to verify that SCORE may give a non canonical coloring to this subgraph. (a) if the nodes are ordered 1< 2 < * * * < 6 as in Fig. 1, the last node receives color 4 but is not contained in a clique of size 4 (b) if the nodes are ordered x1< * < xs, xs receives color 4 but is not contained in a clique of size 4 (c) if the nodes are ordered x1 < - * * < x Z k + , ,x Z k + lreceives color 3 but is not contained in a clique of size 3

Q Fig. 2.

Connected sequential colorings

(d) if the nodes are ordered x1< x 2 k + l < x2 < x 3 < 3 but is not contained in a clique of size 3. 0

57

--<

X2kt X2k

receives color

4. Final remarks

A question that is still open is to know for which graphs G = (V, E) SCORE always provides an optimal coloring for G k ( k = 1, . . . , IV().Any odd cycle with at most one chord, for example, has this property. So such graphs are not perfect. One should observe that connected orders and perfect orders in the sense of Chvital [3] are different concepts. There is no inclusion relation between perfectly orderable (p.0.) and SCORE-perfect graphs: the complement of a chain on five nodes is p.0. but not SCORE-perfect, and the graph in Fig. 2 is SCORE-perfect but not p.0. There is another interesting difference with p.0. graphs which should be mentioned: in a p.0. graph G , there is one order 0 of the nodes which gives a (strongly canonical) coloring of the subgraphs generated by the first k nodes in 0 (for any k ) . Now if we want to find a (strongly canonical) coloring of a collection of arbitrary subgraphs G' of G, we may simply take the order induced by 0 in G'. For an FFP graph G , an arbitrary but fixed connected order % of the nodes will again give a (strongly canonical) coloring of the subgraphs generated by the first k nodes in % (for any k ) . However if we want to find a (strongly canonical) coloring of a collection of arbitrary subgraphs G' of G, then we may not use the order induced by % on G ' ; we may have to consider different orders for the various subgraphs G'. 5

2 b l

1

F-Flag

K*-Kite

1

I

A #

\

#

\

262fi

I : 3

k-1 k-3

5

4

-

ck ( k ) 4 )

P5

chordless cycle on at least 5 nodes

Fig. 3.

5

T-Tent

3 4

58

A. Hertz, D. de Werra

Fig. 4.

The question naturally arises to determine the class of graphs for which we can keep the same (connected) order (e for coloring any subgraph G’. We can state:

Proposition. For a graph G , the following statements are equivalent: (a) every connected order is perfect (b) G does not contain any subgraph isomorphic to the graphs C, ( k s 5 ) , T, F o r K * in Fig. 3.

4,

It is easy to see that the graphs in Fig. 3 have a connected order which is not perfect. Conversely, one shows simply that a graph having a connected order which is not perfect must contain one of the graphs in Fig. 3. There are some variations on connected sequential colorings which might be studied. Instead of taking any connected order x , < . < x p , we could choose xi such that: - N ( x i ) contains as many already colored nodes as possible - N ( x i ) contains as many different colors as possible. But Fig. 4 shows two examples of parity graphs which are not colored optimally with any one of these improvements.

-

Acknowledgement The support of the National Scientific Research Council (Grant no. 2.706-1.85) is gratefully acknowledged. The authors would like to thank Myriam Preissmann and the referees for constructive remarks on an earlier version of this paper.

References [l] C. Berge, Graphs (North-Holland Math. Library, Amsterdam and New York, 1985). [2] M. Burlet and J. P. Uhry, Parity graphs, in: Topics on Perfect Graphs (C. Berge and V. ChvBtal, eds.) Ann. of Discrete Math. 21 (1984) 253-277.

Connected sequential colorings

59

[3] V. ChvBtal, Perfectly ordered graphs, ibid 63-65. [4] E. Olaru and H. Sachs, Contribution to a characterizationof the structure of perfects graphs, ibid 121-144. [5] M. Preissmann and D. de Werra, A note on strong perfectness of graphs, Math. Programming 31 (1985) 321-326.

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Discrete Mathematics 74 (1989) 61-64 North-Holland

61

TWO CONJECTURES ON EDGE-COLOURING A.J.W. HILTON Dept. of Mathematics, University of Reading, Whiteknighrs, P. 0 . Box 220, Reading, RG62AX, U.K. Chetwynd and Hilton have elsewhere posed two conjectures, one a general statement on edge-colouring simple graphs C with A ( G )> f IV(G)l. and a second to the effect that a regular simple graph G with d ( G ) 3 IV(C)l is 1-factorizable. We set out the evidence for both these conjectures and show that the first implies the second.

1. Introduction We are concerned here with simple graphs, that is finite graphs without loops or multiple edges. An edge-colouring of a graph G is a map 4: E ( G ) + %',where %' is a set of colours, such that no two vertices with the same colour have a common vertex. The chromatic index x ' ( G ) is the least value of l%'l for which an edge-colouring exists. A well-known theorem of Vizing [17] states that A ( G ) s x ' ( G )s A ( C ) + 1, where A ( G ) denotes the maximum degree of G. If ~ ' ( ( 3= ) A ( G ) , then G is said to be Class 1, and otherwise G is Class 2. The question of deciding whether or not a graph is Class 1 was shown by Holyer [14] to be NP-complete. However, for certain types of graph, the problem of classifying Class 2 graphs seems to be tractable. If G satisfies the inequality

then G is overfull. Clearly if G is overfull, then IV(G)l is odd. An overfull graph has to be Class 2, since no colour class of G can have more than 1; JV(G)] edges. In [6], Chetwynd and Hilton made the following conjecture (now slightly modified).

Conjecture 1. Let G be a simple graph with A(G) > f IV(G)l. Then G is Class 2 if and only if G contains an overfull subgraph H with A ( H ) = A(G). The graph C obtained from Petersen's graph by removing one vertex is Class 2, but contains no subgraph H with A ( H ) = A ( G ) ; this shows that the figure 3 in Conjecture 1 cannot be lowered. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

62

A.J.W . Hilton

At the time of writing, Conjecture 1 has been proved in a number of cases. Plantholt ( [ 1 5 , 161) and Chetwynd and Hilton ([3-61) have between them established the following.

Theorem 1. Conjecture 1 is true

if A(G) 3 I V(C)l- 3.

In [ l ] , Chetwynd and Hilton posed the following conjecture about regular graphs of even order. First note that a Class 1 regular graph is often called l-factorizable, as it is the union of edge-disjoint l-factors. Also note that a regular graph of odd order is overfull, and so is Class 2 . If a graph G is regular, let d ( G ) denote its degree.

Conjecture 2. Let C be a regular simple graph of even order satisfying d ( G ) 3 1 IV(C)l. Then C is l-factorizable. This conjecture seems to have been known however long before being posed by Chetwynd and myself. When I told Dirac of it, he said it was “going around” in the early 1950s. The figure 12 IV(C)l in the conjecture cannot be lowered, as is shown by the example of a graph C consisting of two K,’s, when n is odd. Chetwynd and Hilton ( [ 1 , 7 , 8 ] )have proved this conjecture in a number of special cases.

Theorem 2. Conjecture 2 is true if either d ( G )3 4

( f i - 1 ) IV(G)l

or d ( G ) 3 IV(C)l- 4. The object of this note is to prove the following theorem.

Theorem 3. If Conjecture 1 is true, then Conjecture 2 is true.

2. Proof of Theorem 3 Let C be a regular graph with IV(C)l = 2n and d ( C ) a a . Suppose that Conjecture 1 is true and that G is Class 2. Let H be an overfull subgraph of G with A ( H ) = d ( G ) . Since H is overfull, it follows that IV(H)I is odd, so H # G. Let def(H) =

2 tJtV(H)

( d ( G )- d H ( V ) ) .

Two conjectures on edge-colouring

63

It is shown in [2] that, if H is overfull, then def(H) S A ( H ) - 2 = d ( C ) - 2. It follows that G has an edge-cut S with )SI G d ( G )- 2 such that G\S = H U J , where V ( H )f Vl ( J )= 9. Since A ( H ) = d ( G ) > n , it follows that H has at least n 1 vertices. Consequently J has at most n - 1 vertices. Thus d ( G ) 1> IV(J)(.Since C is regular, the number of edges joining vertices of J to vertices of H is at least ( d ( G )- IV(J)I 1) IV(J)I. For fixed d ( G ) , ( d ( C )- IV(J)(+ 1) IV(J)I is a quadratic in IV(J)(.In the range 1 s IV(J)(S n - 1, it has two minima, one at each end point, with values d ( G ) and ( d ( G ) - n + 2 ) ( n - 1). But d ( G ) > IS[, and ( d ( C )- n 2)(n - 1) 2 2n - 2 3 d ( G ) - 1> IS(, contradicting the definition of S. Thus C has no overfull subgraph H , and so, by Conjecture 1, is Class 1, or in other words is 1-factorizable. Thus Conjecture 2 is true. This proves Theorem 3.

+

+

+

+

3. A final remark Conjecture 1 has many other implications. Some of these are discussed in [ll-131 by Hilton and Johnson. A survey of the main implications is given in [lo]. See also [9].

Note added in proof A.G. Chetwynd and I have recently proved Conjecture 1 in the case when A G ~ = a ( f i - l)(lV(C)l + 1)+ 1 and IE(C)l= A(G)Li IV(G)ll. See [18].

References [l] A.G. Chetwynd and A.J.W. Hilton, Regular graphs of high degree are 1-factorizable, Proc. London Math. SOC.(3), 50 (1985) 193-206. [Z] A.G. Chetwynd and A.J.W. Hilton, The edge-chromatic class of graphs with large maximum degree, where the number of vertices of maximum degree is relatively small, J. Combinatorial Theory (B) to appear. [3] A.G. Chetwynd and A.J.W. Hilton, Partial edge-colourings of complete graphs or of graphs which are nearly complete, Graph Theory and Combinatorics. Vol. in honour of P. Erdos’ 70th birthday (1984) 81-98. [4] A.G. Chetwynd and A.J.W. Hilton, The chromatic index of graphs of even order with many edges, J. Graph Theory 8 (1984) 463-470. [5] A.G. Chetwynd and A.J.W. Hilton, The edge-chromatic class of graphs with maximum degree at least IVI - 3. Proceedings of the conference held in Denmark in memory of G . A . Dirac, to appear. [6] A.G. Chetwynd and A.J.W. Hilton, Star multigraphs with three vertices of maximum degree, Math. Proc. Camb. Phil. SOC.100 (1986) 303-317.

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A.J. W. Hilton

[71 A.G. Chetwynd and A.J.W. Hilton, The edge-chromatic class of regular graphs of degree 4 and their complements, Discrete Applied Math. 16 (1987) 125-134. 181 . . A.G. Chetwynd and A.J.W. Hilton, 1-factorizing regular graphs of high degree -an improved bound, submitted. [9] A.G. Chetwynd and A.J.W. Hilton, A A-subgraph condition for a graph to be Class 1, J. Combinatorial Theory (B) to appear. [lo] A.J.W. Hilton, Recent progress on edge-colouring graphs, Discrete Math. 64 (1987) 303-307. [ l l ] A.J.W. Hilton and P.D. Johnson, Graphs which are vertex-critical with respect to the edge-chromatic number, Math. Proc. Camb. Phil. SOC., 102 (1987) 211-221. [12] A.J.W. Hilton and P.D. Johnson, Graphs which are vertex-critical with respect to the edge-chromatic class, submitted. [13] A.J. Hilton and P.D. Johnson, Reverse class critical multigraphs, Discrete Math. 69 (1988) 309-311. [14] I.J. Holyer, The NP-completeness of edge-colourings, SIAM J. Computing 10 (1980) 718-720. [15] M. Plantholt, The chromatic index of graphs with a spanning star, J. Graph Theory 5 (1981) 5-13. [16] M. Plantholt, On the chromatic index of graphs with large maximum degree, Discrete Math. 47 (1983) 91-96. [17] V.G. Vizing, On an estimate of the chromatic class of a p-graph (in Russian), Diskret. Analiz. 3 (1964) 25-30. [l8] A.G. Chetwynd and A.J.W. Hilton, Star multigraphs with r vertices of maximum degree, submitted.

Discrete Mathematics 74 (1989) 65-75 North-Holland

65

A NEW UPPER BOUND FOR THE LIST CHROMATIC NUMBER B. BOLLOBAS and H.R. HIND? Dept. of Pure Mathematics, Univ. of Cambridge, 16 Mill Lane, Cambridge, U.K . CB2 1SB For large values of A, it is shown that all A-regular finite simple graphs possess a non-trivial vertex partition. This is then used to show that for finite simple graphs of maximal degree A(G) = A, the list chromatic number is bounded by z;(G) s 7A/4 o(A).

+

Given a graph G and for each edge of G a set (list) of colours, we call an assignment of a single colour to each edge of G a list colouring if the colour assigned to an edge is in the list of colours associated with that edge and no two adjacent edges are assigned the same colour. The list chromatic number of the graph C, x;(G), is defined to be the minimal positive integer such that if for each edge of G , the list of colours associated with that edge has size at least x;(G), then there exists a list colouring for G for any choice of lists. The list colouring conjecture, which has been attributed to various sources, states x;(C)s x’(C). Little progress has been made towards establishing this conjecture, even for special classes of graphs. Bollobfis and Harris [2] have shown that for graphs with maximal degree A ( C ) = A, x ; ( G )S 11A/6 + o(A) and Chetwynd and Haggkvist [3] have shown that for triangle free graphs x;(G)s 9A/5. In this paper we combine the approaches used in the above two papers and the existence of a specific partition for regular graphs to obtain an improved upper bound for the list chromatic number. The term graph will be used to mean a finite simple graph. Before proving the intended result, we need a few definitions. Given a graph G, let A: E ( G ) + P(N) be an arbitrary function. Then we define a A-colouring of graph G to be a function 9 such that

where

and

# ( e ) # @ ( e ‘ ) if edges e and e’ are adjacent.

t Supported by ORS grant ORS/84120 and CSIR grant 9/8/1-2019. 0012-365X/89/$3.50 @ 1989, Elsevier Science Publishers B.V. (North-Holland)

66

B. Bollobis, H . R . Hind

The more general name list colouring is applied to such a colouring where the function A is not specifically mentioned. The function A can be thought of as assigning a list of acceptable colours, A(e), for the edge e. We refer to the function A as the list function for the graph. For a more detailed discussion of list colourings, see [2]. For simplicity we let A, denote a function A, :E( G )+ N(').

For a given list function, A, we define a partial list colouring of G to be a function, I) say, which has an associated edge of G, e* say, such that

q : E(G)\{e*}-. N where V(e)E A(e)

Ve

E

E(G)\{e*},

and

q ( e ) # q ( e ' ) if edges e and e' are adjacent. If we think of the function @ as a colouring of the edges of the graph G, then q is a colouring of all but one of the edges of G. Let E ( v ) be the set of edges incident to vertex v and for a partial list colouring q, with associated edge e*, define

3 :V ( G ) +

6 N(j)

i=l

where

$(v)

= { V ( e ) : eE E(v)\{e*}}.

The definitions given below were first given in [3], but are restated here for convenience. Let G be a graph with list function A and a partial list colouring q. Let the associated edge for q be e*. An edge uv (where uv # e * ) is said to be a floppy edge if

IA(uv>\(3(u>u

3(v))la 1

i.e. if there exists a partial list colouring, q* say, with the same associated uncoloured edge, e*, as q, such that

q*(e)= q(e)

Vefuv

or e*

and

q*(uv)E A ( u v ) \ { V ( u v ) } . We call the colour q*(uv) an escape colour for edge uv. Later, with slight abuse of notation, we refer to the escape colour of a floppy edge; here having assigned a partial list colouring to the graph under consideration, we chose one

New upper bound for list chromatic number

67

(of the one or more) escape colours and assign this fixed colour as the escape colour of the edge. By considering partitions of A-regular graphs, we obtain an upper bound on the list chromatic number. We recall the Erdos-Lovfisz Theorem (see [ l ] , page 22).

Theorem 1. Let A l , A 2 , . . . , A,,, be events with dependence graph F. If F has maximal degree A z=3 and P(AJ s

(A - l)A-l AA

then

We use this theorem to show that A-regular graphs are vertex partitionable with vertex disjoint classes Vl, V2,. . . , V, such that for x E V , there are non-trivial upper and lower bounds on the cardinality of r ( x ) r l V, (denoted by d,,(x)) for all j # i. As is customary, we let Sn,pbe the sum of n independent Bernoulli random variables, with value one or zero, where each Bernoulli random variable has value one with probability p . Then we see P(S,,, = k ) = ( ; ) p * ( l - p y k

Definition. For a given A and r, we call a set of ordered pairs D = {(d,j, Ajj)lsj,jsr,i+j} a (A, 0)-acceptable set if (i) O s S , G A j j s A f o r l s i , j s r a n d i Z j , and (ii) there exists a set P = { p l , . . . ,p r } (called the set of probabilities) with O < p i < l , C I = , p j = 1, such that

The theorem below follows from Theorem 1.

Theorem 2. Let D = {(aij,Ajj)lej,jsr,i+j}be a (A, 0)-acceptable set. Then every A-regular graph has a partition, V ( G )= UTZlK with V, n = 0 if i # j and every vertex in V;: is adjacent to at least 6,j and at most A, vertices in V,. Proof. Let G be a A-regular graph and D = {(aij,Aij),ei,jsp,i+j}be a (A, 0)acceptable set. Let P = { p l , . . . ,p , } be the associated set of probabilities for D.

68

B. Bollobh, H . R . Hind

Take a random partition of the vertex set V ( G ) into disjoint classes V,, . . . , V,, by putting a vertex into class V, with probability pi. Let A, be the event that the condition of the theorem is violated for vertex x , i.e. there exist i, j (i # j ) such that x

E

V , and d v l ( x ) < 6,

or dvl(x)> A,.

We note that event A, is independent of the system {A, :dG(x, y) 2 3}, that is to say the set of events A,, such that x and y are not adjacent and have no common neighbours. The event A, is thus independent of the system of all events {A, :y E V ( G ) }from which at most A2 1 events have been omitted. From the choice of event A, we get

+

r

P(A,) =

r

c c

Pi(P(sA,pl < 6i,) + P(sA,pl > Aij))?

i=l j=l,j#i

but D is a (A, 0)-acceptable set with set of probabilities P, so by condition (ii) of the definition,

It now follows from the Erdos-Lovisz Theorem that 0

Definition. For a given A and r, we call a set of ordered pairs D = {(dij, Aij)l
Using a similar proof to that for Theorem 2, we get

Corollary 3. Let D = {(al,,A,,)lGl,lGr,l+l} be a (A, A/2)-acceptable set and G be a A-regular graph. For each vertex x E V ( G ) let U ( x ) be a set of at least A/2 neighbours of x. Then there exists a partition of the vertex set V ( G ) such that V ( G )= U:=l V , with V, n V, = 0 for i # j and for every vertex x E V, there are at least 6, and at most A, vertices adjacent to x in V,. Also at least one vertex in U ( x ) is in V, for each j E { 1, 2, . . . , r}\{i}.

Proof. The proof follows that of Theorem 3, but with a new event A: defined to include the possibility for x E V;. there is a V, containing no vertex in U ( x ) . 0

New upper bound for list chromatic number

69

In applying Corollary 3, we need a technical lemma.

Lemma 4. For A sufficiently large and r = 2, the set D = ((1,100 log A); (1, A ) } is a ( A , A/2)-acceptable set. Proof. Condition (i) of the definition of (A, A/2)-acceptability is clearly satisfied by the set D. Let P , the associated set of probabilities for D, be chosen to be the set P = {A - 8 log A / A , 8 log A / A}. We now seek to establish the inequality

Consider the righthand side of the proposed inequality; for A 2 3,

We have chosen p 1 = 1 - 8 log A / A , p 2 = 8 log A/A. Thus with aI2= 1, 621= 1, A12= 100 log A and A2, = A , all terms except P(SA,p2> 100 log A), in the left-hand side of the proposed inequality are easily seen to be o ( A - ~ ) An . upper bound for P(SA,,, > 100 log A ) is available (see [l], page 14), namely

and with u = 12.5, it follows that

Thus we have that

and (A2)A2 =0(A-3) (A2 1)'

+

+

So for sufficiently large A , the inequality has been established.

0

Before turning to the main result of the paper, we make two preliminary observations.

70

B. Bollobh, H . R . Hind

Remark 1. In the proof below which produces a list colouring of a graph G, for a given list function A,, we may assume that a partial list colouring exists. Suppose G does not have a partial list colouring. Let El c E ( G ) be a maximally sized edge set such that for each edge e E El we can choose a colour, O(e) say, with O(e) E A , ( e ) such that e ( e ) # e ( e ’ ) if e and e’ are distinct adjacent edges contained in the set E l . Thus GIEl] is the largest edge induced subgraph of G for which a A,-colouring exists. Then choose an edge e’ E E(G)\EI and define a new list function A;: E(G)-+ 9(N) such that A;(f) = A,(f) for all f E E l U { e ‘ } , and for e E E(G)\(E, U { e ’ } ) , the sets A ; ( e ) are disjoint and do not intersect the set Uf.E,u(e’) 4 ( f )* Clearly G has a partial list colouring for the list function A;. Therefore if we show that a list colouring exists for G with list function A;, this implies that our choice of El was not maximal. Thus without loss of generality we may suppose that G has a partial list colouring for list function A,.

Remark 2. In the proof below when showing the existence of a list colouring for a graph G, we may assume G is A-regular. If G is not A-regular we may create a new A-regular graph G’ where G E G’, by adding vertices and edges to the graph G. We define a new function A; :E(G‘)+ N(‘) such that A;lE(G) = A, and for each e E E(G’)\E(C) we choose 1-sets, A ; ( e ) , such that A ; ( e ) f l A , ( e ’ )= 0 for all e’ E E(G’)\{e}. Then we obtain a A-regular graph G’ such that G is list colourable with list function A, iff G’ is list colourable with list function A;.

Theorem 5. If A is sufficiently large, then for every graph G with maximum degree A(G) = A, we have

Proof. Suppose G is a graph of maximal degree A, that 1 > 7A/4 + 125 log A] and A,: E(G)-, N(‘) is an arbitrary function. We shall show that there exists a A,-colouring for G. From the remarks above, we may assume G is A-regular and there exists a partial list colouring for G. We choose V Oto be a partial list colouring with a maximal number of floppy edges. Let the associated uncoloured edge be denoted aobo. For each vertex x E V ( G ) ,let { y l , y2, . . . , Y , , , ( ~ ) be } the set of those vertices y adjacent to the vertex x such that xy is not a floppy edge of the partial list colouring q0. If m ( x ) 3 A/2 define U ( x ) = { y l , y2, . . . ,yrN21} and if m ( x ) < A/2 let U ( x ) = { y l , . . . y m d u { Z , , , ( ~ ) + I , . . . , q A / 2 1 ) , where z , , , ( ~ ) + ~., . . , 2 [ A / 2 1 are any other vertices adjacent to the vertex x.

New upper bound for list chromatic number

71

For a set W c V ( C )and a vertex x , let d,(x) be the number of neighbours of x in the set W. By Corollary 3 and Lemma 4 above, there exists a partition of V ( C ) into V, and V, such that for each x E V,,

1 6 d&)

6

100 log A, and

v, n U ( X )# 0, and for each x E V2, 1 s d,,(x) s A,

and

v,n U ( X )z 0. We distinguish three cases depending on the location of the endvertices of the uncoloured edge sob,. Case (1). a, E V, and b~ E V,.

We create a sequence of partial list colourings I)(,, $,, . . . and their associated uncoloured edges aOb,,,a , b,, . . . . Having defined I),-,and edge a,-, b ,-,, choose q, and the associated uncoloured edge a,b, such that (1) I), is a partial list colouring, (2) a, = b , - , , (3) 6, E VIP (4) I),(e) = I),-,(e) if e E E ( G ) \ { a , - l k l , arb,), and I),(a,-lb,-,) = I),-,(a,h), ( 5 ) v,(a,-,bz-d (2 $,(a,-A (6) $ ~ , ( a , - , b , -is~not ) the escape colour of a floppy edge incident to a , - , , and (7) arbr# a,b, for any j < i. Condition (6) ensures that the number of floppy edges in v, IS not less than that is chosen so that the number of floppy edges is maximal, in I),-,and since equality must hold. Condition (7) implies that for a finite graph the sequence of partial list colourings constructed in this way is finite, say it ends with v,. To simplify the notation let ab = asb, be the associated uncoloured edge. Condition (7) also ensures that if a colour is not in $()(v) but is in $J,(v)for i >0, then it is in $,(v) for all j a i . We note that since I)() has a maximal number of floppy edges, it is sufficient to show that either there is a list colouring of the graph G or there is a partial list Suppose neither case applies. colouring I),with more floppy edges than I)(). Given C with a partial list colouring I),, if e is a floppy edge we let v : ( e ) denote the escape colour assigned to edge e. Then for the partial list colouring v, and a vertex v E V ( G ) ,let F ( I ) , , v ) be the set of colours assigned to floppy edges incident to vertex v, H ( I ) , , v ) be the set of escape colours assigned to floppy edges incident to v, K ( + , , v ) be the set of new colours used to colour edges incident to v , S(q,, v ) be the set of colours always used to colour an edge incident to v, and R ( v , , v ) be the set of colours originally used to colour an edge incident to v , but which have since been removed. Finally let S’(I),, v ) be the set

u,<,

v0

B. Bollobh, H . R. Hind

12

of colours used to colour one edge incident to vertex v in partial list colouring and a different edge incident to v in partial list colouring I/+. More precisely,

F(Vi, v)= {Vi(uv):IA(uv)\(3i(v)u 3i(u))I

3

I),

I>,

H(Vi, v)= {VT(uv):IA(uv)\(3i(v) U $ J i ( u ) ) l a1) and

K(Vi,

21)

= +;(v)\$dv),

S( Vi, v) = 3i(v)n 30(v),

R(Vi, v ) = 3idv)\3i(v), S'(Vi, v) = {Vi(uv): V i ( u v ) E 3 d v ) and Vi(uv)

3idu)).

For simplicity the partial list colouring qi is omitted from the notation for the above sets where no ambiguity can arise. Further we let f ( v ) =f(Vi, v ) = IF(q;, v)l, h(v) = h(Vi, v ) = IH(Vi, v)l and SO on. Since we cannot extend the sequence of partial list colourings beyond vs it follows that all the colours in A/(&) must be in the union of (9 30(a) or 3s(a)9 (ii) the set of colours assigned to edges ajbj with j < s, which are incident to vertex b, (iii) the set of escape colours of floppy edges incident to vertex a, and (iv) the set of colours assigned to edges of the form bx where x E V,. Writing D ( b ) for this last set of colours and d ( b ) for its cardinality (from the choice of the partition, it follows that d ( b ) 6 100 log A), it follows that A/(ab)E ( R ( a )U S ( a ) U K ( a ) ) U ( K ( b )U S'(b))U H ( a ) U D ( b ) , Setting K ' ( b ) = K ( b ) f l A/(ab),we get

Al(ab) E (R(a)U S ( a ) U K ( a ) )U ( K ' ( b )U S ' ( b ) )U H ( u ) U D ( b ) = ( R ( a )U S ( a ) U K ( a ) ) U S ' ( b ) U H ( a ) U D ( 6 ) since K ' ( b ) E (R(a)U $ ( a ) ) = &(a): the edge ab would have been defined as a floppy edge if a colour in Al(ab)was not the colour of an edge incident to either vertex a or vertex b in the partial list colouring V,. We also note that IS'(b)l= IK(b)l= k ( b ) since a colour is added to the set K ( b ) every time the vertex b occurs as the endvertex aj of an edge ajbj in the sequence of uncoloured edges and a colour already used to colour an edge incident to the vertex b is used to colour a new edge incident to vertex b (i.e. added to S ' ( b ) )every time b occurs as the endvertex bj of such an edge ajbj. These events occur in pairs. Thus we get

+

1 6 r ( a ) s(a)

+ k ( a )+ k ( b ) + h ( a )+ d ( 6 )

(5.1)

Clearly

+

r ( a ) s(a) 6 A,

(5.2)

New upper bound for list chromatic number

13

and each colour in A,(ab)must be assigned by qsto a non-floppy edge incident to vertex a or vertex b, so

f ( a ) + f ( b )zs 2 4 - 1. Since the number of escape colours assigned to floppy edges incident to vertex a is at most the number of colours assigned to those floppy edges, a crude upper bound for h ( a ) is

h ( a ) S 2 4 - 1.

(5.3)

It remains to obtain bounds for k ( a ) and k ( b ) . With f ( a ) bounded above by 2 4 - 1 < A/2, our choice of partition for the graph G, ensures that there exists a non-floppy edge ac such that c E V,. At most A of the colours in Al(ac) are in qo(c) and further qo(c)= qs(c) so at least 1 - A colours in A,(ac) must be in qn(a). Since these colours must also be in q j ( a ) for each 0 6 j S s (or we obtain a new floppy edge ac), it follows that

s(a)sl- A, and (with a similar argument for vertex b ) k(a)S 2 4 -1

k ( b ) S 2 4 - 1.

(5.4)

Recalling that d ( b ) S 100 log A = o ( A ) and substituting the bounds (5.2), (5.3) and (5.4) into inequality (5.1) we get 16

A

+ ( 2 4 -1) + ( 2 4 - 1) + ( 2 4 -1) + 100 log A

= 7 A - 31

+ 1001Og A

or

74 1 s - + 25 log A, 4 which is a contradiction. Since qowas chosen to be a partial list colouring for G with a maximal number of floppy edges, this contradiction shows that there is a A,-colouring of G for list function A,.

Case 2. a. E V, and bo E V,. We choose an edge a& such that ah = uo, bh E V, and function

q;: E(G)\{ahbb} + N such that qh(e) = q o ( e ) for all e E E(G)\{aobo, ahbb} and q;(aobn) = qo(a;b[)) and such that qh is a partial list colouring with at least as many floppy edges as for partial list colouring qo.This case then reduces to Case 1. exists. We define the sets F ( q , , v ) We need only show that such an edge and H ( q o , v ) and the cardinalities f(qo,v) and h ( q o ,v ) as in Case 1. Suppose

B. Bollobh, H . R . Hind

74

such an edge a&, does not exist, then

Ar(aobo) E $"(b") u WVO, bo) u W o )

so 1 A + ~ ( V Obo) , + d(ao) or

h(qo, b,)

3 1 - A - o(A).

Since the number of floppy edges incident to vertex 6,) must be at least as large as the number of escape colour assigned to these edges,

f(Vo, bo) 3 (1 - A - 4 A ) ) . Then for sufficiently large A, at most 2 4 - (1 - A - o(A)) = 3 4 - I - o ( A ) < 1 non-floppy edges are incident to vertices a, and bo, so there exists a colour in A,(aobo)assigned to either only floppy edges incident to vertices a, or b,), or not assigned to an edge incident to vertex a,) or vertex 6,. This gives us an immediate list colouring for G.

Case 3. a. E V2and b, E V,. Here we note that there exists a non-floppy edge from u E V, to some IJ E V, for every u E V, such that the number of floppy edges incident to vertex u is at most A/2. We define a sequence of partial list colourings qO,V1, . . . and associated edges a&,, a l b , , . . . , using the conditions listed below. Having I);-, and ~ ; - , b ; - ~ define 3, and aibi such that 9;is a partial list colouring, aI. = b.1-1, V , ( e ) = V ; - , ( e ) if e E E ( G ) \ { U ~ - ~ a~$~; }-, , and , V;(~;-lb;= - ~V) i - l ( ~ ; b ; ) , ~;(a;-lb;-I) Uj
I<-

74 4 '

New upper bound for list chromatic number

75

The 25 log A term which appears in the inequality in Case 1 does not arise in this case since condition (7) implies that each of the colours in the set D ( b ) is an escape colour of an edge incident to the vertex a, or is a colour assigned to be an edge incident to the vertex a, or is not in the set A,(&). This completes the proof. 0 It is possible to give a slightly improved value for the above upper bound for the list chromatic number (namely x;(C)6 12A/7 o(A)). The improvement, however, is modest and has a proof which is more lengthy and does not add significantly to the proof techniques for bounding xi. For this reason this improved upper bound is not included in this paper.

+

References [ l ] B. Bollobhs, Random Graphs (Academic Press, London, 1985). [2] B. Bollobhs and A.J. Harris, List-colourings of graphs, Graphs and Combinatorics 1 (1985) 1 15- 127. [3] A . Chetwynd and R. Haggkvist, A note on list-colourings, manuscript. [4] P. Erdos, A . Rubin and H. Taylor, Choosability in graphs, Congressus Numerantum 26 (1979) 125- 157.

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Discrete Mathematics 74 (1989) 77-84 North-Holland

77

A NOTE ON PERFECT ORDERS C.T. HOANG Department of Computer Science, Rutgers University, New Brunswick, New Jersey 08903,USA

and N.V.R. MAHADEV Department of Mathematics, Northeastern University, Boston, MA 02115, USA Perfectly orderable graphs were introduced by Chvital in 1984. Since then, several classes of perfectly orderable graphs have been identified. In this paper, we establish three new results on perfectly orderable graphs. First, we prove that every graph with Dilworth number at most three has a simplicial vertex, in the graph or in its complement. Second, we provide a characterization of graphs G with this property: each maximal vertex of G is simplicial in the complement of G . Finally, we introduce the notion of a locally perfect order and show that every arborescence-comparability graph admits a locally perfect order.

1. Introduction

It is well known that the problem of determining the chromatic number x ( G ) of a graph G is NP-hard. However, the problem is polynomially solvable for many restricted classes of graphs. One such class of graphs is the class of “triangulated” graphs: a graph is triangulated if each of its cycles with more than three vertices contains a chord. A theorem of Dirac [6] asserts that every triangulated graph contains a simplicial vertex, i.e. a vertex whose neighbourhood forms a clique. It follows from Dirac’s theorem that one can order the vertices of a triangulated graph G into a sequence v l , v 2 , .. . , vn such that each v, is simplicial in the subgraph Gj of G induced by { v l , v2, . . . , vj};such a sequence is called a simplicia1 order of G . Now, an optimal colouring of G can be found by applying the following “greedy” algorithm: (1) Scan the vertices of G in the order given by the sequence v,, v2, . . . , v, (2) Assign to each vi the smallest positive integer assigned to no neighbour v, of vi with j < i. To see that the greedy procedure produces an optimal colouring of the triangulated graph G , one only needs observe that if ui is assigned integer k, then vi belongs to a clique of size k: vi is adjacent to at least k - 1 vertices in {v,, v2, . . . , vi} with colours 1 , 2 , . . . ,k - 1. Since the order is simplicial, it follows that vi and its neighbours in {v,,v2, . . . , v i } form a clique of size precisely k. 0012-365X/89/$3.50 @ 1989, Elsevier Science Publishers B.V. (North-Holland)

C. T. Hocing, N.V.R . Mahadev

78

The above remark shows that for a triangulated graph G we have x ( G ) = w ( G ) (the number of vertices in a largest clique of G). Thus, triangulated graphs are perfect in the sense defined by Berge [2]: a graph C is perfect if x ( H ) = o ( H ) for each induced subgraph H of G. We are interested in studying graphs G with the following property: (*) G admits an order < such that the greedy algorithm, when applied to any induced subgraph H of G, produces an optimal colouring of H. The above property was first studied by Chvhtal [3], and he proved that (*) holds for an ordered graph (G, <) if and only if no chordless path with vertices a , b, c, d and edges ab, bc, cd has a < b and d < c (this ordered path is called an obstruction). Chvhtal proposed to call the order < a perfect order on G , and to call G perfectly orderable. Chvhtal also proved that every perfectly orderable graph G is strongly perfect in the sense of Berge and Duchet [l]:each induced subgraph H of G contains a stable set that meets all maximal cliques of H. In this paper, we shall study several interesting properties of some classes of perfectly orderable graphs.

2. Good orders

-

Let us define an order vl < v 2 < < v, on a graph G to be good if for any induced subgraph H of C, either the largest vertex of (H, <) is simplicial or the smallest vertex of (H, <) is simplicial in H. A graph is good if it admits a good order. (If x is simplicial in G, then we shall call x a cosimplicial vertex of G . )

Remark 2.1. Every good order is a perfect order.

-

Proof. By induction on the number of vertices. Let P = v 1 < v 2 < v2. . < v, be a good order of a graph G. If v I is cosimplicial in G then no obstruction in P can contain v,; by induction v 2 < * . . < v, is perfect, and so P is perfect. Similarly, if v, is simplicial in C, then no obstruction of P can contain v,; but by induction v , < * < v,-, is perfect, and so P is perfect. 0

-

Chvhtal defined a graph C to be brittle if each induced subgraph H of G contains a vertex that is not the endpoint or the midpoint of any P4 (the chordless path on four vertices) of H. It is easy to see that good graphs are brittle. At present no characterization (by minimal forbidden induced subgraphs) of brittle graphs (and good graphs) is known. But by Dirac’s theorem, we know that triangulated graphs and their complements are good and therefore brittle. The purpose of this section is to present a new class of good graphs. Let G be a graph and let x, y be two vertices of G. It is said that x dominates y if {x} U N ( x ) 3 N ( y ) ( N ( t )stands for the set of neighbours of t); if x dominates y or y dominates x then it is said that x and y are comparable. Now, the Dilworth

Perfect orders

79

number of a graph is the largest number of pairwise incomparable vertices of the graph. (Note that the domination relation is a transitive order.) Preissmann first proved that graphs with Dilworth number at most three are strongly perfect. Later, it is proved by ChvAtal, HoBng, Mahadev and deWerra [4] that these graphs are perfectly orderable. Now the following theorem shows that graphs with Dilworth number at most three admit good orders.

Theorem 2.2. ff G is a graph with Dilworth number at most three, then either G contains a simplicial vertex or G contains a cosimplicial vertex. Proof. Let G be a graph with Dilworth number at most three. Then by Dilworth’s theorem [5],the vertex set of G can be partitioned into three sets X o , X , , X , (one or more of the Xi’s may be empty) such that the vertices of each X i can be linearly ordered by <, so that a <, b only if a is dominated by b in G. For each nonempty Xi, define (with respect to <,) (i) a, to be the largest vertex in Xi, (ii) c, to be the smallest vertex in X i , (iii) b, to be the smallest vertex in X , with the property that all vertices (of X,) larger than b, form a clique. Note that within each X,, no vertex smaller than b, is adjacent to b, and that the vertices smaller than b, form a stable set. Also the vertices a,, b,, ci need not be distinct. Let { i , j , k } = (0, 1, 2 ) . If c, is not adjacent to any vertex outside X , then c, is simplicial. Thus, we may assume that c, is adjacent to some vertex in Xi.If either Xk is empty, or ck is adjacent to some vertex in X,, then clearly ai is a cosimplicial vertex of G. From the above remark, we may assume now that each c, has a neighbour in X I + , and no neighbour in X,,, (obviously the subscripts t are taken modulo 3). Now, if some c, is not joined to any vertex smaller than b,, in XI+,, then clearly c, is a simplicial vertex. Hence we may assume that each c, is adjacent to some vertex smaller than b,,, in X,,,. Now, we claim that each a, is a cosimplicial vertex of G. To justify our claim, note that c , + ~is adjacent to some vertex in X I ; therefore a, is adjacent to c,,, and hence to all vertices of X,,,. Clearly, each vertex nonadjacent to a, is either smaller than b,, or b,,,. Thus a, is cosimplicial as claimed. 0

,

Let G be a graph with Dilworth number at most three. Let n and m be the number of vertices, and the number of edges of G , respectively. Suppose we are given a partition of G into sets X,, X , , X , as described in the proof of Theorem 2.2. Then we can construct a good order of G in O ( n . m ) steps as follows. First find a vertex x which is either simplicial or cosimplicial, by testing all six vertices a,, c, (this takes O(m) steps: clearly checking if a vertex is simplicial or

C.T. Hocing, N.V.R. Mahadeu

80

cosimplicial takes O ( m ) steps, and we have to check only six vertices of G). Now, recursively construct a good order x1< * < x , , - ~ of G - x . If x is simplicial then return the good order x1< * * < x n P 1 < x , otherwise return the good order

-

x

<XI

<*

* *

<x,-,.

Perhaps we should remark that the Dilworth number of a graph can be determined in O(n’) steps. Given a graph G , we construct a graph D ( G ) as follows. The vertices of D ( C ) are the vertices of G and two vertices are adjacent in D ( G ) if and only if one of them dominates the other in G. Clearly, D(G)can be constructed in O(n3)steps (testing if a vertex x dominates a vertex y takes O ( n ) steps, and we have to perform O(nz) such tests). Note that D ( C ) is a “comparability” graph (definition will be given later). Also, the Dilworth number of G is the “stability” number of D ( G ) (the stability number of a graph is the largest number of pairwise nonadjacent vertices of the graph); and a partition of the vertices of D ( G ) into a smallest number of cliques gives a partition of the vertices of C into a smallest number of sets such that the vertices in each set are pairwise comparable. Finally, it is well known that the stability number of a comparability graph and a partition of its vertices into a smallest number of cliques, can be determined in O(n3) steps, by solving certain network flow problem (see [ti]).

3. Maximal vertices Define a vertex x of a graph C to be maximal if x is not strictly dominated by any vertex of G , i.e. for each vertex y we have either N ( x ) - N ( y ) - y # 0 or else N ( y ) - N ( x ) - x = 0. We are interested in the following question: (**) what graphs G have the properties that each maximal vertex of G is a cosimplicial vertex? This question arises from the proof of Theorem 2.2: one may ask if it is true that for a graph with Dilworth number at most three, each maximal vertex is cosimplicial, and each minimal vertex (defined in the obvious way) is simplicial. (The answer is CCn0?l: consider the graph 2 K 2 , i.e. the union of two disjoint edges.) Fortunately, it turns out that there is a simple characterization of the graphs described in (**) (in terms of minimal forbidden induced subgraphs). Note that these graphs must be complements of triangulated graphs. As usual we shall let Pk (respectively ck) denote the chordless path (respectively cycle) on k vertices. Define the graph 6 (respectively F2) to be the graph obtained from a P7 with vertices v l , u 2 , . . . , v7 by adding the edge v1v3 (respectively, the edges v1v3and v7vs).

Theorem 3.1. A graph C has the property that each maximal vertex is cosimplicial if and only if G contains no induced subgraph isomorphic to any of the following 4, 8, &. graphs: 2K2, C5,

c6,c7,

81

Perfect orders

Proof. The “only i f ’ part is obvious. To prove the “if” part, suppose that x is a maximal vertex of G that is not cosimplicial. This means that there are adjacent vertices a, b such that both a and b are not adjacent to x . For simplicity, let us say that a vertex r sees (respectively misses) a vertex s if r is adjacent (respectively nonadjacent) to s. Since x is maximal, there is a vertex a‘ that sees x but misses b. To avoid having a 2 K 2 , a must see a‘. Similarly, there is a vertex b’ that sees x , misses a, and sees b. To avoid having a C5,a’ must see b’. Now, since x is maximal, there must be a vertex a” that sees x and misses a ’ ; similarly, there is a vertex b“ that sees x and misses b Next, we claim that a” sees 6’; otherwise, we would have a (if a” sees a, b), or a C5 (if a” sees a and misses b, or if a” misses a and sees b),or a 2K2 (if a” misses both a and b ) . By the same reasoning, we know that a‘ sees b“. Now a must see a” or else the set {a”, x , a ’ , a, b } induces a C5 or it contains a 2K2. Similarly, b must see b” or else the set {b”,x , b’, 6, a } induces a C5 or it contains a 2Kz. Finally, note that all edges in the subgraph H of G induced by { x , a, a ’ , a”, 6, b’, b”} are determined except for the following edges ba”, ab”, a%” which may or may not be present in H. If ar’b”is not an edge, then H contains the complement of a chordless cycle with five, or six, or seven vertices. If a”br‘is an edge, then H contains the complement of a graph isomorphic to a P7J or the graph 4 with i = 1 or 2. 0 I.

c6

4. Local perfection

Let C be a colouring of a graph G , and let x be a vertex of G. Define F ( x , C) to be the number of colours (of C) appearing in the neighbourhood N ( x ) of x . Preissmann [lo] defined a colouring C of a graph G to be locally perfect if F ( x , C ) = o ( N ( x ) )for each vertex x of G. A graph is called locally perfect if each induced subgraph admits a locally perfect colouring. Preissman proved that locally perfect graphs are perfect and that triangulated graphs are locally perfect. Hertz [9] proved that a graph is locally perfect if each of its odd cycles with at least five vertices contains two chords. We propose to call an order < on a graph G locally perfect if the greedy algorithm, applied to any induced subgraph (H, <) of (G, <), produces a locally perfect colouring of H. Observe that a locally perfect order can not contain an obstruction; but (as we shall see later) a perfect order is not necessarily locally perfect. This observation led us to study locally perfect orders that are defined in terms of ordered P3’s. Let (H, <) be an ordered P3 with vertices a, b, c and edges ab, bc. We shall say that (H, <) is of type 1 if a < b
C.T. Hoang, N . V . R . Mahadev

82

be a subset of { 1,2,3}; define <s to be an order such that every ordered P3 is of one of the types belonging to S. Call S admissible if and only if any order < s (if it exists) on any graph G is locally perfect.

Theorem 4.1. A set S is admissible

if and only if

S = (2) or S = (3).

Before proving Theorem 4.1, perhaps we should remark that an order is usually called transitive if it contains no P3 of type 1, and a graph is called a comparability graph if it admits a transitive order. In her pioneering paper on locally perfect graphs, Preissmann constructed a comparability graph that is not locally perfect (see Fig. 3 in [lo]). In our terms, she showed that the set {2,3} is not admissible. Note that the order <{3) is both transitive and simplicial. Wolk [ l l ] proved that a graph G admits a transitive and simplicia1 order if and only if G contains no C4 and no P4. Such graphs are called “arborescence-comparability” graphs [7]. Call a graph locally perfectly orderable if it admits a locally perfect order. Theorem 4.1 shows that arborescence-comparability graphs are locally perfectly orderable. Since these graphs are triangulated graphs, it is natural to ask whether all triangulated graphs are locally perfectly orderable. At present we do not know the answer to the above question. Perhaps a more interesting problem is to characterize locally perfectly ordered graphs by minimal forbidden ordered subgraphs. Now, to conclude our paper, we shall prove Theorem 4.1.

Proof of Theorem 4.1. We remarked previously that the set {2,3} is not admissible. Hence, to prove the “only if” part of the theorem, we only need show that if 1E S then S is not admissible. For this purpose, consider a graph with seven vertices 1 , 2 , . . . , 7 and with exactly three maximal cliques: { 1 , 2 , 3 , 4 } , {4,5,6}, and {5,6,7}. Now order the vertices as 1 < 2 < < 7. Clearly, every P3 is of type 1, but this order is not locally perfect. To prove the “if” part, let (G, <.s) be an ordered graph with S = (2) or S = (3). For simplicity, we shall let < denote <s. Note that < is a transitive order. For each vertex x we shall let A ( x ) denote the set of neighbours y of x with y < x , and let B ( x ) = N ( x ) - A ( x ) .

---

Lemma 4.2. If a vertex x is assigned integer k then A ( x ) contains a clique C with k - 1 vertices such that the vertices of C are assigned integers 1, 2, . . . , k - 1. The above lemma is a special case of a theorem of Chvlital [3]. For the sake of completeness, we shall use Chvlital’s idea to give a short proof of this lemma here. Let i be the smallest integer such that there are vertices v ( i ) , v(i + l), . . . , v(k - 1) in A ( x ) satisfying the following properties: (i) v ( i )< v ( i + 1)<. . . < v(k - l), (ii) each v(j) is assigned integer j (i s j =sk - l), (iii) the vertices v ( i ) ,v ( i l), . . . , v(k - 1) form a clique. If i = 1 then we are done.

+

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If i > 1 then (by our way or colouring) v ( i ) has a neighbour v(i - 1) with assigned integer i - 1 such that v(i - 1) < v ( i ) . But by transitivity, v(i - 1) is adjacent to x and all vertices v ( j ) (i 6 j S k - l), contradicting our choice of i. Hence, the Lemma holds. (Incidentally, note that Lemma shows that the greedy algorithm uses precisely w ( C ) colours.) Now, to prove Theorem 4.1, we shall show that for each x, we have F ( x , C) = w ( N ( x ) ) .For this purpose, let k be the integer assigned to x , and let y be a neighbour of x with the largest (assigned) integer k ' . Note that by our way of colouring, we have k - 1 S F(x, C); furthermore if k' > k then F(x, C ) s k' - 1 (because no neighbour of x can receive integer k ) , and if k ' < k then F ( x , C ) = k - 1 (because k' 3 k - 1). If k' < k then by Lemma 4.2 we have w ( N ( x ) )= k - 1 = F ( x , C ) . Thus we may assume that k' > k. Now, observe that no vertex t of A ( x ) can be assigned an integer n > k , otherwise in A ( t ) there is a vertex z with integer k. But then the ordered P3{z, t, x } is of type 1, a contradiction. From the above observation, we may assume that x < y . If S = ( 3 ) then A ( y ) is a clique (the order is simplicial); thus we have F ( x , C) = k' - 1 = IA(y)l = w ( N ( x)) Now, we may assume that S = (2). By our previous observation, in A ( x ) , only integers m < k can appear, and thus by Lemma 4.2 I w ( A ( x ) ) l = k - 1. Next, observe that (i) t sees z whenever t E A @ ) , z E B ( x ) and (ii) B ( x ) is a clique (if (i) fails then G has a e3of type 1, and if (ii) fails then G has a P3 of type 3). Hence, we have w ( N ( x ) )= ( k - 1) + IB(x)l= F ( x , C ) . 0

-

Acknowledgement We would like to thank an anonymous referee for pointing out several errors in our original manuscript.

References [I] C. Berge and P. Duchet, Strongly perfect graphs, in Berge, C. and V. Chvital, Ed., Topics on Perfect Graphs (North-Holland, Amsterdam, 1984) 57-61. [2] C. Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind (Zusammenfassung). Math.-Natur. Reihe 114 (Wiss. Z. Martin-Luther-Univ., HalleWintenberg, 1961). [3] V. Chvital, Perfectly ordered graphs, in C. Berge and V. Chvital, Ed., Topics on Perfect Graphs (North-Holland, Amsterdam, 1984) 63-65. [4] V. Chvital, C.T. Hoing, N.V.R. Mahadev and D. de Werra, Four classes of perfectly orderable graphs, J. Graph Theory 11, 4 (1987) 481-495. [5] R. P. Dilworth, A decomposition theorem for partially ordered sets, Ann. Math. 51 (1950) 161-166.

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[6] G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961) 71-76. [7] P. Duchet, Classical graphs, in C. Berge and V. Chvital, Ed., Topics on Perfect Graphs (North-Holland, Amsterdam, 1984), 67-96. [8] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980). [9] A. Hertz, Slim graphs, Report ORWP 87/1, Swiss Federal Institute of Technology in Laussane (1987). [ 101 M. Preissmann, Locally perfect graphs, manuscript. [ l l ] E. S. Wolk, The comparability graph of a tree, Proc. Amer. Math. SOC.3 (1962) 789-795.

Discrete Mathematics 74 (1989) 85-97 North-Holland

85

ON THE PENROSE NUMBER OF CUBIC DIAGRAMS Franqois JAEGER Laboratoire de Structures DkcrPtes, IMAG, BP 68, 38402 St Martin d 'HPres Cedex, France

A cubic diagram is a cubic graph G drawn in the plane, possibly with edge-crossings. The drawing defines a sign for each edge-3-coloring of G . The Penrose number of G is the sum of signs of its edge-3-colorings. For plane graphs it coincides with the number of edge-3-colorings. Given a cubic diagram G , we define a sign for every Eulerian orientation of its line-graph L ( C ) and prove that the Penrose number of G is equal to the sum of signs of the Eulerian orientations of L ( G ) . This yields a new recursive scheme for the computation of the Penrose number. Another consequence is a simple formula which gives the number of vertex-4colorings of a loopless plane triangulation in terms of the mappings from the vertex-set to { 1,2,3) which take exactly two distinct values on each triangle.

1. Introduction The Four Color Theorem [l] is equivalent to the statement that every cubic bridgeless plane graph C has an edge-3-coloring. The number T ( G ) of such edge-3-colorings can be computed using a recursive scheme corresponding to the evaluation of the chromatic polynomial of either the dual graph or the line-graph of G. Penrose proposes in [12] another recursive scheme for the computation of T ( G ) . In fact this scheme is not restricted to plane graphs. It works on cubic plane diagrams where edges may cross and computes a signed analogue of T ( C ) (we call this invariant the Penrose number) which coincides with T ( G ) if C is a plane graph. We present Penrose's work in Section 2. Then in Section 3 we establish a simple formula which gives the Penrose number of a cubic diagram in terms of the Eulerian orientations of its line-graph. Finally in Section 4 we show how this formula leads to a new recursive computation scheme for the Penrose number, and we give special results in the case of plane graphs.

2. The Penrose number of a cubic diagram 2.1. General definitions The graphs which we consider here are finite, and may have loops and multiple edges. It will be convenient to view each edge e as subdivided into two half-edges (the two halves of e ) , one incident to each end of e . In particular if e is a loop 0012-365X/89/$3.50 01989, Elsevier Science Publishers B.V. (North-Holland)

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incident to the vertex u , both halves of e are incident to u . The degree of a vertex is the number of half-edges incident to it. A graph is cubic (respectively: 4-regular) if every vertex is of degree three (respectively: four). Let u be a vertex of degree 2 in a graph G , and let h l , h2 be the two half-edges incident to u . For i = 1, 2, hi is a half of some edge ej and we denote the other half of this edge by h i . Assume first that e l f e z . Then deleting u, e l , e2, h l , h2 and creating a new edge with halves h ; , h; yields a graph which is said to be obtained from G by erasing v. If e , = e2 = e (so that e is a loop at v), then h ; = h2 and h; = h l . In this case the erasing process just described replaces the “loop-graph’’ ( { u } , { e } ) by a structure consisting of two half-edges incident to no vertex. By convention we shall consider this as a graph with one edge and no vertex and call it a free loop. For instance a cubic graph or a 4-regular graph may contain connected components consisting of free loops. An orientation of a graph is obtained by choosing for each edge one of its halves as initial and the other as terminal. An orientation is said to be Eulerian if each vertex is incident to an equal number of initial and terminal half-edges. Note that a graph consisting of one loop or one free loop has exactly two Eulerian orientations. A rotation at a vertex u of a graph C = (V, E ) is a cyclic permutation of the half-edges incident to u . A rotation system of C is a family p = (p,,, u E V), where pu is a rotation at u . Rotation systems are the basis of the classical combinatorial description of embeddings of graphs on orientable surfaces known as the “permutation technique” (see [6]). To be more precise, each rotation system p = (p,,, u E V) of the connected graph G = (V, E ) defines an embedding of G on some orientable surface S. If we identify p with the permutation n I v E v p ,and , denote by a the fixed-point-free involution which maps every half-edge to the other half of the same edge, the faces of the embedding correspond to the orbits of pa. The number of these orbits will yield the genus of S by Euler’s formula. Other usual definitions on graphs will be found in [3] or [ 5 ] .

2.2. Diagrams Let G = (V, E ) be a graph. A drawing of G in the plane is obtained by representing each vertex u of C by a point u* of the plane and each edge e of C by a simple Jordan curve e* according to the following rules: - if u,, u2 are distinct vertices then u f # u: - if the edge e has ends v,,u2 (which might be identical) then e* joins uT and v: and contains no u* for u in V - { v l , u 2 } - if e l , e2 are distinct edges then e : n e,,*is finite.

Remarks. (i) Note that if we replace in this last rule “finite” by “contained in V*” (where V * denotes { v * / uE V}) we obtain the usual definition of plane embeddings. (ii) A free loop is drawn as a simple closed Jordan curve disjoint from V*.

Penrose number of cubic diagrams

87

(iii) In graph theory texts (including the present one where we state it explicitly), figures depicting graphs actually depict them as drawn in the plane. Ambiguity is avoided through the use of additional standard conventions which need not be incorporated in our mathematical definition of drawing. The subdivision of an edge e into its halves will be represented in the obvious way by the selection of a point on e* - V * (or two such points in the case of a free loop). To each drawing of the graph G in the plane we associate the rotation system p = ( p u , v E V) of G defined as follows: pv maps each half-edge incident to v to the next half-edge incident to v in the clockwise order around v*. Conversely, it is easy to see that every rotation system of G is associated in this way to some drawing in the plane. Thus plane drawings appear as a convenient description of rotation systems and equivalently of embeddings in orientable surfaces. In the sequel we call diagram a graph drawn in the plane. If the drawing is a plane embedding, the diagram is then called as usual a plane graph. For the sake of simplicity we no longer distinguish between a vertex v (an edge e) and its representative v * ( e * ) .

2.3. Edge-3-coloringsin cubic diagrams An edge-3-coloring of a cubic graph C is a coloring of the half-edges of G with 3 colors such that the two halves of any edge receive the same color (which we call the color of the edge) and the three half-edges incident to any vertex receive three distinct colors. We denote by T ( G ) the number of edge-3-colorings of G. For instance if G has a loop then T ( G )= 0. On the other hand, if G is a free loop, T ( G )= 3. We now define a signed analogue of T ( G )for diagrams which is due to Penrose [12]. Our presentation is slightly different from those given in [8, 91 or [12]. Assume that G is drawn in the plane and consider an edge-3-coloring f of G with the colors 1,2,3. A vertex v of G will be said positive (respectively: negative) with respect to f if the colors of the half-edges incident to v are 1 , 2 , 3 (respectively: 1 , 3 , 2 ) in the clockwise order around v. Let n + ( f ) (respectively: n - ( f ) ) denote the number of vertices which are positive (respectively: negative) with respect t o f . Note that n + ( f ) and n - ( f ) have the same parity since G has an even number of vertices. We write s(f) = 1 if n + ( f )= n - ( f ) mod4, and s(f) = -1 otherwise. The number s(f) is the sign o f f . We define the Penrose number of the diagram G as the sum of the signs of the edge-3-colorings of G , and we denote this number by T ’ ( G ) . Consider now another drawing of G whose associated rotation system is the same as the previous one except at one single vertex v where the rotation is opposite. Let f be any edge-3-coloring of G. The sign of v with respect to f changes, and the sign of the other vertices do not change. Hence the values of n + ( f )- n - ( f ) in the two drawings differ by 2 or -2, and the sign off changes. It

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follows that the sign of T ’ ( C ) changes, but not its absolute value. In other words, IT’(G)l is a graph invariant: it depends only on C , not on the way it is drawn. 2.4. Basic properties of the Penrose number The Penrose number appears as an interesting tool for the study of edge-3colorings of cubic graphs, in particular in the planar case. This stems from the following results (see [8, 9, 121).

Proposition 1. Let G be a cubic plane graph. Then T ( G )= T ’ ( C ) . Proof. All edge-3-colorings of a cubic plane graph have positive sign. This result appeared for the first time (as far as I know) in [13] and was rediscovered by several authors in a number of equivalent forms. An elementary proof is as follows (see also [8, 91). Consider an edge-3-coloring f of C with the colors 1 , 2 , 3 . Replace each edge colored 3 by two parallel edges, one colored a and the other colored b , in order to obtain a plane 4-regular graph. In this new graph the edges colored a or 1 form a family C1of disjoint simple closed curves in the plane, and the other edges form another such family C2. Then it is easy to check that the ends of an edge of C colored 3 have the same signs with respect to f if and only if exactly one of them corresponds to a crossing of a curve of C1with a curve of C 2 . The total number of such crossings is even, thus there is an even number of edges of color 3 whose ends have the same sign, and hence n f ( f ) = n - ( f ) mod 4. 0 Remark. Proposition 1 cannot be extended to non-planar graphs: it is easy to check (see [12] and [7], Section 2.3) that the Penrose number of the Kuratowski graph K3,3is zero. Let G = (V, E) be a cubic graph drawn in the plane with associated rotation system p = (pu, v E V). As before we consider p as a permutation on the set of half-edges. Let e be an edge of G with distinct ends x , y and let h , h’ be the two halves of e . We define two new graphs H’ and H as follows: we first delete the edge e and its halves, and then rearrange the incidences of the half-edges incident to the vertices x and y in the following way. In H‘, ph and p2h‘ are made incident to one of these vertices, and ph’ and p2h are made incident to the other. On the other hand in H”, ph and ph’ are made incident to one vertex of { x , y } while p2h and p2h’ are made incident to the other (see Fig. 1). Note that in the graphs H’, H” all vertices have degree 3 with the exception of x and y which have degree 2. Hence if we erase x and y we obtain cubic graphs (recall that we allow free loops in our definition of such graphs). These graphs will be drawn in the plane, starting with the drawing of G and performing only a local change in the neighbourhood of e as indicated in Fig. 1. Clearly the associated rotation systems can be identified with (pu, v E V - { x , y}). The cubic

Penrose number of cubic diagrams

89

Fig. 1.

diagram thus obtained from H’ (respectively: H”) will be said to be obtained from G by the non-crossing (respectively: crossing) dissolution of the edge e.

Proposition 2. Let G be a cubic diagram, and e be an edge of G with two distinct ends. Let G’ (respectively: G”) be obtained from G by the non-crossing (respectively:crossing) dissolution of the edge e. Then T ’ ( C )= T’(G’)- T‘(G’’). Proof. Let H be the diagram obtained from G by the contraction of e. Thus all vertices of H have degree 3, with the exception of one vertex v which has degree 4.We may extend to H the definitions of edge-3-coloring and of the sign of such a coloring given in Section 2.3 by replacing everywhere the term “vertex” by the term “vertex of degree 3”. Then edge-3-colorings of G , G’, G” can be identified with special edge-3-colorings of H. For any partition P of the set of half-edges of H incident to v , let S ( P ) be the sum of the signs of the edge-3-colorings of H which induce this partition. Then clearly, using the above-mentioned identification: T’(G’)= S ({{p h ,p2h’), {ph’, p 2 h ) ) )+ S ({{p h ,p2h‘, ph’, p 2 h ) ) )

T’(G‘’)= S ( { { p h , ph’), {p2h,p 2 h ’ ) ) )+ S ({{p h ,p2h’,ph‘, p 2 h ) ) ) . Similarly, a simple analysis of the signs of the edge-3-colorings of G yields: T’(G)= S ( { { p h ,p2h’),{ph‘, p 2 h ) ) )- S ({{p h ,p h ‘), {p2h,p 2 h ’))), and the result follows immediately. 0

Fig. 2.

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Proposition 2, together with the easy fact that the Penrose number of a cubic diagram is equal to the product of the Penrose numbers of its connected components, allows the recursive computation of 7”(G) for any cubic diagram G. An example of such a computation is given in Fig. 2, where each diagram stands for its Penrose number.

3. Penrose number and Eulerian orientations of the line-graph 3.1. The line-graph of a cubic graph Let G = (V, E ) be a cubic graph with no loops or free loops and let v be a vertex of G. An angle of G at v is an unordered pair of distinct half-edges incident to v. The line-graph of G , which we denote by L (G), has vertex-set E and contains one edge with ends e, e’ for each angle of G consisting of one half of e and one half of e ’ . It is easy to see that L ( G ) is a 4-regular graph with no loops or free loops. For each vertex v of G we denote by t, and call triangle at v the set of three edges of L( G ) corresponding to the three angles of G at v. Note that {t,, v E V} is a partition of the edge-set of L ( G ) . If G is a diagram, let p = (p,,, v E V) be the associated rotation system. Consider an edge a of L ( C ) with ends e, e ’ , corresponding to an angle { h , h ’ } of G , where h is a half of e and h’ is a half of e ’ . Let us choose the half of a incident to e as initial and the other half of a (incident to e ’ ) as terminal whenever h’ = p h . This defines an orientation of L ( G ) whose restriction to every triangle t,, (v E V) is Eulerian. Thus this orientation of L ( G ) is Eulerian and we call it the canonical orientation of L(C). Consider the embedding of G in an orientable surface S defined by the rotation system p = (p”, v E V) (see Section 2.1). By.representing each vertex of L ( G ) by an interior point of the corresponding edge of G and by drawing each edge of L ( G ) sufficiently close to the corresponding angle of C it is easy to obtain an embedding of L ( G ) on S. This embedded graph is known as the medial graph of G on S (see [ l l ] p. 47 for the planar case). Then each triangle t,, (v E V ) of L ( G ) bounds a face and the canonical orientation of L ( G ) corresponds to a clockwise orientation around each such face. 3.2. Eulerian orientations and the Penrose number Let H be a 4-regular graph and v be a vertex of H. Let h , , h2, h3, h4 be the four half-edges incident to v. A transition at v is a partition of { h , ,h2, h 3 ,h 4 } into pairs. Thus there are exactly three distinct transitions at v. Consider an Eulerian orientation of H. Then each vertex v will be incident to two initial half-edges and two terminal half-edges. The transition at u consisting of the pair of initial half-edges together with the pair of terminal half-edges will be called anticoherent, and the two other transitions at v will be said coherent.

Penrose number of cubic diagrams

91

Consider a cubic diagram G. The anticoherent transitions of the canonical orientation of its line-graph L ( G ) will be said to be crossing, and the other transitions to be non-crossing. This terminology is motivated by the topological interpretation of L ( G ) as a medial graph given in the previous section. Let d be an Eulerian orientation of L ( G ) . We write s ( d ) = 1 if the number of non-crossing anticoherent transitions of d is even, and s ( d ) = -1 otherwise. The number s ( d ) is the sign of d .

Lemma 3. An Eulerian orientation of L ( G ) has positive sign if and only if it is obtained from the canonical orientation by reversing an even number of edges. Proof. Let do be the canonical orientation of L ( G ) and d l be an Eulerian orientation of L ( G ) obtained from do by reversing a set X of edges. It is easy to check that every vertex v of L ( G ) is incident to an even number of halves of edges of X, and that if there are exactly two such half-edges, one of them is initial and the other is terminal (this is true in both orientations). Moreover the anticoherent transitions of do and d l at Y are distinct, that is, the anticoherent transition of d l at v is non-crossing, if and only if v is incident to exactly two halves of edges of X. Since 2 =C i where Y; is the set of vertices of L ( G ) incident to exactly i halves of edges of X, the number lY21 of non-crossing Ci anticoherent transitions of d l has the parity of

1x1

1x1,

1x1.

Proposition 4. Let G be a cubic diagram with no loops or free loops. Its Penrose number T ’ ( G ) is equal to the sum of the signs of the Eulerian orientations of L(G). Proof. Consider a vertex v of G and the associated triangle t, of L ( G ) . Let e l , e2,e3 be the three edges of G incident to Y and let d be an Eulerian orientation of L ( G ) . Two cases may occur for the restriction of this orientation to the edges of t,,. (i) Either it is Eulerian. Then we say that d orients t,, as a circuit. (ii) Or we may assume without loss of generality that e l is a source and e3 is a sink. Then we say that d orients t,, from e l to e3, and that d marks the edges e l , e3 at v. Note that d marks an edge e of G at both of its ends, or none. In the first case we shall simply say that d marks e . We denote by C ( d ) the set of edges of G marked by d . For each subset F of the edge-set E of G we denote by Z ( F ) the set of Eulerian orientations d of L ( C ) such that C ( d )= F, and by z ( F ) the sum of their signs. Thus the sum of the signs of the Eulerian orientations of L ( G ) is equal to CFGEz ( F ) . Let F G E be such that z ( F ) # 0 and consider an orientation d in Z ( F ) . Then d marks zero or two edges at each vertex of G , and hence F is partitioned into vertex-disjoint cycles.

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Assume that some vertex v of G is incident to no edge of F. Then every orientation in Z ( F ) orients the corresponding triangle t, of L ( G ) into a circuit. To every orientation d in Z ( F ) we may associate another such orientation, which we shall denote by d’, by reversing this circuit. Clearly (d’)’ = d. Moreover it easily follows from Lemma 3 that d and d’ have opposite signs. This implies that z ( F ) = O , a contradiction. It follows that every vertex of C is incident to two edges of F. In other words, F is a 2-factor of C . Then for every orientation d in Z ( F ) there exists an unique Eulerian orientation q ( d ) of F such that for every vertex v of G , d orients the triangle t, from the edge of F with terminal half-edge incident to v to the edge of F with initial half-edge incident to v. Moreover, every Eulerian orientation of F is of the form q ( d ) for an unique orientation d in Z ( F ) . Let F,, . . . , Fk be the connected components of F. Let d be an orientation in Z ( F ) and let X be a subset of ( 1 , . . . , k } . We denote by r ( X , d) the unique orientation in Z ( F ) such that q ( r ( X , d)) is obtained from q(d) by reversing the edges of Fx = Uicx8.Note that r ( X , d) is obtained from d by reversing 3 lFxl edges of L ( G ) . If one of the connected components of F, for instance F,, is an odd cycle, we may consider the fixed-point-free involution which associates to every orientation d in Z ( F ) the orientation r({l}, d). These two orientations have opposite signs by Lemma 3. This implies that z ( F ) = 0 , a contradiction. It follows that all components of F are even cycles. In other words, F is an even 2-factor of G. Let d be any fixed orientation in Z ( F ) . Then clearly Z ( F ) = { r ( X , d ) / X r (1, . . . , k}}. It follows that lZ(F)I = 2k. Moreover, Lemma 3 implies that all orientations in Z ( F ) have the same sign, which we denote by s ( F ) . Thus z ( F ) = s(F)2k. Let us say that an edge-3-coloring f of G with the colors 1 , 2 , 3 is compatible with the even 2-factor F if F consists of the edges colored with the colors 1 , 2 by f. Let n = 2p be the number of vertices of G. Consider an orientation d , in Z ( F ) and an edge-3-coloring f compatible with F. There are exactly p vertices v of G such that dl orients the triangle f,from an edge colored 1 (respectively: 2) by f to an edge colored 2 (respectively: 1) b y f . If we reverse the 2p edges of L ( G ) which have both ends in F, we obtain an Eulerian orientation d2 which orients every triangle t,, into a circuit. Exactly p of these oriented triangles meet successively edges of G of colors 1 , 2 , 3 in this circular order. By reversing the 3p edges of the p other triangles we obtain an Eulerian orientation d 3 which orients every triangle t, into a circuit meeting the colors in the order 1 , 2 , 3 . Finally we may obtain the canonical orientation by reversing 3n-(f) edges in d,. Then it follows from Lemma 3 that s(dl) = 1 iff p + n - ( f ) is even. O n the other hand, s(f) = 1 iff n + ( f )- n - ( f ) = 2p - 2 n - ( f ) is a multiple of 4. Hence s(dl) =s(f) and all edge-3-colorings of G compatible with F have sign s ( F ) . Since there are exactly 2k such edge-3-colorings, z ( F ) = s ( F ) equals ~ ~ the sum of their signs. It follows that CFGEz ( F ) = T ‘ ( G )as required. 0

Penrose number of cubic diagrams

93

4. Some consequences 4.1. Another recursive computation of the Penrose number Let H be a 4-regular graph. A transition system of H is a family p = ( p ( v ) ,v E V ( H ) ) ,where p ( v ) is a transition at v which is said to be in p. Let us define the sign (with respect to p ) of an Eulerian orientation d of H as equal to 1 if the number of anticoherent transitions of d not in p is even, and to -1 otherwise. Let S ( H , p ) be the sum of signs (defined with respect to p ) of the Eulerian orientations of H. This concept extends the Penrose number, since Proposition 4 asserts that for a cubic diagram G (with no loops or free loops) T ‘ ( G ) is equal to S ( L ( G ) ,p ) , where, for every vertex v of L ( G ) ,p ( v ) is the crossing transition at this vertex. We now present a further extension in terms of what is known in statistical mechanics as the partition function of an ice-type model (see [ 2 ] , Chapter 8). Let us denote by O(H) the set of Eulerian orientations of H. Consider a mapping W which assigns to every transition t of H a “weight” W ( t ) chosen in some set of numerical variables or constants. The pair (H, W ) will be called a weighted 4-regular graph. For any Eulerian orientation d and vertex v of H , we denote by d, the transition at v anticoherent in d . We associate to (H, W ) the sum S ( H , W) = rIvsv(H) W(d,,). For instance, if p is a transition system of H and W, assigns the weight 1 to the transitions in p and the weight -1 to the others, clearly S ( H , W,) = S ( H , p ) . We now present a recursive scheme for the computation of S ( H , W ) which is based on the “splitting” of transitions. Let v be a vertex of the 4-regular graph H, and let h l , h 2 ,h3,h4 be the four half-edges incident to v. Consider for instance the transition t = { { h l ,h 2 } , {h3,h 4 } }at v. Let us delete from H the vertex v , introduce two new vertices x , y and rearrange the incidences of the half-edges previously incident to v as follows: h , , h2 are made incident to x , and h3,h4 are made incident to y . In the resulting graph, x and y have degree 2 and the other vertices have degree 4. Hence by erasing x and y we obtain a 4-regular graph which we denote by H * t. We now present a result which also appears in a wider context in [8]. Our approach here is different and more direct than in [8].

Proposition 5. Let ( H , W ) be a weighted 4-regular graph and let x be a vertex of H. If t l , t2, t3 denote the three transitions at x , for i = 1, 2, 3 let A ( t , )= - W ( t , )+ (112) C,=1,2.3W(t,). Then: S ( H , W )= r=1,2.3

A(t,)S(H * t,, W l x ) ,

where W l x is the restriction of the weight function W to the transitions at vertices distinct from x .

Proof, For i = 1, 2, 3 let O’(H) denote the set of Eulerian orientations of H whose anticoherent transition at x is t,, and let S,(H, W )= CdEOl(H) IIvEv(H)-(x) W ( & ) . Clearly S W , W )= Cr=1.2.3 W ( W , ( H ,W ) . O n the

F. Jaeger

94

other hand, for i = 1, 2, 3 we may identify O(H * t i ) with the set of Eulerian orientations of H whose anticoherent transition at x is not ti. It easily follows that

Hence

Since

zjE(l,2,3)-(jJ A ( t i ) = W ( t j )for j = 1, 2, 3, the result follows immediately.

0

Proposition 5, together with the obvious fact that a graph consisting of n free loops has 2" Eulerian orientations, yields a recursive computation method for S(H, W ) . The following result is just a special case of Proposition 5.

Proposition 6. Let H be a 4-regular graph and let p be a transition system of H . Let x be a vertex of H . I f t l , t 2 , t3 denote the three transitions at x , with p ( x ) = t l , then : S(H,p ) = (i)(S(H *

fz,

P I X )+ S ( H * t3r P / X )

- 3S(H

* t i , pix)),

where p / x is the restriction of the transition system p to the transitions at vertices distinct from x.

Let us call a transition t of a 4-regular graph H separating whenever H * t has more connected components than H . It is easy to show that if p is a transition system of H and there exists a separating transition not in p , then S ( H , p ) = 0 . This generalizes via Proposition 4 the well-known fact that if a cubic diagram has a bridge, it has no edge-3-coloring and hence its Penrose number is zero. But the converse is false. Fig. 3 depicts a 4-regular plane graph H with no separating transitions (this graph also appears in Fig. 4 of [lo]). It is easy to check, either directly or by repetitive application of Proposition 6 (an interesting exercise), that if p denotes the transition system consisting of the crossing transitions, S ( H , p ) = 0.

Fig. 3.

Penrose number of cubic diagram

95

4.2. Special results in the planar case It is well known (see for instance [2], Section 8.13 and [4], Chapter 3) that the Eulerian orientations of a connected 4-regular plane graph H are in 1-to-1 correspondence with the colorings of its faces with 3 colors 1 , 2 , 3 such that adjacent faces receive different colors (these colorings will be called face-3colorings) and such that the infinite face is colored 1. Indeed, let us consider the colors as elements of Z,.A face-3-coloring c being given, for any oriented edge e denote by lc(e) (respectively: rc(e)) the color of the face lying on its left (respectively: right) side. If we orient each edge e in such a way that rc(e)- lc(e) = 1, it is easy to check that we obtain an Eulerian orientation which we denote by d ( c ) . Conversely, every Eulerian orientation corresponds in this way to a unique face-3-coloring such that the infinite face is colored 1. This can be shown for instance by considering the Eulerian orientation as a tension in the dual graph and then viewing this tension as a potential difference. Note that the face-3-colorings such that the infinite face is colored x ( x in Z,)are obtained from the face-3-colorings such that the infinite face is colored 3 by adding x to all colors. We observe that for a given vertex v, the face-3-coloring c uses 3 colors on the faces incident to v (we shall then say 'that c tricolors v ) if and only if the anticoherent transition at v of the Eulerian orientation d ( c ) is non-crossing. Let us denote by t ( c ) the number of vertices tricolored by c. Then s ( d ( c ) ) ,the sign of d ( c ) (defined with respect to the crossing transitions), equals (-l)'('). We call this number the sign of c and we denote it by s(c). Let C ( H ) be the set of face-3-colorings of H. We have proved the following result.

Proposition 7 . Let H be a connected plane 4-regular graph. The sum of signs of the Eulerian orientations of H is equal to (4) CceC(H) s(c). Now let G be a connected cubic plane graph with no loops or free loops. It follows from Propositions 1 and 4 that T ( G ) , the number of edge-3-colorings of G , is equal to the sum of signs of the Eulerian orientations of L ( G ) , the medial graph of G , which is a connected 4-regular plane graph. By Proposition 7, this is also equal to (4) CcsC(L(G)) s(c). The faces of L ( G ) belong to two different types: those bounded by the triangles t, (v E V ( G ) ) which can be identified with the vertices of G , and the others which can be identified with the faces of G. Thus we may identify the face-3-colorings of L( G ) with the mappings from the set of vertices and faces of G to {1,2,3} such that the value of every vertex is different from the values of the three incident faces. We shall call such a mapping a full 3-valuation of G . For each mapping f from the set of faces of C to {1,2,3} (we shall call such a mapping a face-3-valuation of G), let y ( f ) be the sum of signs of the full 3-valuations of G which extend this mapping. Clearly iff assigns different values to the three faces incident to a vertex v,f cannot be extended into a full

F. Jaeger

96

Fig. 4(i).

Fig. 4(ii).

3-valuation and hence y ( f ) = 0. On the other hand, iff assigns the same value to the three faces incident to v , the full 3-valuations extending f can be partitioned into pairs, two valuations of one pair being identical except for the value of v . Then it is easy to see that two valuations of one pair have opposite signs and hence y ( f ) = 0 also in this case. Finally, if, for every vertex v of G, f assigns exactly two distinct values to the three faces incident to v , we shall say that f is correct. Then we call sign o f f and denote by s(f) the sign of the unique full 3-valuation which extends f . Thus, denoting the set of correct face-3-valuations of G by F ( G ) , we have proved the following result.

Proposition 8. Let G be a connected cubic plane graph with no loops or free loops. The number of edge-3-colorings of C is equal to (4) Cf,F(c,s(f). Now let us make this result more precise. For f in F ( G ) and distinct elements i, j of { 1,2,3} we shall say that the vertex v has type iij in f whenever f assigns the value i to two of the faces incident to v and assigns the value j to the remaining one. On Fig. 4 (i) (respectively: '(ii)) we depict a triangle t,, in L ( G ) when v has type 112 (respectively: 221) in f . We have indicated the face-3-coloring of L ( C ) extending f and the associated Eulerian orientation. We observe that the oriented triangle t,, differs from the canonical orientation on two edges (respectively: one edge) in case (i) (respectively: (ii)). Moreover, if v has type 223 or 331 (respectively: 332 or 113) the situation is similar to the one in case (i) (respectively: (ii)). It then follows from Lemma 3 that the sign o f f is positive if and only if the number of vertices of type 221, 332 or 113 in f is even. Finally, using planar duality and the classical correspondence between edge-3colorings and face-4-colorings in cubic plane graphs, it is easy to reformulate Proposition 8 as follows (the relevant definitions have been dualized in the obvious way).

Proposition 9. Let K be a loopless plane triangulation. The number of vertex-4colorings of K is equal to ($)Cf,F(K*)s(f),where F ( K * ) is the set of correct vertex-3-valuations of K and the sign s( f ) of such a valuation f is equal to 1 if the number of triangles of types 221, 332 or 113 is even, and equal to -1 otherwise. References [l] K. Appel and W. Haken, Every planar graph is four colorable, Part I; W. Haken, K. Appel and J. Koch, Every planar graph is four colorable, Part 11, Illinois J . Math. 21 (1977) 429-567.

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[2] R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, 1982). (31 C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1974). 141 N.L. Biggs, Interaction models, London Math. SOC. Lecture Note 30 (Cambridge University Press, 1977). [S] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (MacMillan, London, 1976). [6] J. Edmonds, A combinatorial representation for polyhedral surfaces, Notices Am. Math. SOC.7 (1960) 646. [7] F. Jaeger, On edge-colorings of cubic graphs and a formula of Roger Penrose, in Graph Theory in Memory of G.A. Dirac, L.D. Andersen, editor, to appear. [8] F. Jaeger, On transition polynomials of 4-regular graphs, preprint (1987). [9] L.H. Kauffman, Map coloring and the vector cross product, preprint (1987). 101 M. Las Vergnas, Le polyn6me de Martin d’un graphe EulCrien, Annals of Discrete Mathematics 17 (North-Holland, Amsterdam, 1983) 397-411. [ l l ] 0. Ore, The Four-Color Problem (Academic Press, New York, 1967). [12] R. Penrose, Applications of negative dimensional tensors, in: Combinatorial Mathematics and its Applications, Proceedings of the Conference held in Oxford in 1969 (Academic Press, London, 1971) 221-244. [13] L. Vigneron, Remarques sur les rCseaux cubiques de classe 3 associks au probltme des quatre couleurs, C.R. Acad. Sc. Paris, t. 223 (1946) 770-772.

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Discrete Mathematics 74 (1989) 99-115 North-Holland

99

ON THE EDGE ACHROMATIC NUMBERS OF COMPLETE GRAPHS Robert E. JAMISON Mathematical Sciences Department, Clemson University, Clemson, South Carolina 29634-1907, U.S.A.

Suppose we wish to color the edges of the complete graph K , with as many colors as possible so that (1) no two edges with a common node get the same color, and (2) for any two colors c , and c2, there are two edges with a common node, one colored c , and the other colored c2. What is the maximum number A ( n ) of colors possible in such a coloring? Coloring problems are notoriously hard and this problem is no exception. In fact, a remarkable theorem of AndrC Bouchet implies that an exact determination of A(n) for all odd n would yield as a corollary all odd orders for which projective planes exist. Thus such a determination is clearly beyond the hopes of this study. The goals here are more modest: (1) to give a careful study of the best available upper bound on A(n), (2) to add to the constructions which give reasonable lower bounds for A @ ) , and (3) to contribute a few more values of n for which A(n) is known exactly.

1. The achromatic index Let G be a simple graph. A proper (vertex) k-coloring of G is a map of the vertices of G onto a set of k “colors” so that any two adjacent vertices of C receive different colors. Moreover, if for each pair of colors c1 and c2 there are adjacent vertices v1 and v 2 so that vi is colored c,, then the coloring is complete. The smallest number k for which a coloring of G exists is the chromatic number x ( G ) of C. Any coloring with x(G) colors is necessarily complete since completeness means that it is impossible to merge any two color classes and still have a proper coloring. The largest k so that there exists a complete k-coloring of (the vertices of) G is the achromptic number v ( G ) of G introduced by Harary and Hedetniemi [8]. An old result of Harary, Hedetniemi, and Prins [9] says that for any k between x(C) and v ( G ) , a complete k-coloring of G exists. Thus the extreme values x(G) and v ( G ) determine the range of possible complete colorings. The achromatic number and the computational complexity of its determination have been studied by various authors. In general it appears that the exact determination of the achromatic number, even for simple structures such as trees, is quite difficult (cf. Lopez-Bracho [lo] and Farber et al. [ 5 ] ) . However, an easy upper bound on +(G) may be obtained as follows. If G has a complete k-coloring, then since there is an edge between each pair of color classes, G must have at least !El 1k ( k - 1)/2 edges. Hence (k - 1)2 G k ( k - 1) < 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

R.E. Jambon

100

The purpose here is to investigate the achromatic number A ( n ) of the line graph L(K,,) of the complete graph K,,. Notationally, A ( n ) = V(L(K,,)). As is well-known, the chromatic index x(L(K,,))is n if n is odd and n - 1 if n is even. That the precise determination of the achromatic indices V ( L ( K , ) ) will be much more difficult is evident from a remarkable result of Bouchet [3] stated below. A complete edge-coloring of K,, with the maximum number A ( n ) of colors will be called an optimal coloring. If r is any color class in an edge-coloring, then the nodes covered by the edges in r will be called the support of r.

Theorem 1.2 (Bouchet). Suppose q is odd and n = q2 + q + 1. Then A ( n ) = qn if and only if a projective plane of order q exists. Indeed, if A ( n ) = qn, then the supports of the color classes in any optimal coloring form the lines of a projective plane with the nodes of K,, as points. Aside from the values given by Bouchet’s theorem, the exact value of A ( n ) is now known only for n 6 11 and n = 25. The best current estimates on A ( n ) for n s 100 are summarized in the last section. In general, since L(K,,) is regular of degree 2(n - 2), inequality (1.1) yields the following upper bound:

A ( n ) 6 v n ( n - l)(n

- 2)

+ 1 6 (n - l)* + 1.

(1.3)

Although this bound can be improved slightly, it is of the right order of magnitude. The proof uses the monotonicity A ( n + 1)Z=A(n) which is a consequence of the following simple lemma.

Lemma 1.4. For any graph G , v ( G )3 Ilt(H)*

if H is an induced subgraph of

G , then

Proof. It suffices to show this if H i s obtained from G by deleting a single node u. Given a complete coloring C of H, extend this to a complete coloring of G by either (1) coloring u with a new color if all the colors of C appear on neighbors of v in G, or (2) coloring v with a color of C not on any neighbor of v otherwise. 0

Theorem 1.5. A ( n ) / n * + 1 as n + w. Proof. The proof depends on a strengthened version, due to Tchebychev, of Bertrand’s “Postulate” which follows from the Prime Number Theorem (cf. Gioia [6]): For any E > 0, there is an N, such that for any real x N,, there is a prime p between x and (1 + E ) X . Now let E > 0 be given, and suppose n > ( N , + 1)*(1 +

Edge achromatic numbers

101

E ) ~ . Set x = [ f i / ( l + E ) ] - 1, so x 2 N,. We may then select a prime p with x s p s (1 E ) X . Note that p 2 p 1 < ( x 1)2( 1 E ) = ~ n. Since projective planes of all prime orders exist (cf. Hall [7]), it follows from Bouchet’s theorem and Lemma 1 . 4 that

+ +

+

+

+

~ ( n ) 3 ~ ( p ~ + p + i ) = p ( +p i~) > + pp 3 a x 3 = ( f i - i - E)3/(1+E)3. Since

E

was arbitrarily small, the result follows.

0

+

Since A(n) grows asymptotically like ng, one might expect to have A(n 1) A(n) =O(fi), but this remains unproved. The best known result on the difference A(n 1) - A(n) is the trivial inequality A ( n ) 3 A(n 1) - n obtained

+

+

by deleting any vertex and the n incident color classes from any optimal coloring of K,,,,. It is also characteristic of the quirks of this problem that no proof of the strict inequality A(n) < A(n 1) is known in general, although this is almost surely the case. The constructions given in Section 3 do confirm this strict inequality for an inflnite class of n, and the result below establishes a two-step strict monotonicity.

+

Theorem 1.6. A(n + 2) 3 A ( n ) + 2 if n > 4. Proof. Consider an optimal coloring of K,. Select a maximal collection r of disjoint edges of different colors, which implies that r m e e t s every color class. Let st be an edge of r. The n - 2 other edges at f all have distinct colors. Since r is a matching, it contains at most n/2 edges. Hence there is an edge tu whose color does not occur in r. Starting with tu, select a maximal collection A of disjoint edges colored with colors not used for r. The subgraph G generated by r U A is bipartite because the bipartition {r,A} is a proper 2-coloring of its edge set. So the vertices of G may be properly colored black and white. Now add two new nodes b and w. Let xy be an edge of G where x is black and y is white. If xy E r, color the edges bx and wy both with the color of xy. If xy E A, color the edges wx and wf both with the color of xy. Since there are at most two edges (one from r and one from A) at each node of G, this is a consistent coloring. Moreover, since all the colors in r U A are different, no color appears twice at either b or w. Now erase the old colors on the edges of r and make r a new color class. Notice for any edge xy of r, its old color class still has x and y as well as b and w in its support in the new coloring. Now erase the old colors on the edges of A and make A* = A U {bw} a new color class. This is a proper coloring by the remarks above. Since the supports of old colors are either left the same or enlarged by b and w , it follows that any two old color classes still meet. Moreover, r meets every old color and A* meets every old color not originally on an edge of r. But edge bw meets all these colors. Finally, r and A* meet on the special edges st and tu. Hence the coloring is complete. 0

R. E. Jambon

102

From the computational viewpoint, Yannakakis and Gavril [ 141 showed the following problem to be NP-complete: Given a graph G and an integer n, is v ( G )an? But for fixed n, Farber et al. [ 5 ] proved there is an algorithm which, for an abritrary graph G, determines in O(IE(G)()time whether v ( G )3 n. Their proof was nonconstructive and they were able to exhibit such an algorithm only for n s 4. Since by Bouchet’s theorem I J J ( L ( K ~ 2~ 3615 ~ ) ) is equivalent to the existence of a projective plane of order 15, one may expect that the constant involved is quite large.

2. Upper bounds

In this section a refinement of the bound in (1.3) is presented and studied. Except for a few values where an improvement by 1 is possible, the result is the best upper bound known for A(n). The following functions play a crucial role: g(x, y ) = 2y(x - y - 1) and h ( x , y ) = x ( x - 1)/(2y).

(2.1)

Note (say, by differentiation) that for fixed x , the quadratic g ( x , y ) in y is increasing for y < ( x - 1)/2.

Lemma 2.2. For any t < ( n - 1)/2, A ( n )s max{g(n, t) + 1, h(n, t + 1)). Proof. Consider any complete k-coloring of L(K,,). Suppose first that there is a color class r with exactly s S t edges of K,, in it. Let S be the set of 2s nodes of K,, covered by the s edges in An edge of K,, is adjacent to an edge of T i n L(K,,) iff it has an endnode in S. There are n - 1 edges of K,, incident with each point of S; there are s(2s - 1) edges of K,, incident with two points of S. Hence the number of edges of K,, not in r but incident with a point of S is 2s(n - 1) - s(2s

- 1) - s = g ( n , s).

Since r m u s t be adjacent to at least one edge of every other color class, it follows that

k
+ 1 s g ( n , t ) + 1.

Now if no color class contains t or fewer edges, then every color class contains at least t 1 edges. In this case, the number of color classes is at most h(n, t + l ) , and the lemma is proved. 0

+

We now wish to give an explicit description of the bound implied by Lemma 2.2. Toward this end, let

M n ) = max{g(n, t ) + 1, LWn, t + 1)J 1

Edge achromatic numbers

103

where 1x1 denotes the greatest integer in x . Now set B ( n ) = min{P,(n): 0 < t < ( n - 1)/2}

Then Lemma 2.2 may be reformulated succinctly as A(n) S B ( n ) . This represents the best known upper bound on A @ ) , except for a sparse set of values (discussed below) where the bound can be improved to B ( n ) - 1. It is therefore convenient to have a direct rather than min-max description of B ( n ) . With t fixed, g ( n , t ) grows linearly in n and h(n, t + 1) grows quadratically. Initially, g is in the lead but at some point, h takes over. Eventually, h(n, t 1) outgrows even g(n, t + 1) and the value of t minimizing PI switches from t to t 1, and the scenario repeats. The exact crossover values are described in the following technical lemma.

+ +

Lemma 2.3 Suppose t 2 2. If 4t2 - t 6 n 6 4t2 + 3t - 1, then B ( n ) = g(n, t ) + 1. If 4t2 + 3t s n s 4(t + 1)’ - t - 2, then B ( n ) = Lh(n, t + I)]. Proof. We need to compare g with [h]. Notice that since g is integral, + 1 s [h(n,t ) ] iff g(n, t) + 1 s h(n,t). By subtraction, this is equivalent to 0 s h(n, t ) - g(n, t) - 1. Multiplied through by 2t, the right side becomes a polynomial in n and t:

g(n, t)

p(n, t ) = n2 - (4t2

+ 1)n + 4t3+ 4t2 - 2t.

+

Thus g ( n , t ) 1s Lh(n, t ) ] iff p ( n , t) 2 0. Similarly, g ( n , t) q(n, t) 2 0 where q(n, t ) = n2 - (4P+ 4t

+ 1)n + 4t(t + 1)2-

2(t

+ 1s Lh(n, t + 1)) iff

+ 1).

Tedious but routine evaluations reveal:

p(4t2 - t - I, t ) = -3t2 p(4t2-t, t ) =

+ t + 1< o

t2- t 2 0

4(4t2 + 3t - 1, t ) = -3t2 - 3t < 0 q(4t2+3t,t)=

t2- t - 2 2 0

if t > o if t > O

if t > 0 if

t22

Now let t 2 2 be fixed. First let us investigate the range between 4t2 - t and 4t2 + 3t - 1. Differentiating, we find D,q(x, t) = 2x - (4t2 4t 1). For x 2 4t2 - I , this is positive, so q(x, t) is increasing. Since q(4t2 + 3t - 1, t) < 0 by (2.4) it follows that q(n, t) < O for all n in the range 4t2 - t S n s 4 t 2 3t - 1. Thus for such an n, = g ( n , t ) + 1. h(n, t 1) < g ( n , t ) + 1, so Now if t < u < ( n - 1)/2, then g(n, u ) + 1 a g ( n , t ) + 1 > h(n, t 1) 2 h(n, u + 1) since g ( x , y) is increasing in y (for y < ( x - 1)/2) and h(x, y ) is decreasing in y. It follows that P U ( n )2 P,(n). Now consider s < t. Differentiating, we find D,p(x, t) = 2x - (4t2 + 1). For x 2 4t2 - t, this is positive, so p(x, t) is increasing. Since p(4t2 - t, t) 2 0 by (2.4),

+

+ + +

+

R.E. Jamison

104

it follows that p ( n , t) 3 0 for all n 3 4t2 - t. Thus for such an n, Lh(n, t ) ] 3 g(n, t) + 1. Hence for s < t, we have &(n) 2 Lh(n, s + 111 3 Lh(n, t ) ] 3 g(n, t ) + 1= Pr(n). It follows that B ( n ) = &(n) = g(n, t) + 1 as desired.

+

+

Now let us consider the range from 4t’ 3t to 4(t 1)’ - f - 2. As noted above, q ( x , t ) is increasing for x 3 4t’ - t. Since q(4t’ 3t, t) > 0 by (2.4), it follows that q(n, t) > 0 and hence Lh(n, t l)] 3 g ( n , t) 1 for n 3 4t’ 3t. Thus for such n , we have &(n) = [h(n, t l)]. Now if s < t, then Ps(n)3 Lh(n, s l)] 3 Lh(n, t l)] = P l ( n ) . To finish the proof, it suffices to show P U ( n )3 P l ( n ) for t 1S u < (n - 1)/2. To this end, consider the derivative D,p(x, t 1) = 211 - 4(t 1)’ - 1 which is positive if x 3 4t’ 3t. Since p(4(t 1)’- (t 1) - 1, t 1) < 0 by (2.4), it follows that p ( n , t 1) < 0 for all n in the range 4t’ 3t S n s 4(t 1)’ - t - 2. Hence for n in this range, we have P l ( n ) = Lh(n, t l)]s g ( n , t 1) 1S g ( n , u ) 1 s Pl,(n). Thus B ( n ) = P r ( n )= Lh(n, t l)] as desired. 0

+

+ +

+

+

+

+

+ +

+

+

+

+

+

+

+

+

+

+ + +

+

Theorem 2.5 Zf t > 1 and n = 4t’ - t, then A ( n ) G B ( n ) - 1. Proof. Let n = 4t2 - t, and suppose there is a complete coloring of L ( K , ) with B ( n ) colors. By Lemma 2.3, B ( n ) = g ( n , t) + 1 in this case. If there were a color class with s < t colors, it would meet only g(n, s) (t’ + t ) / 2 , at least one of these edges, say ab, must come from a class A with exactly t edges. Let au and bv be the edges in I‘incident with ab. But then au and bv are distinct edges having the same color and incident with the t-edge class A. This contradicts the fact that all edges incident with A must have different colors. 0 Bouchet’s Theorem in conjunction with the Bruck-Ryser Theorem (cf. Hall [7], p. 175) also yields (a much deeper!) improvement of the bound B ( n ) for certain n. Indeed, if q is odd and n = q’ q + 1, then setting t = (q + 1)/2 it is easy to see that B ( n ) = h(n, t) = qn. Thus we have

+

Theorem 2.6 (Bouchet, Bruck-Ryser). Suppose q = 1 mod 4 and n = q’ Zfq is not a sum ofsquares, then A ( n ) S B ( n ) - 1.

+ q + 1.

Edge achromatic numbers

105

There remains one other case in which it is known that A ( n ) s B ( n ) - 1. One can easily verify that B(6) =9. However, Bories [2] and later independently Turner [12] both realized that A ( 6 ) =8. As this result was not previously published, it is given here for completeness.

Theorem 2.7 (Bories, Turner). A ( 6 ) S 8. Proof. Suppose on the contrary that L(K,) has a complete 9-coloring. For convenience, call an edge which forms a color class by itself a singleton edge. If there were at most two singleton edges, then in all there would be at least 2 ( 7 ) + 2 = 16 edges, a contradiction. Hence there are at least three singleton edges. Since an edge meets exactly 8 other edges, it follows that the 8 edges incident with a singleton edge must all receive different colors. Moreover, the singleton edges must meet each other. Hence they form either a star or a triangle. Suppose first that the singleton edges are ab, ac, ad and that they are colored 1, 2 , 3. Since bc, cd, db form a triangle, they must receive distinct colors. Since colors 1, 2 , 3 are used just once, say that bc, cd, db are colored 4, 5, 6, respectively. Let e and f be the remaining vertices of K,. Now ae cannot be colored 4 or 6 since ab is already adjacent to these colors. Similarly, ae cannot be colored 5 since ac is already adjacent to color 5. Hence ae (and similarly af) must be colored 7 or 8. Say, ae receives 7 and af receives 8. (See Fig. la). Note that ab is now adjacent with all colors except 5 and 9. Thus the colors assigned be and bf must be 5 and 9, or vice versa. Similarly, ce and cf must receive 6 and 9, and de and df receive 4 and 9. But this forces two edges colored 9 to be incident with one of the vertices e or f, a contradiction. Now suppose the singleton edges are ab, bc, ca and that they are again colored 1, 2, 3. Since all edges incident with ab receive different colors, we may assume the coloring is as shown in Fig. lb. The as yet uncolored edges adjacent to ac are

C

a. Star Case

b. Triangle Case

Fig. 1 . Impossibility of 9-coloring K,. a. Star case. b. Triangle case.

R . E. Jamison

106

cd, ce, cf and these must be colored 5 , 7, 9 (in some order) so that ac is adjacent to all colors. Similarly, the same edges cd, ce, cf are the uncolored edges adjacent to bc, and these must be colored 4, 6, 8, a contradiction. 0

3. Constructions from projective planes

+

Suppose q is the order of a projective plane P. Then P has n = q2 q + 1 points and n lines. We may regard the points of P as the nodes of a complete graph K,,. Each edge lies in a unique line which is a copy of K,,,. Now if q is odd, then K,,, has a l-factorization. Each line may thus be divided into q color classes, each of size ( q 1)/2. All colors of a line are incident with all points of that line. Hence since any two lines meet, the coloring is complete. This yields the easy part of Bouchet’s Theorem: A ( n ) = qn if q is odd and the order of a projective plane. If q is even, then K,,, is not l-factorable, so the argument collapses. However, by adjoining some additional points, it is possible to obtain some good (if not exact) lower bounds.

+

Theorem 3.1. Suppose q is even and the order of a projective plane. Let n = q2 + 2q + 2. Then A ( n ) 3 nq + 1 and A ( n + 1)2 nq q + 2.

+

Proof. Let P be a projective plane of order q regarded as a complete graph. Each line L of P has an odd number q + 1 of points. Hence L may be edge-colored with q + 1 colors. In any such coloring, at each point v of L, there will be exactly one color missing. Moreover, since each color class in L must contain q / 2 edges, different points of L will be missing different colors in L. Using a different set of q + 1 colors for each line L of P, we obtain an edge-coloring of P with ( q + l ) ( q 2 + q 1) = q n + 1 colors. It is a proper coloring, but because of the “missing” colors, it fails to be complete. For each incident point-line pair (v, L) in P, let c(v, L) denote the missing color at v in the line L. These colors are all different and thus comprise the full set of qn + 1 colors. Now arbitrarily order the lines through each point u as L,,(v), L , ( v ) , . . . , L,(v). Let U = { u ~u,l , . . . , u,} be a set of q + 1 new points. Then K,, may be represented with the n points of P U U as node set. Color each edge vui with the missing color c(v, L , ( v ) ) . Since the colors c(v, L ) are all distinct, this remains a proper coloring. Now for each v E L, we have all colors of L occurring on edges at v. Since any two lines of P have a point in common, it follows that any two color classes are incident, so the coloring is complete. The edges between the ui’s have not yet been colored. However, we do have a complete (qn 1)-coloring of a subgraph of L(K,,). Hence by Lemma 1.4, we have A ( n ) 2 qn + 1. Now continue the above construction by adding another new point u * . Set

+

+

Edge achromatic numbers

107

+ + + + + +

U* = U U { u * } . Then U* has an even number q 2 of points and hence is 1-factorable. That is, U* can be edge-colored with q 1 colors, each color occurring at each point of U*. Since all previous qn 1 colors are incident at the points of U , it follows that we may choose these q 1 colors to be new and still have a complete coloring. Thus A ( n 1) 3 nq q 2. 0

+

With n as above, taking t = q / 2 in Lemma 2.3, we see that B(n - 1)= nq - q, B(n) = nq q and B(n + 1) = nq + 3q + [6/(q + 2)1. Thus by the above constructions, we have A ( n - 1) s B(n - 1)< A ( n ) s B ( n ) < A(n + l), so the strict monotonicity of A is established for infinitely many values of n.

+

Theorem 3.2. Let q be a power of 2, and let s be an integer such that $4 + 1 ss s q. Then A(q2 q s) L q2(q + 1) + min{(2s - q - l ) ( q + l ) , A(s)}.

+ +

Proof. As in the above construction, let P be a projective plane of order q and edge-color each line of P with q + 1 colors. Now fix a point p of P and denote the lines of P through p by M,, MI, . . . , M4. (We will call these M-lines and the lines not through p will be called L-lines.) Arbitrarily order the points #p on Mk as w(k, j), j = 1, . . . , q. Now delete p and remove all colors from the M-lines. This leaves q2(q 1) “old” colors. For each point v # p , order the lines through v as L,(v), L,(v), . . . , LJv) where L,(v) is the M-line through v and p. Let U = {ul, . . . , u s }be a set of new points. For each v # p and i with 1s i G s , color edge vu, with the “missing” color c(v, Li(v))at v on line Li(v). There then remain q - s “missing” colors at each point v. Say, v = w(k, j). For each i with s < i G q , color the edge from w ( k , j ) to w(k, j i - 1) with the color c(v, Li(v)). (The sum j + i - 1 is modulo q . ) Now every color on an L-line occurs at each point of that L-line. Since any two L-lines intersect at a point #p, the coloring so far is complete. Morever, the edges in U and certain edges on the M-lines are still available to be colored. Consider some k f k and some i with f q + 1G i s s . The edges w ( k , j ) to w(k, j i - 1) on Mk are as yet uncolored. If i = f q 1, these edges from a 1-factor. Otherwise, these edges form a collection of cycles of length A where A is the least integer such that q divides A(i - 1). Since q is a power of 2, A must be even so this collection of edges may be split into two 1-factors. Thus on each Mk, we have available 2(s - f q - 1) 1 matchings, each of which covers all points of Mk. Since each L-line meets each M-line at a point # p , it follows that if one of these matchings is taken as a new color class, then it will meet all old colors. Therefore the difficulty in adding new colors lies only in making new color classes from different M-lines meet each other. This can be accomplished by repeating the new colors in a complete edge-coloring of U . Cl

+

+

+

+

+

Suppose we fix a constant c 3 1 and take s = $4

+ y where

1s y s c. As q

R. E. Jamison

108

Table 1. A residual edge-coloring of K,.

grows, A(s) grows like a constant times qj by ( l S ) , so A(s) eventually dominates (2r - q - l)(q 1) = (27 - l)(q + 1). Hence the above result yields A(q2 $q + y ) 2 q3 q2 + (2y - l)(q + 1) for large q. This compares favorably with the upper bound B-in fact, well enough to imply strict monotonicity in the range under consideration. It is possible to improve the bound in Theorem 3.2 by taking advantage of the fact that new colors on the same M-line already meet each other. This is illustrated by the following special constructions.

+

+

+

Theorem 3.3. A(24) 3 89 and A(78) 2 591.

Proof. First, take s = q = 4 and proceed with the construction in (3.2) up to the place where the new colors are to be chosen. The edges to receive new colors form six disjoint K4's-the five M-lines and U.Each K4 factors into three pairs of edges. Table 1 shows a method of coloring these edges with 9 new colors A, A', B, B ' , C , C', D, D', and E. A new color standing alone indicates that both edges of a 1-factor are to receive that color. A pair of colors in parentheses indicates that the listed colors are to be distributed over the edges of a 1-factor. Since each new color stands alone on some M-line (and hence has that M-line in its support), each new color meets every old color. It is easily verified that the new colors are pairwise incident. In the same spirit, Table 2 shows a scheme for the new colors in the case q = 8, s = 6. Graphically, the edges to be colored on each M-line form an 8-cycle with Table 2. A residual edge-coloring of K,,

M,: M,: M,: M3: M4: M,:

M6: M,: Me:

.

A B C D E F G H I

A' ( B , B ' , C , C') B' (C, C ' , C, D ' ) C' ( D , D',E , E') D' ( E , E ' , F, F ' ) E' (F, F ' , A , A ' ) F' ( A , A ' , B ,B ' ) ( A , D, A', D ' , A , C, F, C ' ) ( B , E , B ' , E ' , B , C , F', C ' ) (C, C', D , D ' , E , E ' , F, F ' )

'1

' 2

'3

'4

' 5

'6

H

D' t

U

109

Edge achromatic numbers

i$J 2

K 4

K

1

~~~

0

1

2

3

4

0

Fig. 2. A residual edge-coloring of Ku,.

its four principal diagonals. For Mo through M,, imagine the edges split into three 1-factors, each of the first two receiving a single color and the third receiving four different colors as indicated. On M6, M,, and M8,the diagonals receive colors G , H, and I , respectively, and the 8-cycles are colored as indicated. Finally, a special color table is given for the edges of U.The verification that the resulting coloring is proper and complete is easy and left to the reader. 0 To conclude this section, we give two special constructions based on removing a point from an odd order projective plane

Theorem 3.4 A(12) 3 31 and A(30) 3 136.

Proof. The projective plane PG(2,5) over GF(5) may be regarded as consisting of (1) the 25 points of the affine plane AG(2,5)-represented in Fig. 2 by the 5 x 5 square grid, (2) the 5 points on the line at infinity corresponding to “slopes” of nonvertical lines-represented in Fig. 2 by the vertices of K5,the vertex labels being slopes, and (3) one “vertical” point on the line at infinity. Take an optimal coloring of K31 based on the plane PG(2,5). This has qn = 155 colors. Remove the vertical point, the 30 edges and the 30 colors incident with it. This leaves 30 points with 125 “old” colors intact and 60 edges which must be recolored. Geometrically, the uncolored edges lie on the 5 vertical lines of AG(2,5) and the “punctured” line at inflnity. Graph theoretically they from 6 disjoint copies of K,. Fig. 2 provides a scheme for recoloring these edges with 11 “new” colors A, B, . . . , J, K. The colors of the 10 edges on the infinite line are as indicated. The colors on the other vertical lines are assigned as follows. Color “X” in grid position (i, j)

R.E. Jamison

110

0

1

2

Fig. 3. A residual edge-coloring of K , 2 ,

+

means: on the vertical line x = i, color edges (i, j - l ) , (i, j 1) and (i, j - 2), (i, j + 2 ) with color X. The ambiguity in the (0,4) position is to be resolved by coloring edge (0,3), (0,O) with J and edge (0,2), ( 0 , l ) with K. The result is an edge coloring of KW with 136 colors, 125 old and 11 new. Let us check that it is proper. First note that the coloring on the infinite line is a complete (in fact, optimal) edge coloring of K s with 7 colors. That the coloring is proper on the remaining 5 vertical lines follows from the fact that no color appears more than once in any column of the grid representing AG(2,5). The check that this coloring is complete is more tedious and involves verifying that the support of each new color meets every nonvertical line. The details, which are routine, are left to the reader. The data in Fig. 3 may be used in a similar way to define a complete 31 coloring of the edges of KI2. 0 The same procedure may be applied to the coloring induced by any odd order plane. But as the order grows, the recoloring becomes every more tedious. Moreover, the requirement that the new colors “block” all the nonvertical lines, together with results of Aiden Bruen [ l ] on blocking sets, suggests that the new color classes must be too large to yield an efficient coloring in general.

4. Group divisible colorings

In this section we shall further exploit a construction technique introduced by Bouchet [3]. Let G be a group of order n. A Bouchet diagram (over G ) is a (simple) graph D such that

(1) D has a 1-factorization, (2) the nodes of D are elements of G, (3) for any edges [ x , y ] and [u,u] of D , x - l y = u-lu if and only if x = u and y=u, (4) for any g in G, there are nodes x and y of D with xy-I = g. If D has rn nodes and is regular of degree d , then we shall call it an

111

Edge achromatic numbers

0

l

o 5

n

=

9, d = 2

G=Z,

Oo----

415

lo----

4

S*---

-09

2

2

n = 25, d = 4 G=%5

graph is complement of edges shown

14

18

n=19,d=3

n = 23, d = 3

G

G

1

Z,,

= 223

Fig. 4. Bouchet diagrams.

(n,m , d)-Bouchet diagram. Let D* = {x-'y: x and y are adjacent in D}.Note that D * = ( D * ) - l since adjacency is symmetric, that D * does not contain the identity since D is loopless, and D * does not contain any involution by (3). It is often convenient to think of D as a labelled graph: by (2) the nodes of D are labelled by elements of G , and each edge [ x , y ] is labelled by a pair of inverse elements { x - ' y , y - ' x } . Fig. 4 shows four Bouchet diagrams. For n = 9, 19 these are due to Bouchet [3]; for n = 23, 25 they are new. The Cayley graph Cay(G, D * ) has the elements of G as nodes and an edge [g, h] whenever g-'h is in D*.The diagram D induces a complete edge coloring of Cay(G, D*)as follows. For each g E G, let g D denote the translate of D by g-that is, the nodes of gD are of the form gx where x E D and two nodes gx and gy are adjacent in g D iff x and y are adjacent in D.The translates then cover the edges of Cay(G, D*).Condition (3) guarantees that the translates gD are edge disjoint and condition (4) guarantees that any two translates of D have at least one node in common. Select a 1-factorization of each gD and regard this as an edge-coloring with d colors of gD, using disjoint sets of colors for different

R. E. Jamison

112

translates. Since all colors used in any g D occur at all nodes of g D and since any two translates have a node in common, it follows that this coloring is complete. Since dn colors are used and since Cay(G, D*)is an edge-subgraph of the complete graph on G, we have the following result.

Theorem 4.1 (Bouchet). If an (n, m , d)-Bouchet diagram exists, than A ( n ) 3 dn. For n = 9 and n = 25, the colorings generated by the diagrams in Fig. 2 are optimal since B(9) = 18 and B(25) = 100. It is possible that these may be the first in a series for n an odd square but no general constructions are known yet. For n = 19, the coloring is maximal among all colorings with class sizes 3 or more. But it may be possible to obtain more colors by allowing some color classes of two edges. For n = 23, a better coloring was obtained using projective planes (Theorem 3.2 with q = 4 and s = 3). The following result provides a method for augmenting a group divisible coloring.

Theorem 4.2. If an (n, m, d)-Bouchet diagram exists, then for all k, A ( n + k m ) (d + k)n.

Proof. As before, take the induced coloring on Cay(G, D*).Introduce km new points u&) where i = 1, . . . , k and x E D, and introduce kn new colors c,(g) where i = 1, . . . , k and g E G. Now color the edge [g, ui(x)] with color c,(gx-'). For fixed i and x , as g ranges through G, the products gx-', and hence the colors c,(gx-'), are all distinct. Similarly, when g is fixed and x ranges through D, the colors c,(gx-') are all distinct. Thus the coloring is proper. Now any new color c,(g) is assigned to the edge from gx to ui(x)for all x in D. Thus c,(g) occurs at all nodes of gD. Now any old color occurs at all nodes of some translate hD, which meets g D in at least one node. Thus every new color meets every old color. Now suppose c,(g) and cj(h) are two new colors. By condition (4), select x , y E D so that xy-' = g-'h. Then gx = hy. Letting z = gx, we then have zx-' = g and zy-' = h. Thus the edges [z, ui(x)]and [z, u j ( y ) ]are colored c,(g) and cj(h), respectively. Hence all new colors also meet one another, so the coloring is complete. 0 A well-known theorem of Singer [ l l ] says that if q is a prime power and n = q2+ q + 1, then there is a subset D of Z, (the cyclic group of order n ) such that every nonzero element of 2, is expressible uniquely as x - y for x , y in D . This difference set arises from the projective plane PG(2, q) over the Galois field GF(q). Of course, D has q 1 elements, so if q is odd, the complete graph on D is 1-factorable and D is an (n, q + 1, q)-Bouchet diagram. Thus from Theorem 4.2, we obtain the following corollary.

+

Edge achromatic numbers

113

Corollary 4.3. Let q be a power of an odd prime and set n = q2 + q + 1. Then for 0, A ( n + k(q 1)) 3 qn + kn.

+

all k

5. Summary An edge coloring of K,, may be represented by a color table: an n x n matrix whose i, j entry is the color on edge [i, j ] . Of course, such a table is symmetric. The coloring is proper iff each row (and hence each column) contains distinct entries. The coloring is complete iff each pair of colors occurs together in at least one row (column). Color tables for optimal colorings of K 6 , K , , K 8 , Klo, and K I 1 are given in Fig. 5 . (An optimal coloring of K s appears in Fig. 2; the (9,4,2)-Bouchet diagram from Fig. 4 yields an optimal coloring of K 9 . ) The values of A ( n ) for n S 7 were known to Bories [2]. Using differencing methods, Bouchet [3] found A ( 8 ) and A(9). Optimal colorings of Klo and K I 1 were found by Ray Rowley and Craig Turner, respectively, using ad hoc methods described in Turner et al. [13]. Table 3 shows the currently best upper and lower bounds on A ( n ) for n 6 100. The upper bound is B ( n ) except as noted: (a) Theorem 2.5 * 1 2 3 4 5

1 * 5 6 7 3

2 5 ' 7 6 1

3 6 7 * 2 4

4 5 7 3 6 1 2 4 ' 8 8 ;

n=6, k = 8 Bones

1 1 2 3 4 5 6 7 8 9 10

* 7 11 12 13 14 2 15 16 17

2 3 7 11 * 18 18 * 19 24 20 16 21 23 1 8 22 6 23 13 24 19

* 1 2 3 4 5 6

I 2 ' 7 7 ' 5 8 5 6 1 0 9 4 2 8

' 1 2 3 4 5 6 7 1 * 5 8 9 1 0 1 1 4 2 5 * 6 1 2 1 3 8 1 0 3 8 6 ' 7 9 1 4 1 2 4 9 1 2 7 * 1 1 3 1 1 51013 9 1 * 2 1 4 611 8 1 4 1 3 2 * 3 7 4 10 12 11 14 3 *

3 4 5 6 2 8 6 9 1 0 4 8 * 9 1 10 9 * 7 3 1 7 * I1 10 3 1 1 *

n=7, k = l l Bories

4 12 19 24

5 13 20 16 * 9 9 * 15 22 14 4 17 21 5 11 18 23

6 7 8 9 14 2 15 21 1 22 23 8 6 15 14 17 22 4 21 * 12 20 12 * 25 20 25 * 10 26 27 3 27 26

1 0 16 17 23 24 13 19 5 18 I1 23 10 3 26 27 27 26 * 25 25 *

n=8, k=14 Bouchet

$

1

1

'

2 3 4 5 6 7 8 9

5 10 11 12 13 14 15 16

2 3 5 10 * 16 16 * 8 6 17 19 18 4 19 17 20 13 21 22

4 5 6 7 11 12 13 14 8 17 18 19 6 19 4 17 * 22 3 20 22 * 15 1 3 15 * 21 20 1 21 * 14 9 10 5 12 I 1 7 18

n = 10, k = 22 Rowley n = 11, k = 27 Turner Fig. 5. Color tables of optimal k-colorings of L ( K , ) .

8 15 20 13 14 9

9 16 21 22 12 11 10 7 5 18 * 2 2 *

R.E. Jamison

114

Table 3. Best current bounds on A ( n ) for 1 n 100. Lower bounds arising from A ( n + 1) 3 A ( n ) are omitted. n

Upper Lower n

Upper Lower n

Upper Lower n

Upper Lower n

Upper Lower

1 -

2 1

1

n

n

Upper Lower n

n

Upper Lower

18 61

19 65 57 d

20 69

27 117 110 P

28 126

29 135 112

30 145 136

m

S

36 I93

37 199 186 e

38 205

39 21 1 188 m

40 217

45 247 22 1 m

46 258

47 270 223

48 282

49 294 250 e

50 306

55 37 1 288

56 385 343 t

57 399 399 P

58 413

59 427 40 1

60 440

m

a

68 505

69 513 460

70 521

79 616

P

a

m

22 77

23 84 83

24 92 89

25 100 100 d

31 155 155 P

32 165

33 174 157

41 223 190

42 229

C

S

21 73 65 e

P

51 318 252 61 449 403 71 529 462 81 648

91 819 819 P

14

S

34 181

52 331

35 187 159

6 8 8 b, c

7 11 11

8 14 14

C

C

16 53

17 57 52 e

26 108 105 P

m

a, m

43 235 219 e

44 24 1

53 344 254

54 357

m

t

m

62 457

63 465 405 m

64 473

65 481 456 e

66 489

67 497 458

72 539

73 545 513 e

74 553

75 561 515 m

76 570

77 585 583 P

78 600 591

82 664 657 P

83 680 666 9

84 697

85 714 668

86 731

87 748 670

88 765

92 837

93 855 821

94 874

m Upper Lower

10 22 22

15 49 41

m

n

9 18 18 d

5 7 7

44

12 33 31

m

Upper Lower

4 3 3

13 39 39

11 27 27

m

Upper Lower

3 3 3

m

m

m

95 890 823 a, m

m

97 91 1 825 m

98 92 1

80 632 593 m

S

m

96 901

C

89 783 672

90 801 728

m

I

99 93 I 827 m

100

94 1

Edge achromatic numbers

115

(b) Bories-Turner Theorem 2.7 The lower bounds come from these sources: (c) Color tables in Fig. 5 (d) Bouchet diagrams in Fig. 4 (e) Extension of Bouchet diagrams in Corollary 4.3 (m) Monotonicity Theorem 1.6 (A(n 2) 2 A(n) + 2) (p) Projective planes (Theorems 1.2, 3.1, and 3.2) (s) Special constructions from Section 3 (t) The trivial bound A(n) 2 A(n 1) - n Lower bounds which arise simply from the monotonicity of A ( n ) are omitted from the table.

+

+

Note added in proof Improvements on some of the lower bounds in Table 3 using a modification of Bouchet diagrams have come to light since submission of this paper. In particular, A(12) 3 32 and improvements for 46 S n d 49 are known. Details will be reported elsewhere.

References [l] Aiden A. Bruen, Blocking sets in finite projective planes, SIAM J. Appl. Math 21 (1971) 380-392. (21 F. Bories, Sur quelques problCmes de colorations complCtes de sommets et d’argtes de graphes et d’hypergraphes, Thtse de 3tme cycle (Pans, 1975). [3] A. Bouchet, Indice achromatique des graphes multiparti complets et reguliers, Cahiers Centre d’Etudes et Recherche Operationnelle 20 (3-4) (1978) 331-340. [4] R.H. Bruck and H.J. Ryser, The nonexistence of certain finite projective planes, Canadian J. Math. 1 (1949) 88-93. [5] M. Farber, G. Hahn, P. Hell and D. Miller, Concerning the achromatic number of graphs, J. Combinatorial Theory, Ser. B 40 (1986) 21-39. (61 Anthony Gioia, The Theory of Numbers (Markham, Chicago, 1970) 156. (71 Marshall Hall, Combinatorial Theory (Blaisdell, Toronto, 1967). [8] F. Harary and S.T. Hedetniemi, The achromatic number of a graph, J. Combinatorial Theory 8 (1970) 154-161. [9] F. Harary, S.T. Hedetniemi and G. Prins, An interpolation theorem for graphical homomorphisms, Port. Math. 26 (1967) 453-462. (101 R. Lopez-Bracho, Le nombre achromatique d’une Ctoile, Ars combinatoria 18 (1984) 187-194. 1111 James Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. SOC.43 (1938) 377-385. (121 C. Turner, On the edge achromatic number of small complete graphs, Master’s Thesis, Clemson University (1986). (131 C.A. Turner, Ray Rowley, R.E. Jamison and R. Laskar, The edge acrhomatic number of small complete graphs, Congressus Numerantium (1988). [14] M.Yannakakis and F. Gavril, Edge dominating sets in graphs, SIAM J. Appl. Math. 38 No. 3 (1980) 364-372.

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Discrete Mathematics 74 (1989) 117-124 North-Holland

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APPLICATIONS OF EDGE COLORING OF MULTIGRAPHS TO VERTEX COLORING OF GRAPHS H.A. KIERSTEAD* Deparimeni of Mathemaiics, University of South Carolina, Columbia, SC 29208, U.S.A. It is shown that if G is a graph which induces neither K , . 3 nor K I S + , - e and w ( G ) is sufficiently large then z(G) G w ( G ) + s. This result is established by first demonstrating a correspondence between vertex coloring G and edge coloring a certain multigraph and then applying a known result on edge coloring.

1. Introduction Perhaps the two most important results on edge coloring are Vizing’s Theorem [16], which states that the chromatic index x ‘ ( M ) of a multigraph M with maximum degree A ( M ) and maximum multiplicity p ( M ) satisfies A ( M ) s f ( M ) s A ( M ) p(M), and Holyer’s Theorem [8], which states that the problem of determining the chromatic index of even a simple graph is NP-complete. In some sense these two results solve the edge coloring problem for simple graphs. However, the upper bound is quite loose for multigraphs. Unfortunately much of the work on edge coloring has been restricted to simple graphs. One goal of this article is to stimulate research on edge coloring multigraphs by demonstrating how refined bounds on the chromatic index of multigraphs can be used to derive interesting results on the chromatic number of graphs. We will also take the opportunity to pose specific questions about chromatic index. Let S be a set of finite subgraphs and let C ( S ) be the class of graphs which do not induce any subgraph in S. Many vertex coloring theorems give an upper bound on the chromatic number x ( G ) of any graph G in C ( S ) in terms of the clique size w ( G ) of G. GyBrfiis [7] and Sumner [14] independently conjectured that if S = { T}, where T is a tree, then such a bound exists. For stars this follows easily from Ramsey Theory and Brooks’ Theorem [3]. Gyrirfris [7] has shown that the conjecture is true for paths. If S does not have a tree for an element such a statement cannot be true since Erdos [5] has shown that there are graphs with arbitrary high girth and chromatic number and maximum clique size 2. We shall consider classes of graphs which induce neither K1,3nor K2S+3- e, for various choices of s. Vizing’s Theorem can be stated in the above form. Beineke [2] characterized line graphs as those graphs which do not induce any graph in a set S of nine forbidden subgraphs, including K 1 , 3and K s - e. By Vizing’s Theorem x ( G )s

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w ( C ) + 1 for any graph G in C(S). Call a graph linear if it induces neither K1.3 nor K , - e . Kierstead and Schmerl [ l l ] together with Kierstead [lo], improved earlier results of Choudum [4] and Javdekar [9], by proving:

Theorem 1. If G is a linear graph, then x(C) 6 w ( G ) + 1. This theorem was proved in two steps. First the theorem was shown in [ll]to be equivalent to the statement that if M is a multigraph with p ( M ) G 2 such that M does not contain a 4-sided triangle ( K 3 + e ) then x ' ( M ) s A ( M ) + 1. This statement was then proved in [lo]. Kierstead and Schmerl [12] used a similar technique to prove the following theorem, which strengthens Brooks' Theorem for linear graphs.

Theorem 2. Zf G is linear and A(G) 6 2 o ( G )- 5 , then x ( G ) = w ( G ) . In this article we apply the same technique to analyze the effect of relaxing the condition of linearity. We prove the following theorem, where R ( x , y) denotes the least number n such that any graph on n vertices contains K," or K,,.

Theorem 3. Let s be a natural number and let G = ( V , E ) be a graph that induces neither K1.3 nor K2+3 - e. Then x(G) G max(w(G) +s, R(3,4s - 1)). Our notation is quite standard. All graphs and multigraphs are finite. Let = (V, E) be a multigraph. The number of vertices of M is denoted by v ( M ) and the number of edges is denoted by E ( M ) . Let x and y be vertices. If x is adjacent to y we write x y ; otherwise we write x -+ y. The multiplicity p ( x , y) of the pair { x , y} is the number of edges joining x to y. The distance between x and y in M is denoted by d,(x, y.) The open neighborhood of x is N ( x ) = { v E V : v x } . The closed neighbourhood of x is N [ x ] = N ( x ) U { x } . Similarly, we let I ( x ) = {v E V: v + x } . If T is a subset of V, then I ( T ) denotes {v E V: v + x for some x E 7'). The degree of x is 6 ( x ) = IN(x)l. The minimum degree of M is denoted by 6 ( M ) . The complement of M is denoted by M'. The complete graph on n vertices is denoted by K,,. The result of removing an edge from K,,is denoted by K,, - e. The result of adding one parallel edge to K,, is denoted by K,, e. The complete bipartite graph with bipartite sets of size m and n is denoted by Km,".The multigraph M is critcal if x ( M - v ) < x ( M ) , for all v E V, and is edge critical if x'(M - e ) < x ' ( M ) , for all e E E.

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2. The proof The proof of Theorem 3, relies on the following edge coloring lemmas. An s-triple in a multigraph M is a triple of vertices { x , y , z } such that y - z and

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p(x, y ) + p(x, z ) 2 2s + 1. We note that what we are calling an s-triple was called an (s 1)-triple in [lo].

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Lemma 4. [lo] If M is a multigraph such that x ' ( M ) > A ( M ) s then M contains an s-triple.

Lemma 5. If M is an edge critical multigraph such that x ' ( M ) > k 2 A ( M ) + s, then p ( M ) 2 [ ( m - 1)s + k - S ( M ) 2]/m, for some even m, such that 2 6 m s ( S ( M ) - 2)/s. In particular, i f X ' ( M ) > A ( M ) s, x ' ( M ) > 7s - 2, and p ( M ) s 2, then S ( M ) 3 4s.

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Proof. Let x o be a vertex such that S(xo) = S ( M ) , eo be an edge incident to x o , and f be a ( x ' ( M ) - 1)-edge coloring of M - e,. For a vertex v let f (v) be the set of colors not used by f on edges incident to v. In [lo] a path P was defined to be f-acceptable if P = x o , eo, x l re l , . . . , e n - , , x, where e, is uncolored and f ( e i ) E f(xj), for O s j < i < n . It is implicit in the results of [lo] that, since M is edge critical and f ( M ) > A ( M ) , M has a maximal f -acceptable path P of positive even length (number of edges) such that f ( x i ) f l f (xi)= 0, for i # j. By the maximality of P, if color (Y E f (xi) and i < n - 1 then there is an edge colored (Y from x, to xi for some j < n. Let N = U {f ( x i ):i < n } . Then

(k - 6(xo) + 1) + (s + 1) + ( n - 3)s s IN1 S(x,)

6 A(M).

Thus setting m = n - 1 yields 2 s m s ( S ( M ) - 2)/s, where m is even. By the pigeon hole principle, there exists i < n such that

[(m- 1)s + k - S ( M ) + 2]/m

S

INl/m d p ( q , x,)

6p(M),

which proves the first statement of the lemma. For the second statement, suppose for a contradiction that S ( M ) < 4s. Then m = 2 and 2s < [s

+ (7s - 2) - (4s - 1)+ 2 ] / 2 s p ( M ) .

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Next we prove a preliminary proposition.

Proposition 6. Let G be a graph which induces neither K 1 , , nor K,,, - e. (i) If X is a maximal clique and w E V - X , then IN(w) r l XI d 2s. (ii) If X is a maximal clique such that 1x1 3 4s, v E X and y , z E N ( v ) - X , then y -2. Proof. (i) Since X is maximal, there exists x E X such that x + w . Thus ( N ( w )nXI s 2, since otherwise Ka+,- e is induced by a subset of { w , x } u ( N ( w )nX),which is a contradiction.

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Nl ( z ) n XI s 4s - 1. So there (ii) By (i), I ( N ( y ) U N ( z ) ) f l XI G 4s - I N ( y ) f exists x E X,such that x +y and x + z . Thus y z , since otherwise K1,3is induced by {v, x , y, z } , which is a contradiction. 0

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Proof of Theorem 3. The result is trivial if s = 0; so assume s 2 1. We will obtain a contradiction by showing that if there existed a graph C = (V, E) which induced neither K 1 , 3nor K2F+3- e, such that R ( 3 , 4s - 1) < x ( C ) and w ( G ) + s < x(G), then there would exist a multigraph M, such that M contains no s-triple and A ( M ) + s < x ' ( M ) . But this is impossible by Lemma 4. So suppose that G is such a graph and that v ( C ) is minimal over all such graphs. Thus x(G) = max(w(G) + s, R(3, 4s - 1))+ 1, G is critical, and max(w(G) + s, R(3, 4s - 1))zs S(G). It suffices to show that G is the line graph of a multigraph M such that A ( M ) = w ( G ) and M contains no s-triple, for then A ( M ) + s = w ( G ) + s < x ( G )= x ' ( M ) . We begin by proving the following: (1) Every v E V is in some 4s-clique. Since 6 ( v ) s R ( 3 ,4s - 1) and C does not induce K 1 , 3 , N ( v ) contains a 4s - 1-clique K. Thus K U {v} is a 4s-clique. (2) For each v E V there is a unique pair of maximal cliques, X and Y , such that v E X n Y and N [ v ]= X U Y. First we prove the existence of X and Y. Let X be a maximal clique such that v E X and 1 x12 4s. This is possible by (1). By Proposition 6(ii) and the fact that o ( G )< 6(G), N ( v ) - X is a non-empty clique. Let Y be any maximal clique extending N [ v ]- X.Then N [ v ] = X U Y. Now we prove the uniqueness of the pair {X,Y} constructed above. Suppose { W , Z } is a pair of distinct maximal cliques such that v E W n 2 and N [ v ] = W u Z . Then

W u Z 3 X and 4s G I X n WI

+ IXn ZI - IXn W n Z J .

Thus I X n W l 3 2 s + l or IXr7ZIs2s+l. So by Proposition 6(i), X E { W , Z } . Say X = W . Then N [ v ]= X U Z = X U Y. We show that Z = Y by showing that Y U Z is a clique. Suppose y E Y and z E Z . Since Y - X = Z - X , if either y or z is not in X, then y 2. Otherwise both y and z are in X and y z . We will call the elements of the unique pair of maximal cliques {X,Y} for which v E X f l Y and N [ v ]= X U Y, the covering cliques of v. The proof of (2) has actually shown the following is true. (3) If v EX, where X is a maximal clique such that (XI 5 4s, then X is a covering clique of v. Call a covering clique X large if 3 4 s ; otherwise call it small. We next prove: (4) If x -y, then x and y have a common covering clique. Suppose that x and y do not have a common covering clique. We shall show that G is not critical. By (1) and (3) both x and y are in at least one large covering clique; neither is in two large covering cliques, since otherwise the covering clique

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of (say) x , which contains y, is large, and thus, by (2), a covering clique of y. Say X and Y are large covering cliques of x and y, respectively. We define a bipartite graph B = (B,, By,E x y ) as follows:

B, = { v

E X:v

is in exactly one large covering clique}

BY = {v E Y: v is in exactly one large covering clique} Ex, = { x ’ y ‘ E E : x ‘ E B, and y‘ E BY}. Suppose x ’ E B,, y ’ E Y - X, and x ’ - y ’ . Since Y is large, Y is one covering clique of y’; let Y’ be the other. Since Y is large, x ’ f$ Y. So x ‘ E Y‘, and thus Y’ is small. We conclude that y‘ E By.Similarly if y‘ E By,x ’ E X - Y, and x ’ y’, then x ’ E B,. Also by Proposition 6(ii), if x ’ - y ’ in B then N [ x ’ ]- ( X U Y ) = N [ y ‘ ]- (X U Y). Let C = (C,, C,) be the component of b containing x and y. We first show that if v E C then N [ v ]- ( X U Y) = 0. This follows easily by induction on d&, v) once we show it for v = x . So suppose w E N [ x ] - (X U Y). Let X‘ and Y’ be the small covering cliques of x and y. Then w , x , and y are in X‘ and Y’. Since X’ # Y’there exist u E X ’ and v E Y’ such that u + v. Clearly u w v. Since u f$ Y’, u E Y, and similarly v E X. By the remark above, u and v must be in B,U By. Let W and W’ be the covering cliques of w , where W is large. Since neither u nor v is in W, u and v are both in W’. But this contradicts u -/-v. Assume that IC,l S ICvl. We show that there is a matching in B‘ which misses at most s vertices of C,. Otherwise there exists a subset T of C , such that IT1 > ICY n Z(T)I s. Let Z = C yn Z(T) and N = (Cy- Z(T)). First suppose there exist u E N and v E C, such that u + v. Since 6(G) 2 w ( G ) s, IN(v) n NI 2 s - (ZI 3 2s 1- JTI. Thus some subset of { u , v } U T U ( N ( v ) n N) induces K2F+3-e, which is contradiction. Now suppose that every element of N is adjacent to every element of C,. As before, since x and y do not have a common covering clique, there exist v E C, and u E C y such that u v. Since 6(G) 3 w ( G ) +s, ( N ( u )flC,)l 3s. Since IC,l s ICyl, IN1 3 s 1. Thus some subset of {u, v } U N U ( N ( u )n C,) induces K,,, - e, which is a contradiction. Now we are prepared to obtain a contradiction by constructing a ( x ( G )- 1)coloring of G. Partition C, into S and T and partition C yinto S’ and T’ such that (SI s s and there exists a one-to-one function f : T + T’ such that x +f(x). Let qo by a (%(G)- 1)-coloring of G - (T U Cy).The only colored vertices adjacent to vertices in Tare in X.Thus we can fix a set A of IT1 colors that are available for coloring any vertex in T. The only colored vertices adjacent to the vertices in S’ are in S U ( Y - Cy).Thus we can extend qo to a ( x ( G ) - 1)-coloring q1 of G - (T U T’), which does not use any of the colors from A on S‘. The only colored vertices adjacent to vertices in T’ are in ( Y - T‘) U S. Thus there exists a set A‘ of (T’I colors that are available for coloring any vertex in T’. Extend q1to a ( x ( G )- 1)-coloring W of G as follows. For each color /3 E A t l A’ choose an uncolored x E T and color both x and f(x) with @. Color the remaining vertices of T and T’ with the remaining colors in A and A‘.

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We now construct M. Let the vertices of M be the covering cliques. Corresponding to each x E V, there is an edge ex of M which joins X and Y, the covering cliques of x . Using (2) and (4), it is easy to check that L ( M ) = G . Clearly A ( M ) = w ( G ) . Finally we show that M does not contain an s-triple. Since G is critical, M is edge critical. By Proposition 6(i), p ( M ) S 2s. Thus by Lemma 5 and the fact that 7s - 2 S R(3,4s - l ) , 6 ( M ) 3 4s. Thus every covering clique is large. Suppose X , Y,and Z are distinct covering cliques such that IX fl YI + IX n ZI 3 2 s 1 and Y fl Z # 0. Since X, Y, and Z are large covering cliques, X f l Y f l Z = 0. Say w E Y f l Z. Then there exists x E X such that w + x . Then some subset of (X n Y) U ( X fl Z ) U { w , x } induces K2F+3- e, which is a contradiction. 0

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We remark that the theorem cannot be improved when o ( G ) s 3 R(3,4s 1). Let M be the multigraph formed by replacing every edge of K2k+,by s parallel edges, where k > 1. Then f ( M ) 3 2 4 M ) / v ( M ) - 1) = 2sk s = A ( M ) s. If C is the line graph of M, then G induces neither K 1 , 3nor K2F+3 - e, w ( C ) = A ( M ) , and x(C) = x ' ( M) . Thus x ( G )= o ( G )+ s. Whether the upper bound holds for o(G) s < R(3,4s - 1) is not known.

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3. Problems The most important open question concerning edge coloring is the following conjecture of Goldberg [6]. Equivalent conjectures were posed independently by Andersen [l] and Seymour [ 151, but Goldberg's formulation is particularly attractive. Define the density W ( M ) of a multigraph M by

where M Z I H and v(H) is odd. Since no color class can have more than ( v ( H )- 1)/2 edges in H, W ( M )6 f ( M ) .

Conjecture 7. If M is a multigraph such that x ' ( M ) > A ( M ) + 1 then x ' ( M ) = W(M).

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The Petersen graph shows that the hypothesis x ' ( M ) > A ( M ) 1 is necessary. This conjecture has many exciting consequences. For instance, if M is critical and f ( M ) > A ( M ) 1, then v ( M ) is odd and less than A ( M ) . It also implies Theorem 4 and that if p ( M ) = 2 and f ( M ) > A ( M ) + 1 then M contains a 5-sided triangle. It was confidence in the conjecture together with this fact that lead Kierstead and Schmerl to the proof of Theorem 1. Goldberg [6] has proved the conjecture for A ( M ) S 9 . The proof provides a polynomial time algorithm that calculates W ( M ) and colors M with W ( M ) colors, if x ' ( M ) > A ( M ) + 1. It is

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likely that a proof of the whole conjecture would provide a similar algorithm. A reasonable way of attacking the general conjecture is to try to prove interesting results that are implied by it.

Problem 8. Show that if M is a multigraph with p ( M ) = 2 such that x ’ ( M ) > A(M) + 1, then M contains a 5-sided triangle. The next problem was left open in [12]. At first it may seem rather technical, but it has interesting consequences.

Problem 9. Show that if M is a multigraph with no 4-sided triangles such that p(M)s 2 and S ( x ) + S ( y ) S 2A(M) + p(x, y) - 3 whenever x - y , then x ’ ( M ) = A(M). It is shown in [12] that this problem is equivalent to the statement that if G is a linear graph such that A(G) S 2w(G) - 4, then x(C) = o ( G ) and that this would be an optimal result. In particular it would be an improvement on Theorem 2. Theorem 1 is easily seen to be a special case of this formulation. Finally we state the remaining open problem from this article. A positive solution would imply that if G is a graph which does not induce K 1 , 3 , then x(G) s 3w(G)/2, which would be a nice generalization of Shannon’s Theorem

Problem 10. Show that if G is a graph which induces neither K , , 3 nor K,,, - e, then x(G) s o ( G ) s (even when w ( G ) s < R(3,4s - 1)).

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References [l] L.D. Andersen, O n edge colorings of graphs, Math. Scand. 40 (1977) 161-175. [2] L.W. Beineke, Derived graphs and digraphs, Beitrage zur Graphentheorie (Teubner, Leipzig, 1968) 17-33. [3] R.L. Brooks, O n coloring nodes of a network, Proc. Cambridge Phil. SOC.37 (1941) 194-197. [4] S.A. Choudum, Chromatic bounds for a class of graphs, Quart. J. Math. 28 (1977) 257-270. [5] P. Erdos, Graph theory and probability, Canad. J. Math. 11 (1959) 24-38. [6] M.K. Goldberg, Edge coloring of multigraphs: recoloring technique, J. Graph Theory 8 (1984) 123- 127. [7] A. Gyirfils, On Ramsey covering numbers, Coll. Math. SOC.Jinos Bolyai 10, Infinite and Finite Sets, 801-816. [8] 1. Holyer, The NP-completeness of edge coloring, SIAM J. Comput. 10 (1981) 718-720. 19) M. Javdekar, Note on Choudum’s “Chromatic bound for a class of graphs”, J. Graph Theory 4, 3-12. [lo] H.A. Kierstead, On the chromatic index of multigraphs without large triangles, J. Combin. Theory Ser. B 36 (1984) 156-160. [Ill H.A. Kierstead, and J.H. Schmerl, Some applications of Vizing’s theorem to vertex colorings of graphs, Discrete Math. 45 (1983) 277-285.

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[12] H.A. Kierstead, and J.H. Schmerl, The chromatic number of graphs which induce neither K , , 3 nor K , - e, Discrete Math. 58 (1986) 253-262. [13] C.E. Shannon, A theorem on coloring the lines of a network, J. Math. Phys. 28 (1949) 148-151. 1141 D.P. Sumner, Subtrees of a graph and the chromatic number, The Theory of Applications of Graphs (G. Chartrand, ed.) (1981) 557-576. [15] P.D. Seymour, On multi-colorings of cubic graphs, and conjectures of Fulkerson and Tutte, Proc. London Math. SOC.38 (1979) 423-460. [16] V.G. Vizing, The chromatic class of a multigraph, Cybernetics 3 (1965) 32-41.

Discrete Mathematics 74 (1989) 125-136 North-Holland

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INTERVAL VERTEX-COLORING OF A GRAPH WITH FORBIDDEN COLORS Marek KUBALE Institute of Informatics, Technical Universiry of Gdaiisk, 80-952 Cdaiisk, Poland We consider a problem of interval coloring the vertices of a graph under the stipulation that certain colors cannot be used for some vertices. We give lower and upper bounds on the minimum number of colors required for such a coloring. Since the general problem is NP-complete, we investigate its complexity in some special cases with a particular reference to those that can be solved by a polynomial-time algorithm.

1. Introduction The classical model of coloring the vertices of a graph with single colors so that no adjacent vertices have the same color is too limited to be useful in many practical applications. A good illustration of this is the following school timetabling problem. Suppose that we have to arrange the times at which certain lectures are to be given knowing that some particular lectures cannot be held at the same time, since there may be students who wish to attend both of them. This scheduling problem can be represented by a graph in which vertices correspond to lectures and edges correspond to pairs of lectures that cannot be given simultaneously. Thus the timetabling problem is equivalent to the vertexcoloring problem stated in its standard form. However, in practice there are usually more restrictions generated by student and staff requirements which have to be taken into consideration in finding a satisfactory timetable. For instance, we may have to take into account the fact that certain lectures require at least two consecutive hours and that some teachers are not available at certain hours. Due to such restrictions the mathematical models are not simple coloring problems anymore. For this reason we must consider more general notions of graph coloring, and this paper is devoted to one of such generalizations. More formally, let G = (V, E) be a simple graph with the vertex set V = { v l , . . . , v,} and the edge set E = { e , , . . . ,em}. We define a vertexweighting function W :V + N , where N is the set of all positive integers. The pair (G, W) is said to be a weighted graph. By an interval k-coloring of (G, W ) we mean a function C: V + {S G (1, . . . , k}} whose values are sets of consecutive integers satisfying IC(v)l= W ( v ) and C ( u ) fl C ( v ) = 0 whenever {u, v } E E. The interval chromatic number x(G, W ) is the least k for which there is an interval k-coloring of (G, W). Moreover, we assume that with each vertex v E V there is associated zero or more (but no more than a bounded number of) intervals of 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V.(North-Holland)

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forbidden colors, i.e. the colors which cannot be assigned to v in any interval coloring of the graph G. Colors that are not forbidden form intervals of permissible colors. Since our coloring has to be an interval coloring without forbidden colors, it follows that C ( v ) must be consecutive permissible colors for all v. In other words, we are given a forbidding function F: V - , { S c (1, . . . . k}} with the property C ( v ) r l F(v) = 0, which must be satisfied in any interval k-coloring of a weighted graph (G, W). Consequently, an interval k-coloring of (G, W) that avoids F is an interval function C: V - {S (1, . . . , k}} fulfilling IC(v)l= W ( v ) and C ( v )fl F ( v ) = 0 for each v E V, and C ( u )r l C ( v )= 0 whenever ( u , v } E E. By the generalized (interval) chromatic number x(G, W, F) we mean the least number of colors needed to map the vertices of G to appropriate coloring intervals of size given by W in the presence of forbidden colors specified by F. A chromatic coloring is one that achieves the generalized chromatic number. It is not always easy to determine the value of x ( G , W, F). The general decision problem: “Given G, W, F and an integer k, is there an interval k-coloring of (G, W) that avoids F” is NP-complete, since it is already NP-complete to determine the chromatic number of a graph [5]. Hence it is unlikely that the generalized chromatic number can be calculated by a polynomial time algorithm. For this reason in Section 2 we give lower and upper bounds on x(G, W, F) that can be computed efficiently. In the subsequent two sections we investigate how the complexity of the problem is affected by restricting the problem’s domain. We consider some simplified subproblems caused by restrictions involving the form of functions W and F and the structure of a graph G. In this way we arrive at a number of negative results in Section 3 and positive results in Section 4. All the special-case results are then summarized in Section 5. We conclude with some remarks on the complexity of a relevant problem of interval edge-coloring of a graph with forbidden colors.

2. Bounds on the generalized chromatic number Let p,, be the first color forbidden for vertex Y E V and q,, the last color forbidden for v. Obviously, p,, q,, and p,, = q,, = 0 if F(v) = 0. If p,, = 1 then by s, we denote the size of the first interval of forbidden colors for v, i.e. (1, . . . , s,} E F(v). Otherwise we set s,, = 0. Next by W+(U), U 5 V we mean the quantity W+(U)= W ( U ) + m i n { s , : v ~U } ,

(1)

where W(U) = C U E uW ( v ) is the total weight of subset U. Let S ( C ) be the family of all 1-,2-, . . . ,1-element subsets of V that induce complete subgraphs of C. Then by o ( C , W, F) = max(W+(U): U E S ( G ) } (2)

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we mean the weight of the “heaviest appended complete subgraph” of G. Now we are ready to prove

Theorem 1. For any graph G and functions W, F w ( G , w , F ) s x(G, w , F ) .

(3)

Proof. In any interval coloring of graph G the vertices of any complete subgraph induced by U c V must have different colors greater than min{s, : v E U } . By choosing the heaviest of such subgraphs of C we get the lower bound. 0 A straightforward way of finding the value of (2) requires finding of all cliques of G. The number of cliques in an n-vertex graph can be as large as 3n’3 and the best known algorithm for generating all the cliques has the complexity of O(1.44”) [9]. Nevertheless, there are graph (e.g. planar graphs, line graphs) for which w ( G , W, F) can be obtained in polynomial time. If this is not the case we can estimate the value of w ( G , W, F) from below by restricting the calculations to all vertices and edges. We then have m a x { W ( v ) + s , : v E V } s x ( G , W, F) v

max{ W(u) + W(v) + min{su, s,} : {u, v } E E } s x(G, W, F) u.u

(4)

(5)

Now bounds (4) and (5) can be found in time O(m + n ) . In order to give an upper bound on x(G, W, F) by 6(G, W, F) we define the quantity

6(G, W, F ) = m a x { W ( N , ) + q , : v ~ V } ,

(6)

where Nv is the neighborhood of vertex v in graph C, i.e. the set of all vertices adjacent to v plus vertex v. Using the previous terminology we can say that 6(G, W, F) is the weight of the “heaviest appended neighborhood” of any vertex of G.

Theorem 2. For any graph G and functions W, F

Proof. To every vertex vi of (G, W) we link one new vertex ui of weight W‘(ui) = qvj. Thus (C, W) is a subgraph of a weighted graph (G’, W’), where W‘ is the extension of function W to set V U { u l , . . . , u,}. Now we apply the following coloring procedure to the list of vertices arranged in order u l , . . . , u,, vl, . . . , v,. First, each vertex ui gets colors C(ui)= { 1, . . . , W‘(u,)}. This makes it possible to avoid all forbidden colors for the vertices of G. Then for every c = 1,2, . . . we check if there is a yet uncolored vertex v i , i = 1, . . . , n that can

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128

be assigned the set {c, . . . , c + W ’ ( v i )- 1). If so, we color such a vertex vi and continue checking on the list starting from v ; + ~If. not, c is increased by one, and so on until all vertices have been colored. Now observe that when coloring vertex vi with set C ( v , ) all colors preceding C ( v J are occupied by neighbors of vi. For lack of gaps between intervals and because the obtained coloring of (G’, W ’ ) omits forbidden colors of (G, W ) , coloring of vi can be postponed no longer than we get to color W’(N,,,- {v;}) + 1. Thus the last color assigned to vi is less than or equal to W’(N,,,). Since W’(N,,,)= W(N,,,)+ W’(ui), so x(G, W , F ) 6 max{ W(N,,,) q,,,}, i = 1, . . . , n, and the upper bound follows. 0

+

In contrast to the lower bound (2) the upper bound (6) can be obtained in time proportional to the size of G, i.e. O ( m + n). Finally note that the upper bound is tight in the sense that there are graphs for which the generalized chromatic number is equal to S(G, W , F). A simple example is an arbitrary weighted complete graph with a common set of forbidden colors (1, . . . ,q } = F (v ) for all v. Nevertheless, bound (6) can be improved in some special cases of unweighted graphs. For example, if is a complete bipartite graph with unit weights and arbitrary forbidden colors then X ( K ~ - , ,W~ ,,F ) 6 max{g,, :

E V},

(8)

where g,, is the gth permissible color at vertex v E V. Also, every planar graph is properly colorable within the first six permissible colors at each vertex [l].

3. NP-completeness results Let CN1, CN2, CN3 stand for the decision problems for the chromatic number x(G), interval chromatic number x(C, W ) , and generalized chromatic number x(G, W, F), respectively. In what follows, in order to distinguish those features

that make CN3 an intractable problem, we consider the complexity of this problem in some of its more interesting special cases. For brevity, we say that the weighting function is binomial if W : V + { 1, L}, where L is any integer greater than 1. The function W : V - . {L} is said to be uninomial if L > 1, and unary if L = 1. As we know the general problem CN3 is NP-complete. This implies the NP-completeness of all special cases of CN3 where general graphs are allowed, since all of these include the CN1 problem as a subproblem. It turns out, however, that CN3 remains an NP-complete problem even if every pair of vertices is joined by an edge.

Theorem 3. CN3 is NP-complete even when restricted to complete graphs with at most one forbidden color per vertex.

Interval vertex-coloring of a graph

129

Proof. Our reduction uses the well-known partition problem [ 5 ] :“Given a set of positive integers A = { a , , . . . , a,) such that al + a , = 2b; is there a partition P of A such that C i p pa; = C i e pai = b?”.

+- -

For a given set A we construct a complete graph K,, on n vertices v l , . . . , v, with weights W ( v i )= a i , i = 1, . . . , n. Additionally, we assume F(v,)= { b 1) for all i. Suppose that there is an interval (26 1)-coloring of (K,, W) in which color b + 1 is not used. This means that some of the vertices are colored completely within 1, . . . ,b and the others within b 2, . . . , 2b + 1. Conversely, the existence of partition P implies that there is (2b 1)-coloring of K, such that b + 1$ C ( v i )for i = 1, . . . , n. Thus x ( K , , W, F) S 2b 1 if and only if there is a partition P of A, and the claim of theorem follows. 0

+

+

+ +

+

It is worth mentioning that the above subproblem becomes polynomially solvable if function W is binomial (cf. [ 4 ] ) .If, however, there are at most three intervals of forbidden colors per vertex then CN3 is NP-complete even if W is uninomial.

Theorem 4. CN3 is NP-complete even when G is a complete graph, W is uninomial, and F is such that there are at most three intervals of forbidden colors per vertex.

Proof. We transform to CN3 the following multiple choice scheduling within intervals (MCSWI): “Given m processors, a set of n tasks T = { t , , . . . , t , } , a common task length I , and for each task ti E T a collection { [ q ( i ) ,d j ( i ) ]:1s j s k ( i ) } of permissible scheduling intervals with integer release time/deadline endpoints and such that 0 S q ( i )S d j ( i )- 1 for j = 1, . . . . , k ( i ) . The question is whether there exists an rn-processor nonpreemptive schedule for T that places each task i in one of its permissible intervals”. MCSWI was shown to be NP-complete even if m = 1, 1 = 2 and there are at most two intervals (one of length 4 ) per collection [S]. Without loss of generality we assume that in a given instance of MCSWI each task t, has exactly two permissible intervals: [ d , ( i )- 2, d l ( i ) ] , [ d 2 ( i )- 4,d 2 ( i ) ] . Let d be the latest deadline of all collections of the intervals. Then the corresponding instance of CN3 is graph K,, function W such that W ( v , )= 2 for i = 1, . . . , n , and function F such that there are three intervals of forbidden colors per vertex, namely F(vi)= (1, . . . , d , ( i )- 2 ) U { d l ( i )+ 1, . . . , d2(i)- 4 ) U { d 2 ( i )+ 1, . . . , d}. Now it is easy to see that there exists a solution to the single-processor case of MCSWI if and only if x ( K , , W, F) S d . 0 The MCSWI problem remains NP-complete even if m = 1, three task lengths

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are allowed and there is just one permissible interval per task (cf. [ 4 ] ) .Hence, taking advantage of the transformation used in the proof of Theorem 4 one can prove that CN3 is an NP-complete problem if G is a complete graph, W is a 3-valued function, and there are at most two intervals of forbidden colors per vertex. Our last negative result deals with unary bipartite graphs.

Theorem 5 . CN3 is NP-complete even when restricted to unary bipartite graphs with at most two intervals of forbidden colors per vertex. Proof. Our reduction uses the well-known 3-satisjiability (3SAT) problem: “Given a collection C = { c , , . . . , c,} of clauses on a set X = { x , , . . . ,x,} of Boolean variables such that Icj(= 3 for j = 1, . . . , m . Is there a truth assignment for X that satisfies all the clauses in C?”. Assume that we have an instance of 3-SAT with n variables and m clauses. We must construct a bipartite graph G and a positive integer k such that x(G, W, F) s k if and only if there is a satisfying truth assignment for X.For each variable xi E X we create three vertices: vertex xi and two other vertices x & - , and x; adjacent to xi and called literals. We think of x + as unnegated and x - as negated form of x for all x . In addition, every triple of vertices has two permissible colors in common, namely 2i - 1 and 2i for i = 1, . . . , n. Then for each clause cj E C we add one vertex cj adjacent to three literals according to the corresponding variables that occur in cj in their unnegated or negated form. Next, to each cj, j = 1, . . . , m we link 2n pendant vertices named p i , , . . . ,pjZn.There are no forbidden colors for vertices cj. Also vertex pjh, h = 1, . . . , 2n has no forbidden colors if cj is adjacent to x l or x i . Otherwise it has just one permissible color h. The construction of our instance of CN3 is completed by setting k = 2n. It is easy to see that G is bipartite and all the vertices have at most two intervals of forbidden colors. Suppose that the Boolean formula is satisfiable and let a , , . . . , a, be any assignment of variables that evaluates the problem instance to true. With each value a, we associate an integer b, such that b, = 2i if a, =false and b, = 2i - 1 if a,=frue. Each vertex xi can be colored with b,. Also, each pair of literals adjacent to xi can be assigned the other permissible color, i.e. 4i - b, - 1. Next, by the construction, each vertex cj can get color b, corresponding to value a, of xi which evaluates that clause to true. Finally, each pendant vertex pjh, 1 < j =sm , 1 s h s 2n can be colored with h or any other color different from that of cj. Thus the satisfiability of the formula implies that graph G is 2n-colorable. Conversely, suppose that there exists 2n-coloring of G that avoids F. From the way in which 2n - 3 of the pendant vertices pjh on cj are preassigned forbidden sets of colors it follows that each of them must be colored with value h equal to the index of a literal not adjacent to vertex cj. Thus each cj is colored with some color 6, in common with vertex x, whose literal is adjacent to cj. If b, is odd we

lnterual vertex-coloring of a graph

131

may regard variable xi as being assigned ai = true, otherwise ai =false. Now consider the whole formula of the 3-SAT instance. By the construction, each clause cj is satisfied by assignment ail, aj2, aj3 for variables xi,, x,,, xi, contained (either negated or unnegated) in it. Therefore conjunction of all the clauses in C is also satisfiable. Thus the 2n-colorability of G implies the satisfiability of the problem instance, which completes the proof. 0 The proof of Theorem 5 can be probably simplified and strengthened.

4. Polynomial solvability results In this section we consider the cases where chromatic colorings can be found efficiently. Our first positive result involves unary complete graphs. In its original formulation it deals with scheduling unit-length tasks with multiple release time/deadline intervals [8], but can be easily transformed into a graph coloring problem.

Theorem 6. If all vertices of a complete graph K,, have the common weight 1 and an arbitrary number of forbidden colors then a chromatic coloring can be found in time O(n4). Proof. See Simons and Sipser [8]. 0 Note that the complexity of this special case depends on the maximum number of prohibited intervals per vertex. For instance, if the number of permissible intervals is one then it is possible to achieve a chromatic coloring of K,, in time (cf. 131). The remaining special-case results deal with bipartite graphs.

Theorem 7. If G is a bipartite graph, W is unary, and F is such that left endpoints of prohibited intervals are odd and right endpoints are even, then G can be optimally colored in time O ( m + n ) . Proof. Let B = B(V,, V2)be a unary bipartite graph with two independent sets V, and V,, V, U V2= V. A chromatic coloring of B can be obtained in two phases. In the first phase each vertex u E V, is colored greedily, i.e. with s, + 1. In the second one, each vertex v E V2is colored greedily. If the number of colors used is odd, say 2; + 1, its optimality follows from the fact that the greedy algorithm first attempts to assign odd values to all vertices. If the number is 2; + 2 then there exists an edge { u , v } E E such that s, = s, = 2;. Hence u must be colored with 2; + 1 and v with 2; + 2. Thus x ( B , W, F) = 2; + 2, as desired.

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'6

'8

'7

'9

'10

'11

'12

"13

Fig. 1. A caterpillar with 5-vertex body and 8 hairs.

Since the determining of sets V, and V, as well as the greedy coloring can be done in time proportional to the size of B, the theorem is proved. 0 The following theorem involves a caterpillar, i.e. a tree in which the removal of all pendant vertices results in a path. These pendant vertices can be thought of as hairs attached to the body of the caterpillar, i.e. a path of non-pendant vertices (see Fig. 1).

Theorem 8. Zf G is a unary caterpillar and each vertex v E V has at most one interval of forbidden colors such that F ( v ) = (1, . . . , s"}, then G can be optimally colored in linear time.

Proof. Let s be the maximum color among all colors forbidden for the vertices of G. If G contains two adjacent vertices with s forbidden colors each (s-vertices, for short) then G is (s + 2)-chromatic. Thus all we need to do is to divide V into independent sets V, and V,, and color elements of V, with s + 1 and V, with s + 2. So suppose that no two s-vertices are joined by an edge. In this case we apply the following limited backtracking algorithm. Step 1. Color all s-vertices with s + 1. Step 2. Color uncolored segments of the body starting from its left endpoint v,. Namely, succeeding vertices are colored greedily until an (s - 1)vertex requiring color s + 2 is encountered; otherwise we go to Step 3. If vj is such a vertex then we move sequentially back in search of the first vertex vi that can be recolored with the color of value one greater than the previous color (but less than s + 2), however not farther than to the last resumption point (initially, vertex v, is called the last resumption point). Next we increase the color of vi and perform the forward step once more. If the backward step has made it possible to color vj with s then vertex vj is stored as the last resumption point and _ -

-U

even

n

Fig. 2. Illustration for Theorem 8, Case 1; white nodes denote (s - 1)-vertices, black nodes denote s-vertices.

Interval vertex-coloring of a graph

133

& Fig. 3. Illustration for Theorem 8, Case 2; white nodes denote (s - 1)-vertices, black nodes denote s-vertices.

the sequential coloring of the next body segment is realized. Otherwise, two following cases are possible. Case 1. No such vertex vi has been encountered. This case is illustrated in Fig. 2, where the remaining body and hair vertices are left out. Case 2. Such a vertex vi has been encountered but the next forward step failed to reach vj. This case is depicted in Fig. 3. Observe that both the cases imply that G has two s-vertices, say u and w, such that the path between u and w consists of even number of (s - 1)-vertices. Conversely, if there are no vertices u and w as described above then the forward step will reach vj. In both cases the path between and including L(, w requires s 2 colors. Hence the whole graph G is (s 2)-chromatic and can be colored by the method described at the beginning. Step 3. Color the remaining pendant vertices in the greedy way. Since any vertex of G is colored at most three times (by two methods) within constant time, the coloring of all n vertices can be done in time O ( n ) . 0

+

+

Theorem 9. If G is an n-vertex star and functions W , F are arbitrary except that each vertex has O ( n ) intervals of forbidden colors then a chromatic coloring can be found in time O(n2log q ) , where q = max{q, :v E V } . Proof, Let S,, be a star on n vertices numbered as in Fig. 4. Our algorithm consists of two phases. In the first phase we color the vertices in order v l , . . . , v,, in the greedy way. If the number of colors thus used is equal to the lower bound (4) or (5) then the algorithm terminates. Otherwise we proceed to the second

v47k "3

"2

Fig. 4. The star S,.

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M. Kubale

phase in which we apply a binary search to try and improve the best interval coloring found so far. Initially, let k be equal to a half of the number of colors used in the first phase. Starting from k downwards we color u, with the highest permissible interval while the leaves are colored greedily in order v,, . . . , v,-, starting with color 1. If for some i vertex vi cannot be colored within the limit then v, is assigned the next permissible interval, i.e. one that makes it possible to , presumed limit k color vi as well. However, if there is no such interval for u , ~the is increased appropriately and the process is repeated with the new value of k. Otherwise, with the same k, the greedy coloring is verified for those leaves among v l , . . . , vi-l that have not been assigned colors greater than the current coloring C(v,) and after that it is continued for vi+,,. . . , v,-,. Finally, if for a prespecified coloring of v, all its neighbors have been colored with nonoverlapping intervals, the interval k-coloring of S, that avoids F is stored, the limit k is diminished appropriately, and the process is repeated with the new limit, etc. The correctness of the method follows from the exhaustness of the binary search for a minimal k, so let us estimate the time complexity of the algorithm. In the first phase each prohibited interval is processed at worst once. Since there are O(nz) many of them, the initial phase can be implemented to run in O(n’) time. In the second phase, for a fixed k the central vertex is colored at most n times and the time spent for any leaf is also O(n). Hence in this stage the algorithm performs O(n2) steps. Since the verifying of k-colorability is executed [log, q1 times, the overall running time is O(n2log q ) , as claimed. 0 Observe that we have ignored forbidden intervals in our analysis. Surely these have to be inputs to the problem. Since there are O ( n z )many of them, the input size is at least O(n210gq). Thus our algorithm can be regarded, in the wide sense, as a linear-time algorithm. Moreover, given the intervals of prohibited colors as part of the input, it is possible to remove all restrictions on the number of forbidden intervals. Theorem 9 implies the following positive result concerning bipartite graphs.

Theorem 10. If G is a bipartite graph B(V,, V,) with a linear number of forbidden i = 1, 2 such intervals per vertex and there is a vertex u of maximum weight in that for every v E V, we have F (v ) c F(u) and N ( v ) c N ( u ) U {v}, then a chromatic coloring of G can be found in time O ( n 2log q ) .

v,

Proof. Assume, without loss of generality and to simplify the notation, that B(Vl, V,) is connected and that the vertex u belongs to V , . We first color the vertices of the star induced by Nu by means of the method described in the proof of Theorem 9. As we know, this can be done in O(lV,12log q ) time. Let C ( u ) be the interval of colors assigned to u in the optimal coloring of the star. Then we color the remaining uncolored vertices in Vl with colors belonging to C ( u ) in time O(lV,l).Thus the time complexity is O(n2log q ) . 0

Interval vertex-coloring of a graph

135

Finally, note that Theorem 10 implies polynomial solvability of all special cases of complete bipartite graphs I?(&, V,) in which all vertices in V, (or V,) have identical forbidden colors, since the heaviest of such vertices can be regarded as vertex u.

5. Complexity classification

In the previous sections we have considered the complexity of CN3 assuming restrictions imposed on vertex weights, the number of prohibited intervals per vertex, and the structure of a graph. In this way we have identified several classes of highly structured graphs for which chromatic colorings can be obtained in polynomial time. The main results of this investigation are summarized in Table 1. Entries in the table are either “NPC” for NP-complete, “?” for “open”, or O ( . ) for an upper bound on the complexity derived from the best polynomial optimization algorithm known for the corresponding subproblem. In addition, in the column CN3 we have placed signs “ - ” to indicate that there is known a negative result (NP-completeness proof) for a special case of the corresponding to indicate that there is a positive result (polynomialsubproblem, and signs time algorithm) for a special case of that subproblem. We conclude with some remarks on the complexity of an analogous problem of interval edge-coloring of a graph with forbidden colors. First of all note that interval edge-coloring is harder than interval vertex-coloring even when there are no forbidden colors at all (cf. [6]). If, however, G is an edge-weighted graph with forbidden colors on the edges then interval edge-coloring is NP-complete even if G is a star with just one forbidden color per edge. This follows from the fact that coloring the edges of a graph G is equivalent to coloring the vertices of a graph L ( G ) , the line graph of G. Using Theorem 3 and by the fact that L(S,) = K , - , we get the desired result. Another interesting result is due to Even et al. [2] and states that it is NP-complete to decide the edge 3-colorability of an unweighted bipartite graph with single forbidden colors. Although there are also some positive results concerning bipartite graphs (e.g. those arising from completing

“+”

Table 1. Complexity classification for interval vertex-coloring ~~

Graphs

CN 1

General graphs Complete graphs Bipartite graphs Trees Stars +

CN2

CN3

References

NPC

NPC

O(n) O(m n ) O(n) O(n)

a n ) O(m n ) a n ) O(n)

NPC NPC-‘+ NPC-++

~ a r 151 p Theorems 3 , 4 , 6 Theorems 5 , 7 , 1 0 Theorem 8 Theorem 9

+

NP-completeness proof. Polynomial-time algorithm.

+

?+ O(n2log 9 )

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partial latin squares), the interval edge-coloring of a graph with forbidden colors is NP-complete for all classes of graphs for which the corresponding problem CN3 was open or polynomially solvable (in Table 1). More details on the subject can be found in [7].

Acknowledgments The author is indebted to Dr. W. Kubiak for his helpful discussions and to the referees whose suggestions have improved the presentation of this paper.

References [l] P. Erdos, A. L. Rubin and H. Taylor, Choosability in graphs, in: Proc. West Coast Conf. on Combinatorics, Graph Theory and Computing (Humboldt State University, 1979) 125-157. (2) S. Even, A. Itai and A. Shamir, On the complexity of timetable and multicommodity flow problems, SIAM J. Comput. 5 (1976) 691-703. [3] G.N. Frederickson, Scheduling unit-time tasks with integer release times and deadlines, Inf. Process. Lett. 16 (1983) 171-173. [4] D.S. Johnson, The NP-completeness column: An ongoing guide, J . Algorithms 4 (1983) 189-203. (51 R.M. Karp, Reducibility among combinatorial problems, in: R.E. Miller, J.W. Thather, eds., Complexity of Computer computations (Plenum Press, New York, 1972) 85-103. [6] M. Kubale, The complexity of scheduling independent two-processor tasks on dedicated processors, Inf. Process. Lett. 24 (1987) 141-147. [7] M. Kubale, Graph coloring, in: A. Kent, J.G. Williams, eds., Encyclopedia of Microcomputers (Dekker, New York) (to appear). [8] B. Simons and M. Sipser, On scheduling unit-length jobs with multiple release time/deadline intervals, Oper. Res. 32 (1984) 80-88. [9] E. Tomita, A. Tanaka and H. Takahashi, The worst-case time complexity for finding all the cliques, Technical Report VEC-TR-CS (University of Electro-Communications, Tokyo, 1988).

Discrete Mathematics 74 (1989) 137-148 North-Holland

137

HADWIGER'S CONJECTURE (k = 6): NEIGHBOUR CONFIGURATIONS OF 6-VERTICES IN CONTRACTIONCRITICAL GRAPHS Jean MAYER Universite' Paul Vale'ry, Montpellier, France

1. Definitions; purpose of the study The graphs considered here are simple (without loops or multiple edges). A vertex colouring is such that two neighbour vertices (joined by an edge) are of different colours; colours are designated by numbers 1 , 2 , 3 . . . A k-coloration of a graph G is a colouring using k different colours; when it is possible, G is said to be k-colourable. If G is k-colourable, but not (& - 1)-colourable, it is said to be k-chromatic; k is the chromatic number of G. To contract a graph G consists in deleting the vertices and edges of a connected subgraph H of G which is replaced with a new vertex h ; the edges of G having one end in H are replaced with edges joining G - H with h, the other end remaining unchanged; multiple edges or loops occasionally resulting from this operation are eliminated. It is possible to carry out a contraction as a sequence of elementary contractions bearing upon one edge at once. Thus contraction is a transitive operation. Let us now suppose G connected. It is possible to continue its contraction until K , , the trivial graph, is obtained. Previously a graph r will be reached, such that its chromatic number is k, but every graph contracted from r has a chromatic number strictly lower than k:r is said to be k-chromatic contraction-critical (shortly &-c.c.); this type of graphs was introduced and studied by Dirac [4,5,7,81. Let us call K , the complete graph of n vertices. According to Hadwiger's conjecture [9], every connected &-chromatic graph can be contracted to Kk.In other words: The only k-c.c. graphs are the complete graphs (Kk).This conjecture which is obvious for k 6 3 was proved by Dirac [4] for k = 4. For k = 5, Wagner [14] showed that it is equivalent to the four colour conjecture which was demonstrated by Appel and Haken, with the help of Koch [l, 21. As for k 3 6, it is undecided. However, Dirac [7] showed that every 6-chromatic graph is contractible to K, - E (K, with one edge deleted). Analogous results were found for k z-7 by Jakobsen [ 101 and Mader [12]. 0012-365X/89/$3.50@ 1989, Elsevier Science Publishers B.V. (North-Holland)

J . Mayer

138

We shall consider here the case k = 6. Let r be a 6-C.C.graph, not isomorphic to K6. We shall prove that, if rexists: 1. the configuration induced by the neighbours of a vertex of degree 6 (configuration designated here by &) can only be one of the eight types presented in Fig. 2 (type H being excluded, as reducible); 2. if an edge joins two v6, these are of the same type (namely A, C or J ) . Notations used

A graph or subgraph is designated by a capital letter, a point or vertex by a small letter (italics are reserved for numbers or indices named by letters). Usual signs for inclusion or belonging to a set are used. Moreover we shall denote: vj:vertex of degree (or &vertex). (a, b ) : edge joining a and 6. P(a, b): path connecting .a and b. K,: complete graph of 0 vertices. SF): K , with one edge subdivided into a path of length m. C > HIC is contractible to H. r:6-chromatic contraction-critical graph, not isomorphic to K 6 . r(a): subgraph of r induced by the neighbours of the vertex a (or neighbour configuration of a ) . If a is a v,, one may write <. Colours used in a m-coloration of a graph are designated by numbers 1 , 2 , 3, . . . )m.

2. Properties of 6-C.C.graphs If

r is 6-C.C.and non-isomorphic to K6: Every vertex of r is of degree 2 6 (Dirac). r # K 5 (Dirac). Hence, for every vertex x , T ( x ) $ K4. r is 6-connected (deleting any five vertices of r does not destroy connectivity) (Mader, [ll]).

(2.1) (2.2) (2.3)

For every vertex of degree i, r; does not contain 1: - 3 independent vertices. In particular & has stability number p = 2 (Dirac). (2.4) From the precedent statement Dirac deduced:

& contains either two vertex-disjoint triangles or K4 - E (K4 with one edge deleted). Menger-Dirac theorem [6]: in a m-connected graph, two disjoint sets of vertices A and B are connected by a set Y of at least m two-by-two independent paths (except for the end-vertices which

(2.5)

Hadwiger’s conjecture

(k = 6 )

139

may be common). Moreover, if A (resp. B) contains at least m vertices, every vertex of A (resp. B) is the end of at most one path of the set Y. (2.6)

3. Structure of

r,

One can strengthen statements (2.2) and (2.5). Let us designate by Sr’ a KO, an edge of which has been subdivided by insertion of a vertex of degree 2 (See Fig. 1 for 0 = 5).

r $ Sk2);by consequence, for every x , T(x)$ Sp).

(3.1)

Proof. r being 6-connected by (2.3), r - S i 2 ) = U includes a connected component C (possibly C = U ) attached at the six vertices of Si2).By contracting Si2) to K S and C to a single vertex, one obtains a K 6 : thus r i s not 6-c.c., contrary to the hypothesis. The consequence is immediate. 0

& (neighbour configuration of a u6) is isomorphic to one of the graphs of Fig. 2, except H , which is reducible. (34 Proof. By (2.5), & contains either the diagram A or the subgraph r ( a , b , c , d ) of the diagram G. In the former case, on account of (2.2) and (3.1), one can add at most three edges (not all concurrent) to A : thus one obtains B, C, D, E or F. In the latter case, on account of (2.4), e is joined with a or d (say with a ) and with f ; i f f is joined with a, we have the first case again; thus f is joined with d: thus one obtains G , H or J. 0 Proof of the reducibility of H Let xo be a v6 in r with neighbour configuration isomorphic to H (Fig. 2, H). If we contract P ( x s , x g , x 6 ) to a vertex, we get a graph I“, 5-colourable as contracted from r. Every 5-coloration of r‘ induces a 5-coloration of r - xo with x s and x6 of the same colour, say colour 5. Since r is 6-chromatic, each of the 5 colours needed in colouring r - x o must be present in I&), that is to say, x l , x2, x 3 , x4 use four different colours, none of which can be removed by a “Kempe interchange” between two colours. Thus there are two

Fig. 1.

1. Mayer

140

4

G

A

6

C

D

E

F

H

J

(reducible)

Fig. 2.

bicoloured chains P ( x l , x,) and P ( x z , x,) in r - x,, without a common vertex. Contract each of them to an edge and ( x , , x s ) to a vertex: you obtain a K 6 , against the assumption made about r. 0

Every marked vertex in Fig. 2 is of degree 3 7 in

r.

(3.3)

Proof. One can verify directly that every graph in Fig. 2 (except A) contains five distinct vertices, say h, j, k, I , m, such that { h , j } and {k, m } are independent sets and P(k, I, m ) is a path of length 2, I being any marked vertex in the diagram considered. Let x, (Fig. 3) be a 6-vertex in r and 1 a marked vertex in the neighbour configuration of xo. Suppose 1 of degree 6 in r. x,, U T ( x , ) includes two paths P(h, x o , j) and P(k, I, m),every one of which we contract to a vertex; thus we obtain a graph I" which is 5-colourable as a graph contracted from r. Every 5-coloration of I" induces a 5-coloration of r - x , , in which the colour of h is repeated in j, and the color of k is repeated in m. The colouring of I (which is of degree 5 in r - x,) does not encounter any obstacle, since two neighbours of I are of the same colour. The 5-coloration of r - x , extends directly to r, since T ( x o ) uses four colours at most. Then if I is of degree 6 in r, T i s not 6-C.C. 0

Hadwiger’s conjecture (k = 6 )

141

xO Fig. 3.

If T(xo) is isomorphic to A, B, C , . . . , we shall say that x,, is of type A, B, C . . . We now intend to prove that, if an edge of r joins two v6, both of them are of the same type, and this type can only be A, G or J . Firstly two new reductions will be stated.

Let G be a 6-connected graph with a subgraph G ( a , b, c, d , e, f ) made of an edge (a, b ) and four common neighbours c, d , e, f of a and b; if G contains a K4disjoint from {a, b } , G can be contracted to K6 (Fig. 4). (3.4)

Proof. Let c’, d ’ , e’, f ’ be the vertices of the K4;they are different from a and b, but not necessarily from c, d , e, f . G - a - b is 4-connected, because G is 6-connected. Without loss of generality, by (2.6) we can assume that there are four (i, i’)-paths for i = c, d, e, f . Possibly i = i’ for some i ; if for example c = c’, the graph G - a - b - c is 3-connected and there are three (i, i’)-paths for i = d, e, f . In every case, we contract each of the (i, i‘)-paths to a vertex to produce the required K6. Cl Let G be a 6-connected graph, xo a 6-vertex of G with G(x,) consisting of two triangles (a, c, d ) and ( b , e, f ) connected by the

Fig. 4.

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edge (a, b ) (and possibly other edges); if G contains a K4 disjoint from { a , b } , G can be contracted to K,. (3.5)

Proof. x o U G(x,) contains the subgraph of Fig. 5 . G - a - b is 4-connected and there is no path from xo to the K4 independent from {c, d , e, f } . Therefore one can apply the same contraction as in (3.4) and then contract (a, b) to a vertex to produce the required K,.

If T contains a v 6 of type J , say x o , T(xo)being a wheel W, with centre c: (1) every v 6 E T is joined with c by an edge, ( 2 ) a v, of T i s necessarily of type G or J , ( 3 ) in particular, a v6 joined with x o is of type J (with the same (3.6) centre c as xo). Proof. (1) The edge (xo, c) belongs to five triangles; therefore, by (3.4), every clique K4 c T must contain xO and/or c. Since a v, is contained in two K 4 or in a K s - E, it must be joined with x 0 or c ; but every neighbour of x g is also joined with c. (2) If x1 is a v 6 of type A , B, C, D,E or F, it belongs to two K4, one of which contains x o and the other one contains c. Therefore, in T ( x , ) , the edge ( x O , c) joins two K 3 , hence A is excluded. Furthermore such an edge cannot have an end-vertex of degree 6, which excludes the other types above-mentioned, by contradiction with the degree of xo. On the other hand, if x , is of type J, c can be the centre of T ( x , ) as well as of T(xO):if x l is of type G (see Fig. 2, G), c can occupy the position b (or c) in T ( x , ) . (3) If x , is a u6 joined with xO by an edge, it has three common neighbours with xo; hence x o is of degree 3 in T ( x l ) ;for a 6-vertex, that condition is satisfied only if T(xJ is the wheel J. Evidently, the centre of this wheel is the same as that of

mJ. The v, of types B , C , D , E , F have no neighbours of degree 6.

xO Fig. 5.

(3.7)

Hadwiger’s conjecture (k = 6 )

143

Proof. Let xo be a v, of one of the above types and v a v, joined with x,,. Because of (3.3), v cannot be a marked vertex. And if v is a non-marked vertex, three cases are to be considered: (1) T(v)contains two disjoint triangles: thus v is contained in two K4, one of which is disjoint from {a, b}, (a, b) being an edge joining the two triangles existing in T(xo).But in this case T > K 6 , by (3.5). (2) 21 is of type G, with xo E K5 - E c v U T ( v ) . But in this case xo is a marked vertex in T ( v ) and cannot be a 216: contradiction. (3) v is of type J: this case is excluded by (3.6.) 0 A (v,, v,)-edge joins two vertices of the same type, namely A , G or J . (3.8)

Proof. Let xI and x, be the two 6-vertices. They can have only two or three common neighbours. If they have two common neighbours joined by an edge, both x, and x, are of type A by (3.7); if the two common neighbours are not joined by an edge, both x1 and x, are of type G; if x I and x, have three common neighbours, both of them are of type J (and their neighbouring configurations have the same centre). In strong contrast with type J , the presence in the cliques K4 extremely rare, so we can state:

r of a v6 of type C or F makes

If there is in T a v, of type C or F, I’does not contain any other 216. (3.9) Proof. Let xo be a v6 of type C with T ( x O )consisting of two triangles (a, b, c) and ( a ‘ , b’, c’) joined by two edges (a, a ’ ) and (b, b’). By ( 3 . 3 , a K4 disjoint from xg necessarily contains a or a’ and b or b‘, i.e. a and b or, symmetrically, a’ and b’. If it contains a and b and not c, (a, b) belongs to four triangles and T(xo,a ’ , b ’ , c’) is a K4 disjoint from { a , b}, a case which is excluded by (3.4). Therefore the K, must contain c and is, say, r(a,b, c, d); r - x , ] can include only one more K4, namely T ( a ’ ,b’, c’, d’). But under these conditions d (or d ’ ) cannot be a v,, by (3.5). Since xo has no 6-neighbours by (3.7), xo is the only vh in r. For xo of type F, i.e. with (c, c’) E r, the proof is strictly the same. 0

4. G-vertices

If r contains non-isolated 6-vertices of type G, they can only form K,- or K,-components (Figs 6 , 7 , 8 , left). In the present section we shall prove that: (1) A K, made of three G-vertices is a reducible configuration. (2) If two G-vertices form a K,, r d o e s not contain any other 6-vertex.

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Fig. 6.

By construction, if two G-vertices x1 and x 2 of r f o r m a K 2 , r(xl) U r ( x 2 ) = X is necessarily isomorphic to Fig. 6 (left) or to Fig. 7 (left). A third G- vertex can be added to Fig. 6 (in position e ( = x 3 ) ) to give the Fig. 8 (left) with X = T ( x l )U r ( x 2 ) U r ( x g ) . In each of these three cases let us designate by U the complement of X in r, i.e. U = r - X. Every vertex of X - x1 - x 2 (resp. X - x l - x 2 - x 3 ) has neighbours in U , by (2.1). These vertices are the attachments of U in X. The attaching sets of U are represented in Figs 6, 7, 8 (right). U can be connected or can consist of connected components U,, U2,. . . , each of which has at least six attachments in X,because of (2.3).

The configuration of Fig. 6 is reducible by r>K 6 , unless U = U l + U2, the connected components U, and U2 being respectively attached at {a, b , , d l , e, d 2 , b 2 } and at {a, c1, d l , el d 2 , c2). (4.1)

Proof. There are three cases, the first two of which are reducible: (1) U, is attached at b l , c l , b2 (or equivalently c2) and at least three vertices

Fig. 7.

Hadwiger's conjecture (k = 6 )

145

a.I

bl

b3

83

a2

Fig. 8. E P(a, c2, d2, e, d,). Let us contract P to an edge ( a , d , ) so that a and d , remain (or become) attachments of Ul; let us then contract U, U b, Ux 2 to a vertex: we get the result r > K 6 . This case includes the case of U connected. (2) U, is attached at a, b,, c,, d , , e, d2. By exclusion of Case 1, U2 isattached at a and d,: therefore the contraction of U, provides the edge ( a , d , ) ; then contract U, U d2 U x , to a vertex: you get a K6 again. (3) U, is attached at a, b , , d , , e, d 2 , b2; U, is attached at a , c , , d , , e , d,, c,. This is the irreducible case of the statement (other irreducible cases differ from that only by the notations, e.g. by the permutation of bl and cI). Of course, if there exists a component U3 providing the edge ( a , d , ) , U, U (b,, c,) U U, can be treated as a connected component of I'- q x , ) and we get a K 6 again. 0

The configuration of Fig. 7 is reducible by r >K 6 , unless U = U , + U2, the connected components U, and U, being respectively attached at {al, b l , d l , a2, b,, d,} and at { a , , C I , d l , a21 c2, d2). (4.2) The proof is quite similar to that of (4.1). A triangle of G-vertices is reducible.

(4.3)

Proof (see Fig. 8). By (4.1), there remains only one case to be considered, in which U = U1 + U,, the attaching sets being respectively { a , , b , , a2, b,, a3, b3} and {al, c,, a,, c2, a3, c3}. Then contract P ( a , , x 3 , a3, c2, u2) to an edge ( a , , a2); contract U, U c3 to a vertex to obtain the edge (c,, c3);contract P ( x l , x,, b2)and P(b3, c3, c,) each to a vertex; finally contract U, to a vertex f : you get a K 6 (see Fig. 9). 0 The irreducible cases of (4.1) and (4.2) are concerned by the following

J . Mayer

146

Fig. 9.

statement:

If r contains two G-vertices x , , x2 joined by an edge, r does not contain any K4 except those included in T ( x , )U T(x2). By consequence, there are no other 6-vertices in r than x , and x2. (4.4) Proof. We shall treat only the case of Fig. 6. The case of Fig. 7 is very similar. U consists of two connected components UI and U2 with respective attaching sets { a , b , , d l , e, d 2 , 62) and { a , cl, d , , e, d 2 , c2}. Let us suppose that rcontains a 4-clique K included in (say) U , U a U e. There are three cases: (1) K contains neither a nor e. By ( 2 . 3 ) , r - a - e is 4-connected, so there are four paths from the vertices of K to { b l , d l , d 2 , b 2 } :contract each of them to a vertex; contract ( x , , x 2 ) ; contract a U U2U e to a vertex: you get a K 6 , against the assumption made about r. (2) K contains a ; it does not contain e, because ( a , e ) F#I'.Since r - b2- e is 4-connected, there are three paths from K - a to { b , , d , , d 2 } : contract each of them to a vertex; contract (a, b2) and ( x l ,x 2 ) ; contract ( b l , cl) and (b2,c2); finally contract U2 U e to a vertex; you get a K6 again.

Hadwiger's conjecture

( b = 6)

147

(3) K contains e (and not a ) . Since r - d2 - a is 4-connected, there are three paths from K - e to { b , ,d l , b 2 } : contract each of them to a vertex; contract (e, d2) and (xl, x 2 ) ; contract ( b l , c l ) and ( b2 ,c2); finally contract U,U a to a vertex: again, you get a K 6 . Since a v6 of r belongs to a clique K4, a v6 different from x1 or x 2 cannot exist in r. 0

5. Concluding remarks It follows from the two preceding sections that, except when it consists of a K 2 made of two G-vertices, the 6-vertex subgraph of r can consist only of: (1) connected components, all made of A-vertices of J-vertices (or is exclusive here, by (3.4)) and (2) isolated vertices. It is important to remark that r may contain no vertex of degree 6. Mader [12] has proved that if a graph has p vertices and q edges, p 2 6 and q 2 4p - 9, then it can be contracted to K6. T h i s for r q G 4 i - 10 and r must contain a v6 or a v,; if it contains no 6-vertex, it must have at kast twenty 7-vertices. But, after our examination, & can have more than a hundred different types. We are greatly indebted to the referee for precise remarks on the paper, and to Mrs Florence Depraz for carefully reading over the English text.

Note added in proof Types F and J are reducible.

References [I] K. Appel and W. Haken, Every planar map is four colorat : Part 1, Disc arging, Illinois J. Math. 21 (1977) 429-490. [2] K. Appel, W. Haken and J. Koch, Every planar map is four colorable: Part 2, Reducibility, Illinois J. Math. 21 (1977) 491-567. [3] C. Berge, Graphes et Hypergraphes (Paris, Dunod, 1973, 28me Cd.), 516 pp. (English translation: North-Holland, Amsterdam, 1973). [4] G.A. Dirac, A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc. 27 (1952) 85-92. [5] G.A. Dirac, The structure of chromatic graphs, Fund. Math. 40 (1953) 42-55. [6] G.A. Dirac, Extensions of Menger's Theorem, J . London Math. SOC.38 (1963) 148-161. [7] G.A. Dirac, On the structure of 5- and &chromatic abstract graphs, J. Reine Angew. Math. 214 (1964) 43-52. [8] G.A. Dirac, Homomorphism theorems for graphs, Math. Ann. 153 (1964) 69-80. [9] H. Hadwiger, Uber eine Klassifikation der Streckenkomplexe, Vierteljahresschr. Naturforsch. Ges. Zurich 88 (1943) 133-142.

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[lo] I.T. Jakobsen, A homomorphism theorem with an application to the conjecture of Hadwiger, Studia Sci. Math. Hungar. 6 (1971) 151-160. [ l l ] W. Mader, Uber trennende Eckenmengen in homomorphiekritischen Graphen, Math. Ann. 175 (1968) 243-252. 1121 W. Mader, Homomorphiesatze fur Graphen, Math. Ann. 178 (1968) 154-168. [ 131 J. Mayer, Conjecture de Hadwiger: Un graphe k-chromatique contraction-critique n’est pas k-rkgulier, in Graph Theory in Memory of G.A. Dirac, Annals of Discrete Mathematics. 1141 K. Wagner, Uber eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937) 570-590.

Discrete Mathematics 74 (1989) 149-150 North-Holland

149

ABOUT COLORINGS, STABILITY AND PATHS IN DIRECTED GRAPHS Henry MEYNIEL C.N.R.S., Paris, France

In [2] Berge asked: In a digraph, does there exist an optimal coloring and a path meeting each color class exactly once? We give a negative answer to this question when the chromatic number is >5. H e also asked: Does every directed graph G have a maximum stable set (with IS1 = a ( G ) , the stability number of the digraph) and a partition of vertex-set into paths p,, p,, . . . , pn(c) such that lpi f l SI = 1 for all i? We give here a negative answer to this question.

1. Introduction: definitions and notations Definitions and notations are classical. See [l]for instance. In [2], Berge asked whether every directed graph has an optimal coloring and a path meeting each color exactly once. He proved this for perfect graphs, symmetric graphs and graphs with chromatic number 3. We show here that the answer is negative for graphs with chromatic number 2 5 and we suggest some new problems. Also he asked the following question: In every 1-graph, does there exist a maximum stable set S (a set with IS1 = a ( G ) ,the stability number of the digraph) .and a partition of the vertex-set into paths p , , p2, . . . , such that [pin SI = 1 for all i? We give here an example where the answer is negative.

2. The examples (a)Example of the first problem It is not difficult to see that there is a unique optimal coloring in 5 colors (as indicated on Fig. 1) and that every path of length 5 (hence containing either (a, b, c ) or (a', b ' , c ' ) ) meets the same color twice. (By adding new vertices connected to all the preceding, we can obtain examples for every value of y with y 3 5).

Further questions Problem 1. Does the same property hold true for graphs with chromatic number 4? 0012-365X/89/$3.50 @ 1989, Elsevier Science Publishers B.V. (North-Holland)

H.Meyniel

150

a l l arcs going from Ci t o C2 except ( d , d ’ )

Fig. 1.

( p ) Example of the second problem Consider a collection of ten antidirected cycles of length five (by antidirected cycle of length five, we mean a cycle of length five directed in such a way that there exists a unique path of length 3) and let C, = (a,, b,, c,, d,, e l ) , 1s i < 10 be these antidirected cycles. For each cycle we will choose an orientation such that (a,, b,, cJ, 1s i < 10, is the unique path of length 3. Furthermore, we add all directed edges with initial vertex on the C,’s, 1G i s 5 and terminal vertex on the C,’s 6 sj s 10, except for the edges (a,, a,), (al, c,), 6 s 1 s 10. It is easy to check that there is a unique maximum independent set S with IS1 = n ( G )= 11 that is the set of vertices {ul, a6, c h , a 7 , c7, a8, c8, a9, c9, a,,,, Cm>.

Now, we prove that there is no partition of V ( G ) into 11 paths meeting S only once. Suppose the contrary. The paths of such a partition cannot contain the subpaths (u,, b,, c,), 6 S i S 10, otherwise these paths would intersect S twice and so the partition must contain at most 5 paths of length at most 5 (containing (a,, b,, c,), 1s i s 5), the remaining paths being of length at most 4. Hence the partition can cover at most 5 X 5 + 6 X 4 = 49 vertices, while the graph has 5 x 10 = 50 vertices. Please note that in every graph there exists a maximal (for inclusion) stable set S’ and a partition of the vertex-set into paths p,, p 2 , . . . , pk such that lp,n S’I = 1, 1 s i s k. This is a trivial consequence of Theorem 11, Corollary 3 in [l], see also [2]. We are pleased to thank Professors Berge, Duchet and Hamidoune for their helpful comments on this paper.

References [ 11 C. Berge, Graphs (North-Holland Publishing Company, second revised edition, 1985). [2] C. Berge, Diperfect graphs, Combinatorica 2 (3) (1982) 213-222. [3] N. Linial, Covering digraphs by paths, Discrete Math. 23 (1978) 257-272.

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Discrete Mathematics 74 (1989) 151-157 North-Holland

ON THE HARMONIOUS CHROMATIC NUMBER OF A GRAPH John MITCHEM Mathematics and Computer Science Department, San Jose State University, San Jose, California 95192, U.S . A.

The harmonious chromatic number of a graph C , denoted by h ( C ) , is the least number of colors which can be assigned to the vertices of G such that adjacent vertices are colored differently and any two distinct edges have different color pairs. This is a slight variation of a definition given independently by Hopcroft and Krishnamoorthy and by Frank, Harary, and Plantholt. D. Johnson has shown that determining h ( G ) is an NP-complete problem. In this paper we give various other theorems on harmonious chromatic number and discuss various open questions.

The harmonious chromatic number of a graph C , denoted by h ( G ) , is the least number of colors which can be assigned to the vertices of G such that each vertex has exactly one color, adjacent vertices have different colors, and any two edges have different color pairs. Here a color pair for edge e is the set of colors on the vertices of e. This parameter was introduced by Miller and Pritikin [6]. It is, however, only a slight variation of the parameter h '(G) introduced independently by Frank, Harary, and Plantholt [ l ] and Hopcroft and Krishnamoorthy [2]. The definition of h'(G) is the same as h ( G ) except that adjacent vertices are not required to have different colors. Thus it is clear that h'(G)s h ( G ) for each graph G. Since the edges must have different color pairs it is apparent that if h ( G ) = h, then the binomial coefficient C(h, 2) 2 IE(G)(.Unless stated otherwise in this paper we let k be the smallest integer such that C ( k , 2) 2 IE(C)l, where G is the graph currently under consideration. Finding the harmonious chromatic number of a graph is apparently quite difficult. In an appendix to [2] David S. Johnson gives an elegant proof that determining h ( C ) (or h '(G)) is NP-complete. However, the harmonious chromatic number of P,,, the path with n vertices has been determined.

Theorem 1 [5]. If k is odd or if k is even and n - 1 = C ( k , 2) - j , j = (k k J 2 , . . . , k - 2, then h(P,) = k. Otherwise, h(P,,) = k + 1.

- 2)/2,

An obvious next problem is to find the harmonious chromatic number of trees. In fact one might guess that for any tree T, h ( T ) is close to k. We show that this is not true, rather a tree T of order p can have h ( T ) be any value between k and 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

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p inclusive. Before formally stating this fact as a theorem we state an obvious proposition and prove a lemma.

Proposition 2. For every graph G h ( C )2 D ( G ) + 1, where D ( G ) denotes the maximum degree of G.

Lemma 3. Let k be the least integer such that C ( k , 2) 3 p - 1. Then C ( k - 1, 2) 3 p - k, and k - 1 is the smallest integer t such that C(t, 2) 2 p - k unless C ( k - 1,2) = p - 2. In that case k - 2 is the smallest such t. Proof. Since k ( k - 1)/2 2 p - 1, we have k ( k - 1)/2 - ( k - 1) 3 (p - 1) - ( k - 1) and thus C ( k - 1,2) 2 p - k. Note that p and k are fixed integers, and we suppose that k - 1 is not the smallest integer t such that C(t, 2) 3 p - k. Thus C ( k - 2,2) 2 p - k. However, from our choice of k we have that p - 1> C(k - 2 , 2 ) + ( k - 2) 2 ( p - k ) + ( k 2). Thus p - 1> C ( k - 1, 2) 3 p - 2. I t follows that C ( k - 1, 2) = p - 2, and furthermore C ( k - 2,2) = C ( k - 1,2) - ( k - 2) = ( p - 2) - ( k - 2) = p - k , which completes the proof of the lemma. 0

Theorem 4. Let k be the least integer such that C ( k , 2) 2 p - 1. Then for each t, k G t G p , there is a tree T of order p such that h( T ) = t. Proof. Let T be the tree of order p which consists of a path vo, u & - ~v,k , . . . , vpp2, together with vertices vl,v2, v 3 , ,. . , vk-2 each of which is adjacent to vo. We note that by Proposition 2, h ( T )3 k. Now for each i, 0 c i s k - 1, color vi with i. Since k is the least integer such that C ( k , 2) > p - 1 we have, by Lemma 3, that k - 1 is the least integer such that C ( k - 1, 2) 3 p - k unless C ( k - 1, 2) = p - 2. The subgraph T’ of T induced by v & - v~k ,, . . . , is p p - k + l . According to Theorem 1 and Lemma 3, h ( T ’ ) = k - 2 or k - 1 or k. If h ( T ’ )S k - 1, then we color T’ with colors 1, 2, . . . , k - 1, where vkPl receives color k - 1. It follows that h ( T ) = k . Similarly if h ( T ’ ) = k, then we obtain h ( T ) = k 1. Now by removing the vertex farthest from vo and joining it to uo we successively create trees with harmonious chromatic number k + 1, k + 2, . . . ,p. The proof is thus = k , we have not constructed a tree T” complete except when h(T‘) = h(Pp--k+l) of order p with h(T”)= k. In this case k - 1 was the least integer t such that C(t, 2) 2 p - k. From Theorem 1 we have that k - 1 is even and thus k is odd. Hence h(Pp)= k and for each t, k s t S p , we have constructed a tree Tp with h(T,)=k.

+

Corollary 5. Let p S q S C ( p , 2). Let k be the least integer such that C ( k , 2) 2 q. Then for any t, k S t S p there exists a graph G with p vertices and q edges such that h ( G ) = t.

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Harmonious chromatic number of a graph

Proof. Let T be the tree of order p given in the theorem such that h ( T ) = t. Now C(t, 2) 3 C ( k , 2) 2 q and thus C(t, 2) - ( p - 1) a q - ( p - 1). That is the number of pairs of non-adjacent colors is at least q - p + 1, which is the number of edges we need to add to T to form a graph with q edges. Hence we can insert q - p + 1 edges joining non-adjacent colors of T. The resulting graph G has p vertices, q edges, and h ( G ) = t. 0 We now determine the harmonious chromatic number of 2-regular graphs with at most 2 components.

Theorem 6. Let r + s = p where 3 C r G s, and k be the least integer such that C(k, 2) " p . If k is odd and p # C(k, 2 ) - i , i = 1 , 2 , then h(C,) = h(C, U C,) = k . q k is even andp # C ( k , 2) - i, i = 0, 1, . . . , k / 2 - 1, then h(C,) = h(C, U C,) = k. Otherwise, h(C,) = h(C, U C,) = k + 1. Proof. Suppose that C, can be harmoniously colored with 1 , 2 , . . . ,t. Then that coloring generates an Eulerian circuit which is a subgraph with p edges of k,. Conversely if K , has as Eulerian subgraph H with p edges, then any Eulerian circuit with these p edges corresponds to a t-coloring of C,. We use these observations to first show that the appropriate k or k + 1 is the value of h(C,,). Case i. k is odd. If p = C ( k , 2), then K k , which I will call El, is the required Eulerian subgraph. If p = C ( k , 2 ) - i , i = 3, 4, . . . , k - 2, then the required Eulerian graph, denoted EZ,is Kk - C1. If p = C ( k , 2) - i , i = 1 or 2, then Kk has no Eulerian subgraph with p edges. Then we consider subgraphs E3 and E4 of K k + , . Let E3 be the graph formed by removing the edges of a P4 from Kk and joining its endvertices to a vertex u which is not in K k . So E, is an Eulerian subgraph of K k + , with exactly C ( k , 2) - 1 edges. Thus h(C,) = k 1 when p = C ( k , 2) - 1. Similarly if p = C(k, 2 ) - 2, we see that h(C,) = k 1 by forming E4 by removing the edges of Ps from Kk and joining the endvertices to a new vertex u.

+

+

Case ii. k is even. In this case Kk has all vertices of odd degree, so the removal of i lines i = 0, 1, . . . , (k/2) - 1, cannot result in an Eulerian graph. Thus if p = C ( k , 2)-i, i = O , 1 , . . . , ( k / 2 )- 1, h ( C , ) > k + 1. In order to see that equality holds, consider the Eulerian graph K k + l . Let t = C(k + 1, 2 ) - p . Now if t s k 1, remove C, from K k + l . If t > k + 1, remove t edges which form two edge-disjoint cycles using all k + 1 vertices of & + l a In either case the result is an Eulerian graph, which we call E5, so that h(C,) = k 1.

+

+

+

If p = C(k, 2) - i, i = k / 2 , . . . , k - 2, then let i = (k/2) t. We remove i edges from Kk as follows. Remove the edges of the complete bipartite graph Kl,zt+l together with (k/2) - f - 1 independent edges which are also independent from

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the edges of K1,2r+l.Thus ( k / 2 )- t - 1+ 2t + 1= ( k / 2 )+ t = i edges have been removed from Kk creating an Eulerian subgraph E6 with C ( k , 2 ) - i edges. It follows that h(C,) = k. We now use the various Eulerian graphs El described above to show h(C, U C,) = h(C,+,). First we observe that it is routine to verify that h(C, U C,) =h(C,+,) for k c 8 . Thus assume k > 8 . In each Ei, a maximum degree vertex q,is non-adjacent to at most 2 other vertices. Find an Eulerian subgraph H consisting of r edges of El - vo. Then consider the graph H , consisting of all of Ei - H except its isolated vertices. Then H I is Eulerian with s edges. Hence h(C, U C,) = h(C,+,), and the theorem is proved. 0 It is obvious that the proof technique used above can be used to show that many graphs which are the union of vertex disjoint cycles have harmonious chromatic number k or k + 1. It is simply necessary to show that the appropriate Ei can be partitioned into appropriate Eulerian subgraphs. A number of researchers have investigated the partitioning of E ( K 2 , + l )into s edge disjoint copies of C,. It follows, for example, that whenever such a partition exists then the graph which consists of s vertex disjoint copies of C,, has harmonious chromatic number k = 2t + 1. The most recent work in partitioning complete graphs into cycles is by Jackson [3]. In that paper it is shown that if 2t + 1= qr where q and r are odd or t = q r , then E ( K 2 , + l )can be partitioned into disjoint copies of C,. These results improve earlier results due to Kotzig [4] and Rosa [7]. We now give another easy lower bound for h ( G ) , where G is regular.

Proposition 7 . Let h ( G ) = h where G is r-regular of order p . Then h a [ p l h l r + 1. Proof. Harmoniously color G with h colors. Now the average color class size is p l h , so one color class has size at least [ p l h ] . But that color occurs on [ p / h l vertices each of degree r. Thus that color is adjacent with at least [ p / h l r other colors, and the total number of colors h a [ p / h l r 1.

+

Corollary 8. Let G be r-regular, k be the least integer such that C ( k , 2 ) 2 IE(G)I, and h be the least integer such that h 2 [ l V ( G ) l / h ] r+ 1. Then h ( G ) 2 max{h, k } . In [6] Miller and Pritikin showed that if B, is the complete binary tree on n levels, then h(B,) = O(2,"). Later, in [ 5 ] Lee and Mitchem proved Theorem 9, which trivially implies that h(B,) = 0(2n'2).

Theorem 9. [4]. For any graph G , h ( G ) 6 ( D 2+ 1) [GI. We give a significantly more efficient coloring of B, in the two theorems below. Before proving the theorems, we state the following easily verified lemma.

Harmonious chromatic number of a graph

155

Lemma 10. Let k be the least integer such that C ( k , 2) 3 IE(B2r+l)l, where r 3 2, then k = Y+' 1.

+

Theorem 11. For each r 2 5 , h(BZr+,)6 2('+') + 2(r-4) + (r - 2) = k (r - 3) where k is the smallest integer such that C ( k , 2) 3 lE(B2r+l)l.

+ 2('-') +

Proof. We first color B7, use that coloring to color B,, and use B9 to show how to color B2r+3given a coloring of B2r+l,r 3 4. Color B7 with 24 3 colors as shown in Fig. 1. The colors of the form C, are called special colors. Note that the root is colored with special color C3 and no two special colors C, and C, are adjacent. We now construct B9 with a coloring with 25 + 4 colors. Take two copies, H1 and H2, of B7.Color H1 as shown in Fig. 1. For each vertex of Hl colored with a number n we color the analogous vertex of H2 with n 24. Furthermore, in H2 interchange the use of colors C2 and C3 from H I . Now construct B9 by adding a root v adjacent to the roots of H1 and H 2 , and color v with C4. Also join 2 new vertices to each endvertex of HIU H2. Now the endvertices of B, which are adjacent to vertices of H1 will be colored with 17,. . . ,32 and the other endvertices will be colored with 1, . . . , 16. Specifically assign colors 17, . . . ,24 to the 8 endvertices adjacent to the vertices of each color i , 1 s i G 8 in H,. Also assign colors 25,26,. . . ,32 to the 8 endvertices adjacent to each color j , 9 < j s 16. Similarly use colors 1, . . . , 8 on the 8 endvertices adjacent to each i, 25 s i G32, in H2 and colors 9 , . . . ,16 on the endvertices adjacent to each i, 17 s i G 24. Thus we have a proper coloring of B9 with Z5 + 4 colors. In general suppose we have a coloring of B2r+lwith 2' 4 colors if r = 4, and 2('+') 2(r-4) ( r - 2) colors if r 3 5. We now color B2r+3.Let H , and H2 be copies of BZr+,.Color H1 as &+, was originally colored. If vertex 2) of H1 is colored with number n, color the corresponding vertex of H2 with color n + Yfl.

+

+

+

+

+

The vertices on the tap level o f this tree are labeled consecutlvely: 1 ; - ~ 4 1 ; - 3 4 1 ~ 3 4 i i 3 4 5 6 7 8 ~ ~ 7 8 ~ ~ 7 a 5 ~ 7 ~ 3 i o i i i 1 1 1;- 9 1 0 1 1 1- 7 1 0 1 1 1s 13 1 4 15 16 13 14 15 1 € 13 14 15 16 13 14 15 16

Fig. 1. Binary tree 8,.

1. Mitchem

156

If a vertex of H1 is colored with Cilet the corresponding vertex of H, also have color C,. Now form BZr+3 from HI and H,, a new vertex which is adjacent to the two roots of HI and H,, and 2"+' new endvertices such that 2 are adjacent to each endvertex of H I U H,. We now color the endvertices of BZr+3. Note that the endvertices of HI are colored with 1 , 2 , . . . , 2 ' + l and the endvertices of H, with 2'+' 1, . . . , 2'+'. Furthermore each color is used exactly T-'times on the endvertices of HI U H2. Assign colors 2'+' 1, . . . ,2'+l+ 2' to the 2' endvertices adjacent to each color i, 1s i S 2' in Hl. Also assign colors 2'+l+ 2' + 1, . . . , 2'+' to the 2' endvertices adjacent to each of the colors i, 2'+ 1Si ST+' in Hi.Similarly use colors 1, . . . , 2' on the 2' endvertices adjacent to each color i, 2'+' 2' + 1 s i G T+', and use colors 2'+ 1 , . . . ,T+' on the endvertices adjacent to each color i, y+' + 1 s j s 2'+' + 2'. Finally we color the root v of BZr+3 and recolor certain vertices of H,. If r = 4, then change the color of the root of H2 to Cs and use 26 on v. Then h(B,,) = h(Bzr+3)S 26 + 5 = 2'+' + 2'-3+ ( r - 1). If r 2 5, then corrresponding vertices on level r - 3 of HI and Hz have the same special colors and are adjacent to the same special colors on level r - 2. Use 2r-4 new special colors on these 2'-4 vertices on level r - 3 of H,. Now if the roots of HI and H2 are colored with numbers we color v with yet another new special color. If the roots of H1 and H, are colored with a special color, then recolor the root of H2 with another new special color and color v with the number 2'+' - ( r - 5). Now we have used T+* numbers and 2'-4+ T4 + ( r - 2 ) + 1 = r3 + ( r - 1) special colors on &+3. We now check that the assignment of 2'+' + 2'-3 ( r - 1) colors to B2r+3is harmonious. By inductive assumption the coloring of H1 is harmonious, and it follows that the coloring of H2 is also. Now H I U H2 is harmoniously colored because the numbers used in H2 are not used in H I and the only time special colors are adjacent in H1 we have introduced new special colors in H,. Furthermore, it is clear that we have harmoniously colored the root and the endvertices of B2r+3.Thus the theorem is proved. 0

+

+

+

+

Theorem 12. For r 2 6, h(Bzr) 3(2'-')

+ 2'-' + ( r - 3).

Proof. Color BZrp1with 2'+2'-'+ ( r - 3 ) colors as in Theorem 11. Now the endvertices of B2'-' are colored with 2' colors, each occurring 2'-' times. Join two new vertices to each endvertex. For each color i on level 2r - 1 color one of its level 2r neighbors with each of the colors 2' + 1, . . . , 2' + 2'-'. Thus we have h(B,,) S 2' + 2'-s + ( r - 3) + 2'-' = 3(2'-') + T P 5+ ( r - 3). 0 22r-2

Although the determination of h ( T ) when T is a complete n-ary tree with t levels is apparently difficult we have found the exact value of h ( T ) when T has t = 3 levels.

Harmonious chromatic number of a graph

157

Theorem 13. Let T be a complete n-ary tree on 3 levels. Then h ( T ) = [3(n + 1)/21. Proof. Color the root v with A and its neighbors vl,.. . , v, with 1 , . . . , n respectively. Let t = [ ( n - 1)/2]. For each level 2 vertex color n - t of its level 3 neighbors with color n + 1, n + 2, . . . , n + (n - t) respectively. Then color the remaining neighbors of vj with j + 1, j +2, . . . ,j + t (modn). Now color j , l 6 j s n is adjacent to colors A, n + 1 , . . . , 2n - t and colors j + 1, j 1, . . . ,j + t, j - t (mod n ) . It follows that we have a 2n + 1 - t = [3(n + 1)/21 harmonious coloring of T. We show now that there is no harmonious coloring of T with fewer colors. The color A , used on the root v, cannot be used on any other vertex of T. There are n2 edges which are not incident with v. At most C(n, 2) of these edges can join vertices colored 1, . . . , n. Thus at least n2 - C(n, 2) = n ( n + 1)/2 of the edges must have its level 3 vertex colored with a new color. Hence there must be a level 2 vertex which is adjacent to at least [ ( n + 1)21 new colors and h ( T ) n + 1 + [ ( n 1)/2] = [3(n + 1)/2]. 0

+

References [l] 0. Frank, F. Harary and M. Plantholt, The line-distinguishing chromatic number of a graph, Ars Combinatorica 14 (1982) 241-252. [2] J . Hopcroft and M.S. Krishnamoorthy, On the harmonious colorings of graphs, SIAM J . Alg. Disc. Math. 4 (1983) 306-31 1. [3] B. Jackson, Some cycle decompositions of complete graphs, preprint, 24 pages, [4] A. Kotzig, On the decomposition of the complete graph into 4k-gons, Mat. Fyz. Casopis 15 (1965) 229-233 (in Russian). [5] S . Lee and J. Mitchem, An upper bound for the harmonious chromatic number of a graph, J. Graph Theory 11 (1987) 565-567. [6] Z. Miller and D. Pritikin, The harmonious coloring number of a graph, preprint, 24 pages. [7] A. Rosa, On the cyclic decomposition of the complete graph into (4m +2)-gons, Mat. Fyz. casopis, 16 (1966) 349-353.

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Discrete Mathematics 74 (1989) 159-171 North-Holland

159

WEAK BIPOLARIZABLE GRAPHS Stephan OLARIU Department of Computer Science, Old Dominion University, Norfolk, V A 23508, U.S.A .

We characterize a new class of perfectly orderable graphs and give a polynomial-time recognition algorithm, together with linear-time optimization algorithms for this class of graphs.

1. Introduction A linear order < on the set of vertices of a graph C is perfect in the sense of Chqatal [3] if no induced path with vertices a, b, c, d and edges ab, bc, cd has a < b and d < c . Graphs which admit a perfect order are termed perfectly orderable. Recognizing perfectly orderable graphs in polynomial time seems to be a difficult problem. Quite naturally, this motivated the study of particular classes of perfectly orderable graphs. Such classes have been studied by Golumbic, Monma and Trotter [7], ChvAtal, Hoang, Mahadev, and de Werra [4],Hoang and Khouzam [9], and Preissmann, de Werra and Mahadev [12]. Recently, Hertz and de Werra [8] proposed to call a graph G bipolarizable if G admits a linear order < on the set V of its vertices such that b < a and c < d whenever { a , b, c, d } induces a path in G with edges ab, bc, cd. They characterize bipolarizable graphs by forbidden subgraphs and prove that both bipolarizable graphs and their complements are perfectly orderable. In this paper we first define and characterize the class of weak bipolarizable graphs which properly contain the class of bipolarizable graphs. This characterization can be exploited to obtain a polynomial-time recognition algorithm for weak bipolarizable graphs. Finally, given a weak bipolarizable graph G, we show how an algorithm of Rose, Tarjan and Lueker [13] can be used to obtain efficiently a linear order on the vertices of G. As soon as this is done, an algorithm of ChvAtal, Hoang, Mahadev and de Werra [4] can be used to optimize weak bipolarizable graphs in linear time. Given a graph G, we shall let G denote the complement of G; if x is a vertex in G , then N,(x) stands for the set of all the vertices in G which are adjacent to x; N&) denotes the set of all the vertices in G which are adjacent to x in G (whenever possible, we shall write simply N ( x ) and N ' ( x ) ) . We shall let GHstand for the subgraph of C induced by H ; Ck(Pk) will stand for an induced chordless cycle (path) with k vertices. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V.(North-Holland)

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El

Fl

F2

Fig. 1

Fig. 2.

A graph G is called triangulated if every cycle of length greater than three in G has a chord. Dirac [5] proved that every triangulated graph contains a simplicia1 vertex: this is a vertex w such that N ( w ) is a clique. A proper subset H (IHI 2 2 ) of vertices of G will be referred to as homogeneous if every vertex outside H is either adjacent to all the vertices in H or to none of them. A graph G will be called a weak bipolarizable graph if G has no induced subgraph isomorphic to C, ( k 3 5 ) , p5 or to one of the graphs F,, 6 in Fig. 1. Since every forbidden subgraph of a weak bipolarizable graph is also a forbidden subgraph of a bipolarizable graph it follows that every bipolarizable graph is also weak bipolarizable. In addition, note that the graph in Fig. 2 is a weak bipolarizable graph but not a bipolarizable graph. Therefore, the class of weak bipolarizable graphs properly contains the class of bipolarizable graphs. As it turns out, the class of weak bipolarizable graphs also contains all triangulated graphs, all Welsh-Powell opposition graphs (see Olariu [lo]), all superbrittle graphs (see Preissmann, de Werra, and Mahadev [12]) and all superfragile graphs (see Preissmann, de Werra, and Mahadev [12]).

2. The results

The following theorem provides a characterization of the class of weak bipolarizable graphs.

Weak bipolarizable g r a p h

161

Theorem 1. For a graph G the following three statements are equivalent: (i) G is a weak bipolarizable graph (ii) Every induced subgraph H of G is triangulated, or H contains a homogeneous set which induces a connected subgraph of G (iii) Every induced subgraph H of G is triangulated or H contains a homogeneous set. Proof. To prove the implication (i)+ (ii), consider a graph G = (V, E) that satisfies (i). Assuming the implication (i)+ (ii) true for graphs with fewer vertices than G, we only need prove that G itself satisfies (ii). If G contains a homogeneous set with the property mentioned in (ii), then we are done. We shall assume, therefore, that G contains no such homogeneous set. We want to show that, with this assumption, G is triangulated. For this purpose, we only need show that G has no induced C4. Suppose not; now some vertices x , y , z , t induce a C4 with edges xy, y z , zt, tx E E. Consider the component F of the subgraph of G induced by N ( y ) n N ( t ) , containing x and z . By assumption, F is not a homogeneous set, and thus there exists a vertex u in V - F, adjacent to some but not all vertices in F. By connectedness of F in G, we find non-adjacent vertices x ’ , z‘ in F such that ux’ E E and uz‘ $ E. Trivially, u is not in N ( y ) n N ( t ) , and hence u is adjacent to at most one of y , t. If u is adjacent to precisely one of y , t then { u , x ’ , y , z‘, t } induces a p5,a contradiction. Now u is adjacent to neither y nor t. Write N ( x ’ )fl N ( z ‘ ) = U, U U, in such a way that every vertex in Ul is adjacent to u, and no vertex in Uo is adjacent to u. By the above argument, y and t belong to U, and thus 1 &,I 2 2. Observe that every vertex in Ul is adjacent to every vertex in U,, for otherwise { u , p , q, x ’ , z ’ } induces a 4, for any non-adjacent vertices p in U, and q in U , . Consider the connected component H of the subgraph of G induced by U, that contains the vertices y and t. Since H is not homogeneous, there must exist a vertex v in V - H adjacent to some but not all vertices in H. Trivially, v is not in { x ’ , z’, u } U U, U U , . By connectedness of H in G, we find non-adjacent vertices y‘, t‘ in H such that vy’ E E, vt’ $ E. Now v is adjacent to at most one of the vertices x ’ and z’. If v is adjacent to precisely one of them, then { v , x ’ , z ’ , y ’ , t ’ } induces a p’, a contradiction. Thus, v is adjacent to neither x ’ nor z’. By definition of U,, u is adjacent to neither y ’ nor t‘. However, this implies that {u, v, x ’ , y ’ , z’, t ’ } induces either an Z$ or an F , , depending on whether or not uv E E. This proves that G is triangulated, as claimed.

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162

The implication (ii)+ (iii) is trivial. To prove (iii)+ (i) we only need observe that if a graph G does not satisfy (i), then (iii) fails. This completes the proof of the theorem. 0 Consider a graph G1 and a graph G2 containing at least two vertices, and let v be an arbitrary vertex in G1. It is customary to say that a graph G arises from C , and G, by substitution if G is obtained as follows: (*) delete the vertex v from C , , and (**) join each vertex in G2 by an edge to every neighbour of v in GI. If G arises by substitution from graphs G1 and G,, then we shall say that G is substitution-composite. It is a simple observation that a graph G is substitutioncomposite if and only if G contains a homogeneous set. Now the equivalence (i) @ (iii) in Theorem 1 can be rephrased as follows.

Corollary la. A graph G is weak bipolarizable if and only if every induced subgraph of G is either triangulated or substitution -composite. Let Y be the class of graphs defined as follows: (v1) if G is triangulated, then C is in Y. (v2) if G' is obtained from a graph GI in Y and a triangulated graph G, by substitution, then G' is in Y.

Theorem 2. Y is precisely the class of weak bipolarizable graphs. Proof. To begin, we claim that every graph in Y is weak bipolarizable. For this purpose, let C be an arbitrary in Y. Assuming (1) to be true for all graphs with fewer vertices than G, we only need prove that G itself is weak bipolarizable. This, however, follows immediately from the observation that G is either triangulated or it contains a homogeneous set. Now Theorem 1 guarantees that G is weak bipolarizable. Conversely, we claim that every weak bipolarizable graph is in Y.

(2)

Let G be a weak bipolarizable graph. Assume that (2) holds for all graphs with fewer vertices than G. If G is triangulated, then G is in Y b y (1$1).Now we may assume that C is not triangulated. Theorem 1 guarantees that G contains a homogeneous set. Let H be a minimal homogeneous set in C (here, minimal is meant with respect to set inclusion, not cardinality). By Theorem 1, H must be triangulated. By the induction hypothesis, the graph induced by ( V - H) U { h } is in Y, for any choice of h in H. Hence, by (v2), G itself is in Y, as claimed. 0

Weak bipolarizable graphs

163

We shall refer to a graph G which contains no homogeneous set as substitution-prime. For later reference we shall make the following simple observation, whose justification is immediate.

Observation 1. If a graph G with a homogeneous set H contains an induced substitution-primesubgraph F, then either every vertex of F belongs to H or else F and H have at most one vertex in common. Let E be a class of graphs such that all forbidden graphs for E are substitution-prime.

Theorem 3. If G arises by substitution from graphs GI and G2in 2,then G is also in 2. Proof, Suppose not; now G must contain an induced subgraph F isomorphic to a forbidden graph for the class E . By assumption, F is an induced subgraph of neither Gi (i = 1, 2). By Observation 1, F has precisely one vertex in common with G2. However, this implies that GI has an induced subgraph isomorphic to F, a contradiction. 0 Theorem 1 and Theorem 3 provide the basis for a polynomial-time recognition algorithm for weak bipolarizable graphs. In addition, we shall rely on algorithms to recognize triangulated graphs (see, for example, Rose, Tarjan and Leuker [13]), as well as polynomial time algorithms to detect the presence of a homogeneous set in a graph (see Spinrad [ll]). The following two-step algorithm recognizes weak bipolarizable graphs.

Algorithm Recognize( G ) ; {Input: A graph G = (V, E ) . Output: ‘Yes’ if G is weak bipolarizable; ‘No’ otherwise.} Step 1. Call Check(G) Step 2. Return(‘Yes’); stop. Procedure Check( G ) ; begin if G is not triangulated then if G is not substitution-composite then begin return(‘N0’); stop end

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else begin {now C contains a homogeneous set H ; let H' stand for the set of all the remaining vertices in G.} Check( G H ;) pick an arbitrary vertex h in H; Check(G{h)"H,) end end; {Check} The correctness of this algorithm follows directly from Theorem 1 and Theorem 2. Furthermore, its running time is clearly bounded by O(n3):to see this, note that Check is invoked O(n) times for a graph G with n vertices. Each invocation of Check runs in O(n') time since the recognition of triangulated graphs [13] and the detection of a homogeneous set [ l l ] are both performed in 0(n2)time. Given a P4with vertices a, b, c, d and edges ab, bc, cd, the vertices a and d are called endpoints and the vertices b, c are called midpoints of the P4. We shall say that a vertex x in a graph G is semi-simplicial if x is midpoint of no P4 in G. Trivially, every simplicia1 vertex is also semi-simplicial, but not conversely. A linear order < on the vertex-set V of G is said to be a (semi-)perfect elimination if the corresponding ordering x , , x 2 , . . . ,x, of the vertices of G with xi <xi iff i <j satisfies xi is a (semi-)simplicia1vertex in G , , , , ,,..., + ,..) for every i.

(3)

It is immediate that every graph G with a semi-perfect elimination is brittle in the sense of Chv6tal [2]: every induced subgraph H of C contains a vertex which is either midpoint or endpoint of no P4 in H. Furthermore, it is an easy observation that every brittle graph is also perfectly orderable. Hertz and De Werra [8] demonstrated that bipolarizable graphs are brittle; we extend their result by showing that weak bipolarizable graphs are also brittle. Actually, we also exhibit a linear-time (and thus optimal) algorithm that finds a perfect order for any weak bipolarizable graph. The details are spelled out in Theorem 4. Rose, Tarjan and Lueker [13] proposed a linear-time search technique which is referred to as Lexicographic Breadth-First Search (LBFS, for short). They prove that a graph G is triangulated if, and only if, any ordering of the vertices of G produced by LBFS is a perfect elimination. We shall use their algorithm to obtain a perfect order on the set of vertices of a weak bipolarizable graph. To make our exposition self-contained, we give the details of LBFS.

Weak bipolarizable graphs

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procedure LBFS(G); {Input: the adjacency list of G ; Output: an ordering u of the vertices of G} begin for every vertex w in V do label(w) -0; for i t n downto 1 do begin pick an unnumbered vertex v with the largest label; u(v) c i ; {assign to v number i } for each unnumbered w E N ( v ) do add i to label ( w ) end end; Note that we can think of the output of LBFS as a linear order < on V by setting u
whenever

u ( u ) < u(v).

It is immediate (see Golumbic [6]) that every linear order produced by LBFS satisfies the following property. (P) a < b, b < c, ac E El and bc $ E imply the existence of a vertex b' with bb' E E, ab' $ E and c < b'. We are now in a position to state our next result.

Theorem 4. If G is a weak bipolarizable graph, then every ordering of the vertices of G produced by LBFS is a semi-perfect elimination. Our proof of Theorem 4 uses the following result of an independent interest.

Proposition 1. Let G be a graph with no induced Ps, Ck( k 3 5 ) and no 6, and let < be a linear order on the vertex-set of G satisfying the property ( P ) . Then for every vertices a, 6 , c, d with a
(4)

Proof of Proposition 1. Write G = (V, E), and let < be a linear order on V satisfying the hypothesis of Proposition 1. If < is a semi-perfect elimination, then the conclusion follows trivially. We may, therefore, assume that < is not a semi-perfect elimination. If the statement is false then we shall let a stand for the last vertex in the linear order < for which there are vertices b, c, d with cd $ E satisfying (4). Next, we let c stand for the largest vertex in N ( a ) for which there exist vertices b and d with cd I$ E

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satisfying (4). Further, with a and c chosen as before, let b stand for the largest vertex in < for which there is a vertex d , cd $ E , such that (4) is satisfied. Finally, with a, b, c chosen as above, we let d be the largest vertex in the linear order < for which (4)is satisfied. For the proof of Proposition 1 we shall need the following intermediate results which we present as facts.

Fact 1. c
Fact 2. b and c have no common neighbour w with a < w and aw $ E. Proof of Fact 2. Let w be a common neighbour of b and c with a < w and aw $ E. We shall let w be as large as possible. Trivially, dw $ E (else {a, b, c, d, w } induces a 4); Fact 1 implies b < d ; furthermore, d<w (5) [Otherwise, either b or w contradicts our choice of a.] Apply property (P) to the vertices b, d , w : by (9,we find a vertex d’ with b d ’ $ E , d d ’ E E a n d w
Weak bipolarizable graphs

167

Write x E B whenever there exists a path

b = w o , w l ,. . . , w , = x

(~20)

joining b and x , with W ~ - ~ < W , and

awi$E, ( l s i s s ) .

(7)

Similarly, write y E C whenever there exists a path c=WJ,v1, . . . , v , = y

(t20)

joining c and y, with zli-l

< vi and m i$ E, (1 =si s t ) .

(8)

We note that Fact 1 implies that B # {b}. Furthermore, it is easy to see that we can apply property (P) to the vertices b, c, d; we obtain a vertex x adjacent to c but not to a, and such that d < x . By Fact 1, c < d and so c < x . This shows that c# {c}. Let b‘, c ) stand for the largest vertex in < which belongs to B, C , respectively. By the definition of B, we find a chordless path

b = bo, b l , . . . , b, = b’ in B, joining b and b‘, with the bis satisfying (7) in place of the w,’s. Similarly, the definition of C guarantees the existence of a chordless path c = c o , c ] ,. . . , c q = c ’ in C , joining c and c‘, with the ci’s satisfying (8) in the place of the vi’s. For further reference, we note that

cbi $ E, (0 =si G p ) .

(9)

[To justify (9), let i stand for the smallest subscript for which cb, E E. Since bc @ E, we have i 2 1; by Fact 2, we have i 2 2. But now, {a, c, bo, b , , . . . , bi} induces a C, with k 3 51. By Fact 1, c < d E B, and so c
( 10)

Now for the following Fact 3, symmetry allows us to assume that b’ < c’.

(11)

Fact 3. B fl C # 0.

Proof of Fact 3. Clearly, we may assume that no edge in G has one endpoint in B and the other in C, for otherwise we are done. Let i be the subscript for which ciVl< b’ < ci (such a subscript must exist by virtue of (10) and (11) combined).

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168

Property (P) applied to the vertices ci-l,b’, ci guarantees the existence of a vertex b” with b’b“E E, E and ci < b”. We must have a b ” E, ~ else we contradict the maximality of b ‘. The shortest path joining b“ and b with all the internal vertices in B, together with {a, b, b”} determines a chordless cycle I‘.By assumption, I‘contains at most four vertices. Next, note that

cob”E E [If not, then clb” E E, or else b” contradicts our choice of c. But now, {a, co, cl, b”} U I‘induces a P5 or an &.I Since cob”EE and c i P l b ” $E, it follows that co and ci-l are distinct vertices. Let j be the first positive subscript such that cjb”$ E (such a subscript must exist $ E). Note that cj+,b’’eE, for otherwise cj-, contradicts our choice since of the vertex a. But now, {z, cjW1,cj, c ~ + b”} ~ , induces a P’ with z = a or z = c ~ - ~ This . completes the proof of Fact 3. 0 Let w be the first vertex in the linear order < which belongs to B n C. By the definition of B, there exists a chordless path Q B in B joining w and b satisfying (7); similarly the definition of C implies the existence of a chordless path Q, in C joining w and c, and satisfying (8). By our choice of the vertex w , Q B fl Q, = {w}.By Fact 2, w is adjacent to at most one of the vertices b and c, and thus C must contain a chordless cycle of length at least five induced by {a, b, c } together with Q B U Q,. With this the proof of Proposition 1 is complete. 0

Proof of Theorem 4. Write G = (V, E). If the statement is false, then some linear order < on V produced by LBFS is not a semi-perfect elimination. We shall let a stand for the last vertex in the linear order < which contradicts (3). Write x E A whenever a < x . Let c be the largest vertex in N ( a ) nA for which there exist a vertex b in N ( a ) rl A with bc $ E, and a vertex in ” ( a ) rl A which is adjacent to precisely one of the vertices b and c. Our choice implies, trivially, that b < c. Since every ordering produced by LBFS satisfies property (P), Proposition 1 guarantees that every vertex w in N ( b ) n “ ( a ) n A is adjacent to c. Therefore by our choice of a, we find a vertex d in A with cd E A and ad, bd $ E. We shall let d be as large as possible. Property (P) applied to vertices a, b, c guarantees the existence of a vertex 6‘ such that ab’ $ E, bb‘ E E and c < b’. By Proposition 1, we must have b‘c E E. Obviously, b’d $ E, or else {a, b, b’, c, d} would induce a P,. We claim that d < b’.

(12)

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[To prove (12), assume b ' < d , and apply property (P) to the vertices c , b ' , d ; there exists a vertex b" with cb"$ E, b'b" E E and d < b". Proposition 1 guarantees that b"d E E. By Proposition 1, again, we must have bb" @ E. Clearly, ab" E E, else {a, b, b ' , b", c, d } induces an F2. But now, b" contradicts our choice of c.] Next, we claim that b
(13)

[To justify (13), assume d < b, and apply property (P) to the vertices a, d , b. We find a vertex d' with ad' @ E, dd' E E and b < d'. Note that bd' @ E, for otherwise {a, b, c, d , d ' } induces a p5 or a C5. Our choice of d guarantees that cd' @ E. Further, b'd'E E, or else d contradicts our choice of a. But now, {a, b, b', c, d, d ' } induces an 4.1 By virtue of (12) and (13) combined, we can apply property (P) to the vertices b, d , b'. We find a vertex d' with dd' E E, bd' @ E and b' < d'. Note that since c < d ' , we must have d' # c. Clearly, ad' @ E, for otherwise d' contradicts our choice of c. Furthermore, cd' @ E, or else d' contradicts our choice of d. Now b'd' E E, for otherwise either c or d contradicts our choice of a. It follows that { a , b, b ' , c, d , d ' } induces an F2, a contradiction. With this the proof of Theorem 4 is complete. 0

Note. The proof of Theorem 4 does not use the forbidden graph 4,and thus Theorem 4 provides a new proof of the main result of Hoang and Khouzam [9]. This result also characterizes the graphs for which the LBFS gives a semi-perfect elimination. In the remainder of this paper we shall point out how Theorem 4 can be used to find linear-time solutions for the four classical optimization problems for weak bipolarizable graphs, namely: find a minimum colouring of G (a colouring of the vertices of G using the smallest number of colours), find a largest clique (standing for a set of pairwise adjacent vertices) in G, find a largest stable set (standing for a set of pairwise non-adjacent vertices) in C,and find a minimum clique cover of G (a partition of the vertices of G into the smallest number of cliques). To solve all these problems in linear time, we shall rely on the following result.

Proposition 2 (Chvital, Hoang, Mahadev, and de Werra [4]). Given any graph G = (V, E ) , along with a perfect order on G , one can find in time O(lV1 + [ E l )a minimum colouring of G and a largest clique in G. Given any graph G, along with a perfect order on its complement G, one can find in time O( I V J + 1 E I) a minimum clique cover and a largest stable set in G. Furthermore, we shall need the following easy observations.

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Observation 2. If < is a semi-perfect elimination of a graph G , then the linear order <’ defined by x

<’ y if, and only if y < x

is a perfect order on G.

[To see this, consider vertices a, b, c, d with ab, bc, cd E E , and such that a <’ b and d <’ c. This implies that b < a and c < d, and so either b or c contradicts the assumption that < is a semi-perfect elimination.] Observation 3. If < is a semi-perfect elimination on graph G , then < is a perfect order on the complement G of G. [Let a, b, c, d be vertices of G with ab, bc, cd $ E , and such that a < b and d < c. But now, either a or d contradicts the assumption that < is a semi-perfect elimination, depending on whether or not a < d . ] Let G be a weak bipolarizable graph. The following algorithm will produce a minimum colouring, a largest clique, a largest stable set and a maximum clique cover for G. Step 1. Let < be the linear order produced by LBFS with G as input. Step 2. Call CoIour(G, <); Step 3. Call Max-Clique(G, <); Step 4. Let <’ be obtained by reversing <; Step 5. Call Colour(G, <’); Step 6. Call Max-Clique(G, <’). Here, Colour and Max-Clique are algorithms which, given a graph G along with a perfect order on G return a minimum colouring of G , and a largest clique in G, respectively. Their existence, as well as their running time, is guaranteed by Proposition 2. In addition LBFS takes linear-time to return an ordering of the vertices of an arbitrary graph. Theorem 4 guarantees that, with a weak bipolarizable graph G as input, LBFS will return a semi-perfect elimination. Hence the above algorithm correctly solves the four optimization problems in linear time.

References [l] C. Berge and V. ChvBtal, Topics on Perfect Graphs, Annals of Discrete Math 21 (Norh-Holland, Amsterdam, 1984). [2] V. Chvital, Perfect Graph Seminar (McGill University, Montreal, 1983). [3] V. ChvBtal, Perfectly ordered graphs, in Berge and ChvBtal [l]. [4] V. ChvBtal, C. Hoang, N.V.R. Mahadev and D. de Werra, Four classes of perfectly orderable graphs, J. Graph Theory 11 (1987) 481-495. [5] G. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg. 25 (1961) 71-76.

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[6] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, 1980, New York). [7] M.C. Golumbic, C.L. Monma and W.T. Trotter Jr., Tolerance graphs, Disc. Appl. Math. 9 (1984) 157-170. [8] A. Hertz and D. de Werra, Bipolarizable graphs, Ddpartment de mathematiques, Ecole Polytechnique Fddtrale de Lausanne, Suisse, O.R. W.P. 85/13, to appear in Discrete Mathematics. [9] C. Hoang and N. Khouzam, A new class of brittle graphs, School of Computer Science, McGill University, Montreal, Que, Tech. Report SOCS-85-30, [ 101 S. Olariu, Doctoral thesis, McGill University, Montreal (1986). [ l l ] J. Spinrad, P,-trees and module detection, manuscript. (121 M. Preissmann, D. de Werra and N.V.R. Mahadev, A note on superbrittle graphs, Disc. Math. 61 (1986) 259-267. [13] D. Rose, R. Tarjan and G. Leuker, Algorithmic aspects of vertex elimination on graphs, SIAM J . Comput. 5 (1976) 266-283.

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Discrete Mathematics 74 (1989) 173-200 North-Holland

Pd-COMPARABILITY GRAPHS C.T. HOANG Department of Computer Science, Rutgers University, New Brunswick, NJ 08903, U . S .A.

and B.A. REED Dbcrete Mathematics Group, Bell Communications Research, Morrbtown, NJ 07960,U.S.A.

In 1981, Chvhtal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs. In this paper, we introduce a new class of perfectly orderable graphs, the P4-comparability graphs. This class generalizes comparability graphs in a natural way. We also prove a decomposition theorem which leads to a structural characterization of P,-comparability graphs. Using this characterization, we develop a polynomial-time recognition algorithm and polynomial-time algorithms for the clique and colouring problems for P,-comparability graphs.

1. Introduction

A natural way to color the vertices of a graph is to put them in some order

v, < v 2 < - . < v, and then assign colors in the following manner: (1) Consider the vertices sequentially (in the sequence given by the order) (2) Assign to each v i the smallest color (positive integer) not used on any neighbor vj of v, with j < i. a

We shall call this the greedy coloring algorithm. An order < of a graph G is perfect if for each subgraph H of G, the greedy algorithm applied to H, gives an optimal coloring of H. An obstruction in a ordered graph (G, <) is a set of four vertices { a , b , c, d } with edges ab, bc, cd (and no other edge) and a < b , d < c. Clearly, if < is a perfect order on G, then (G, <) contains no obstruction (because the chromatic number of an obstruction is two, but the greedy coloring algorithm uses three colors). In 1981, Chvital [l] introduced perfect orders and perfectly orderable graphs. A graph is perfectly orderable if it admits a perfect order. He proved that (3) An order is perfect if and only if it contains no obstruction, and (4) All perfectly orderable graphs are perfect. (A graph G is perfect if for each induced subgraph H of C , the chromatic number of H equals the number of vertices in a largest clique of H.) An orientation U of a graph G is a anti-symmetric directed graph obtained from G by assigning a direction to each edge of G. To an ordered graph (G, <), there corresponds an orientation D (G, <) of G such that ab E D ( G , <) if and only if ab E E ( G ) and a < b. Similarly, given an acyclic directed graph F we can 0012-365)3/89/$3.50 0 1989, Elsevier Science Publishers

B.V.(North-Holland)

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Fig. 1.

construct an ordered graph (G, <) such that D ( G , <) = F. Thus, we can restate (3) as: a graph is perfectly orderable if and only if it admits an acyclic orientation which does not contain an induced subgraph isomorphic to the graph F of Fig. 1, or equivalently, a graph is perfectly orderable if and only if it admits an acyclic orientation in which each induced path of length three is of one of the three types shown in Fig. 2. A graph (G = V, E) is a comparability graph if G satisfies the following three equivalent conditions: (i) There is a partially ordered set corresponding to G such that two vertices are adjacent in G if and only if the corresponding elements are comparable in the partially ordered set. (ii) G permits an order < on its vertices so that no subgraph with vertices a, b, c and edges ab, bc (and no other edge) has a < b , b < c (a transitive order). (iii) G admits an acyclic orientation which contains no induced subgraph isomorphic to the graph shown in Fig. 3 (a transitive orientation). Comparability graphs have been studied by many researchers (see GhouilaHouri [3] and Gilmore and Hoffman [4]). Important results on comparability graph are discussed in the book of Golumbic [ S ] . Observation 1.1. Every comparability graph is perfectly orderable.

Proof. Clearly, in a transitive orientation, every P4is of type 3. 0 The following theorem of Ghouila-Houri [3] is the key to a polynomial-time algorithm to recognize comparability graphs.

Theorem 1.2. A graph is a comparability graph if and only if it admits an orientation which contains no induced subgraph isomorphic to the graph depicted in Fig. 1 (a semi-transitive orientation).

-+-++jd

Fig. 2.

type3

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Fig. 3.

Ghouila-Houri used the following two lemmas to prove Theorem 1.2. (A set C of vertices of G is homogeneous if C has at least two vertices, there is at least one vertex outside C and each vertex outside C is adjacent to either all or no vertices of C . )

Lemma 1.3. If a comparability graph G does not contain a homogeneous set, then every semi-transitive orientation of G is acyclic (and thus, transitive). Lemma 1.4. No minimal noncomparable graph contains a homogeneous set. In this paper, we introduce a new class of perfectly orderable graphs. This new class of graphs generalizes comparability graphs in a natural way. We shall say that a graph G is P,-comparability if it satisfies the following three equivalent

a

b Fig. 4.

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conditions: (i) G admits an order which is transitive when restricted to any P4 (a P4-transitiveorder). (ii) G admits an acyclic orientation which is transitive when restricted to any P4 (a P4-transitiveorientation). (iii) G admits an acyclic orientation in which every P4 is of type 3. Obviously every comparability graph is a P4-comparability graph and every P4-comparability graph is perfectly orderable. The graph depicted in Fig. 4b is P,-comparable but not comparable and the graph of Fig. 4a is perfectly orderable but not P4-comparable. The main result of this paper is to prove an analog of Ghouila-Houri’s lemma for P4-comparability graphs. Using this result, we develop polynomial-time algorithms to: (i) determine if a given graph is a P4-comparabilitygraph, (ii) color a given P4-comparabilitygraph, and (iii) find a largest clique in a P4-comparability graph (a clique is a set of pairwise adjacent vertices).

2. Definitions For a graph G , G denotes the complement of C . Given a graph G = (V, E), we define G * = ( V , X ) to be a directed graph such that xy E E if and only if x y , yx EX. Let c k (respectively Pk) be the chordless cycle (respectively path) with k vertices. We are especially interested in the P4. If G is a graph, then EP4G denotes the set of edges of G which are contained on some P4 of C . By the P4 abcd we shall always mean a P4 with vertices a, b , c, d and edges ab, bc, cd; the edge bc is called a rib, the edges ab, cd are called wings, the vertices b, c are called joints, and the vertices a, d are called tips. An edge is inactive if it does not belong to any P4. For vertices, x , y of a graph G, if xy E E ( G ) then we say that x sees y, otherwise we say that x misses y. Recall that a set of vertices H of a graph G is homogeneous if 2 G IH( < ICl and each vertex outside H sees either all vertices of H, or no vertex of H. A component C of a graph is big if C has at least two vertices. A k-circuit is a directed cycle with k vertices. We shall refer to the 3-circuit as a directed triangle.

--

3. P.-comparability graphs Given a graph G, we define a relation L on the arcs of G* as follows. First, ab L a b . Secondly, ab Lcd if either -.+

---.-I

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(i) c = a, b # d and wbcd or bcdw is a P4 for some vertex w of H, or (ii) b = d, a # c and wadc or adcw is a P4 for some vertex w of G. Consider any P4-transitive orientation U of G. Clearly & E U and & L z implies 2 E U (otherwise there will be an improperly directed P4).Now let M(&) be the equivalence class of directed edges under the transitive closure L* of L which contains 2. Let N ( 2 ) be the elements of M ( 2 )but with no direction. ~ and ) only if z ~ M ( g and ) therefore N ( z ) =N(G). Note that Z E M ( if Clearly, E(G) is partitioned by L* into disjoint equivalence classes Nl = N ( a l b l ) , N2 = N(a2b2),. . . , N k = (a,b,). By the preceding remarks, a P4transitive orientation U restricted to some Nimust be either M(&) or M(biaj). C l e 3 U restricted to N, must be acyclic. Trivially M(&) is acyclic if and only if M(b,a,) is. Thus if a graph permits a P4-transitive orientation, each M(&) must be acyclic. In fact this necessary condition is also sufficient.

-

-

-

Thoerem 3.1. Let G be a graph such that for each 2 of G * , M(Z),the equivalence class of 2 under L , is acyclic. Then G is a P4-comparability graph. Proof of Theorem 3.1. We shall say a P4 is proper under some orientation if the Orientation is transitive when restricted to that P4. We define a proper orientation of EP4(G) to be one in which every P4is proper. A graph G is interesting i f M ( 2 ) is acyclic for every pair of adjacent vertices a and b in G. Note that this implies that G has a proper orientation. We shall show that if U is a proper orientation of the P4-edge set of an interesting graph G, then one of the following holds: (A pyramid is a set of vertices {sl, cl, c2, s2, p , r } such that slclc2s2is a P4, p sees all of sl, c,, c2, s2 and r sees cl, c2, but misses sl, s2). (i) G has a homogeneous set. (ii) G contains a pyramid. (iii) U is a acyclic and thus can be extended into a P4-transitive orientation of c.

Definition 3.1. Let G be a graph. We call a partitioning of G into five disjoint sets C , S, P, Q, R good if the following properties hold. (1) C is a clique with at least two vertices, S is a stable set. (2) x E P + x is adjacent to all of C U S. (3) x E R j x is adjacent to all of C and none of S. (4) x E Q + x is nonadjacent to all of C U S U R. (5) P U Q U R is nonempty. We shall show that if (ii) holds while (i) and (iii) do not, then the pyramid is directed as in Fig. 5, and furthermore this implies that G permits a good partition. Thus, we actually prove the following lemma.

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r

Fig. 5.

Lemma 3.2. Let U be a proper orientation of the P4-edge set of a graph G , then one of the following four conditions holds. (i) G has a homogeneous set (ii) G has a good partition (iii) G is not interesting (iv) U is acyclic and thus can be extended into a P4-transitiveorientation of G. If Theorem 3.1 fails, then there exists a minimal counter-example G. Without loss of generality, we may assume that G is minimally P4-incomparable. Also, G is interesting since the orientation of each equivalence class is anti-symmetric and acyclic. We know that neither (iii) nor (iv) holds on G. We show that neither (i) nor (ii) can hold on any minimally P4-incomparable graph. But this contradicts Lemma 3.2 and thus Theorem 3.1 must hold. Therefore, Theorem 3.1 follows from Lemma 3.2 and the following two lemmas:

Lemma 3.3. No minimally P,-incomparable graph contains a homogeneous set. Lemma 3.4. N o minimally P4-incomparablegraph contains a good partition. To prove Lemma 3.2, we need the following two lemmas.

Lemma 3.5. If a proper orientation of an interesting graph is cyclic, then it contains a directed triangle. Lemma 3.6. If an interesting graph G admits a proper orientation which contains a directed triangle, then G contains a homogeneous set or a good partition. Proof of Lemma 3.3. Assume that the lemma is false and let G be a counter example with a homogeneous set H. First let us observe that if a P4 has a vertex in H and a vertex not in H , then this P4 has precisely one vertex in H. Furthermore, if three vertices of G - H form a P4 with a vertex in H , then these three vertices form a P4 with every other vertex of H.

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Now, let y be an arbitrary vertex of H. We know that both the graphs G - (H - y ) and H are P4-comparable. Consider P4-transitive orientations U, and U, of G - (H - y ) and H, respectively. We can obtain a P,-transitive orientation of G by orienting G - (H - y ) as in U,,zrienting H as in U,, and for each vertex z of H, x of G - H, we direct x to z if xy E U, and we direct z t o x if GEU , . By our observation in the last paragraph, U is interesting; it is also easy to see that U is acyclic. Thus U is a P4-transitive orientation of G, a contradiction. 0

Lemma 3.4. N o minimally P4-incomparable graph permits a good partition. Proof of Lemma 3.4. Consider a good partition of a minimal non-P4-comparable graph G into a clique C, stable set S, and sets P, Q, R as in Definition 3.1. If every element of S saw every element of C then C would be a homogeneous set in G, contradicting Lemma 3.3. Thus we can choose s E S and c E C such that s misses c. Now, since G is minimally non P4-comparable, the graph H induced by the vertices of G - (C - c ) - (S - s) permits a P4-transitive orientation W,. Similarily the graph F induced by the vertices of C U S permits a P4-transitive orientation W,.Now, it is a tedious but routine matter to verify that every P4, not entirely in C U S and not entirely in G - (C U S) must be of one of the following forms: (i) p l , x I p z , q where X E S , P ~ , P ~ E q EP Q, . (ii) p l , x, p2, r where x E S, p l , P , E P, r E R. (iii) x , p , 41, q 2 where x ES,q l , q 2 E Q , p E P. (iv) x, p, r,, r, where x E S, r , , r, E R, p E P. (v) q, p, Y , r where Y E C, q E Q, p E P, r E R. (vi) Y , P , 41, q z where Y E C ,q i , q z E Q , p E P (vii) p I , Y , p Z , q where y E C, p I , p 2 P,~ q E Q. (viii) x, p , y, r where y E C , x E S, p E P, r E R. Given a P4of types (i) to (iv) we can find a corresponding P4 in H by replacing x by s. Similarily given a P4 of types (v) to (vii), we can find a corresponding P4in H by replacing y by c. Finally, given a P4 of type (viii), we can find a corresponding P4 in H by replacing x by s and y by c. These remarks imply that EP?= E P Y U EPCUA U B, where A = { x z : x E S , z E P, sz E EPY} and B = { y z :y E C ; z E P U R, cz E EPf}. We construct an orientation W, of EPF as follows: (i) Orient EPY as in W,. (ii) Orient EPC as in W,. (iii) Given an edge xz of A, with x in S and z in P, we have

Z e W 3 if

Z EW,

and

if

ZEW,.

(iv) Given an edge yz of B, with y in C and z in P U R, we have ~

E

ifW

ZE ~ W,

and $E

W, if

Z Ew,.

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We want to show that every P4 in G is proper under W,. Since W, and W, are P4-transitive, this is clearly true for any P4 contained in F or in H . Any other P4 that is partially but not entirely in C U S, by the preceding remarks, corresponds to a P4 in H. Now the algorithm used to orient the edges of A and B ensures us that since the P4 in H is proper, the corresponding P4 in G must be. Thus, every P4 in G is proper. We claim that W3 is also acyclic. Assume there is a circuit K in W3. Since W, and W, are acyclic, K is partially but not entirely in C U S. It follows that we can find, in K, a path from a vertex x of C U S to a vertex y of C U S , such that all the midpoints of the path are in G - (C US). We replace x (resp. y) by c if x (resp. y ) is in C and we replace x (resp. y) by s if x (resp. y ) is in S. This replacement gives a corresponding path in H. Now this path cannot have both endpoints the same since W, is acyclic. Thus in H we have either a directed path from c to s or a directed path from s to c. In fact, these cases are the same as if the second case occurs we can simply consider the orientation W ; where GEW ; if and only if W3. Thus, we need only consider the first possibility. Let z be the vertex before s in this c to s path P. Now, sz must be in a P4of one of the seven types enumerated earlier. Clearly sz cannot be the edge of a P4 of types v, vi, or vii, as P4’sof these types contain no vertices of S. If sz is in a P4 of type viii, szyr, then since the corresponding P4, szcr must be proper, zc must be in W,. But then P - s induces a circuit in W,, a contradiction. Also, if sz is in a P4: szq,q, of type iii, then c is in a P4, czq,q, and again P - s must induce a circuit in W,, a contradiction. Similarly, if sz is in a P4, zsp,q or p l s z q (type i), then by replacing s by c we obtain a new P4, from which it follows that zc E W, and P - s is a circuit in W,, a contradiction. We can apply the same method to obtain a contradiction if sz is in a P4of type iv, szr,r, (consider the P4szcr,). Now, consider the case where sz is in a P4of type ii. This P4 is either zsp2r or p , s z r . In the first case, we obtain the usual contradiction by considering the P4, szcr. In the second case, we know that p l s and zr are in W, (since p , s z r is a P4). Now, since rcp,s must be proper, it follows that rc E W,. This implies that P - s U { r } is a circuit in W,, a contradiction. Thus, no matter which type of P4sz is in we obtain the desired contradiction. We have shown that W, is an acyclic proper orientation of EPF. We can therefore extend W3 into a P4-transitive orientation of G (first order the vertices vl,v2,. . . , v, such that if viujE W, then i <j ; now, for each undirected edge v,vj, direct vi to vj if and only if i <j ) . This contradicts the non P4-comparability of G implying the desired result. 0

FE

Lemma 3.5. If a proper orientation of an interesting graph contains a directed cycle then it contains a directed triangle. Proof of Lemma 3.5. Let U be a proper orientation, of an interesting graph G, which contains a circuit. Let K be a circuit of minimum length in G. If G contains

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a directed triangle, then we are done; so we may assume K has order at least 4. We shall enumerate the vertices of K as vo, u l , . . , , vk-1 in such a way that vivi+lis in U for all i 6 k - 1 (where addition is taken modulo k). We note that, by the minimality of K, no other edge between two vertices of K extends into a P4.We will call vertex vi a switch if vi-lvi and ~ l ; v ; +are ~ in different equivalence classes. Since G is interesting, the edges of K cannot all come from one equivalence class and therefore some vertex of K must be a switch. If vi is a switch then both the edges U ; - ~ V ;and vivi+lmust extend into P4s.We will first show that at a switch these P4smust have a special nature.

Step 1. vi-l must see v ; + ~ . Assume that viis a switch and vi-l misses v ; + ~We . shall show that this leads to a contradiction. Case 1. vivi+l is a rib. Label the tip of the P4which sees vi+]x and label the other tip y. Now x sees uiPl as otherwise xvi+lvivi-lwould be an improper P4. But then y sees vjVlor x v j - , v j y would be an improper P4.Since v j + l x v j - l yis a proper P4, it follows that is in U. However, this implies that yv,-,v, is a directed triangle, a contradiction.

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Case 2. V ; V ; + ~is a wing with joint v ; + ~We . label the other vertices of the P4, so that the P4is x y ~ l ; + ~Now v ~ . y must see vi-] as otherwise yui+luiui-lwould be ; be an improper P4. an improper P4.But then x must see vi-l or x y ~ , - ~ vwould However, it follows that xvi-lvivi+l is an improper P4,a contradiction. Case 3. v,v,+]is the wing of a P4 with joint v,. We label the vertices of this P4 so that the induced path is x y v , ~ , +We ~ . know that x misses v,-] as otherwise x ~ , - ~ v , vwould , + ~ be an improper P4.Now y must see v , - ~ or x y ~ , v , would -~ be an improper P4. At this point we must consider the vertex v , + ~Clearly . this . fact, t ~ , +must ~ also see v,. To see this note that if u , + missed ~ vertex sees u , + ~In v, then it would see y as otherwise y u , ~ , + ~ vwould , + ~ be improper. It would also have to see x as otherwise x y ~ , + ~ v ,would + ~ be improper. But then x ~ , + ~ v , + ~ v , would be improper. So, we know that v , +sees ~ v,. Now, if u , + missed ~ v , - ~it, would also miss x as otherwise xv,+2v,v,-Iwould be a P4 having u , + ~ vas, an edge. Also y would see u , + ~ or the edge v , + ~ vwould , extend into a P4with vertices x and y. But this would lead to a contradiction as ~ , + ~ v , +would ~ y x be an improper P4.This contradiction implies the existence of the edge ~ , + ~ 2 1 , - ~ Furthermore, . by the minimality of K either this edge is in no P4 or it is oriented from I J , + ~to u , - ] . It follows immediately that y sees v , + as ~ , + ~ exist and it clearly cannot be proper. otherwise the P4 y ~ , - ~ v , + ~ vwould Finally, we know that x sees u , + ~as otherwise the P4 x y ~ , + ~ v ,would +~ be improper. Now the edge between v , + ~and ulc2extends into a P4.By considering the three

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vertices vi+l, vi+z,vi-l and using the argument of Case 2, we can see that this edge is not a wing with joint vi+l. Also, using arguments similar to those of Case 1, we can show that vi+1vi+2is not the rib of any P4. Thus there exists an induced , can path zavi+2vi+l.Just as we have shown that x sees vi+2and misses v ; + ~we also show that z sees vi and misses viVl.Now x sees z or the edge ujvi+2would be the rib of a P4 with x and z as tips. However, it follows that both xzuivi+,and xzvivi-, are P4s and at least one of them must be improper. This contradiction . will now examine this case in implies that if vi is a switch then v i P 1sees u ; + ~We more detail. Step 2. lf ui is a switch then either: (a) vi-lvi is a rib and vivi+l is a wing with tip v,, or (b) V , V ~ +is~ a rib and vi-lv is a wing with tip v,. We have just shown that if vi is a switch then viP1must see v i + l .We note also that, by the minimality of K, the edge between viPland vi+l does not extend into a P4. We shall show first that, in this case, at least one of the edges, uivi+l,and viP1vi is not the rib of a P4. If this is not true then both edges must be the ribs of P4s. Thus we have induced paths x v i - , v i y and zvi+lvia.Now, if y misses v , + ~ then it must see z or ~ v , + ~ vwould , y be an improper P4. In fact, it would also see a as otherwise zyvia would be improper. But, in this case, since ayzvi-, must be properly directed, we know that ayv, forms a directed triangle. This contradiction implies that y sees v , + ~Using . symmetrical arguments, we can show that a sees uipl. Also z sees vi-l as otherwise z v i + l v i - l awould be a P4 containing an inactive edge. Similarly, x sees v,+~.Now a sees y as otherwise the inactive edge vi-luj+lwould be the rib of a P4 which has these vertices as tips. Furthermore, if x missed a then, because of the three P4s, x v j - l a y , xvi+lvia,x v i , , y a the edge ya would be in the equivalence class of both V , V ; + ~ and of vi-lvi. Since this contradicts the fact that vi is a switch, x must see a. Similarly z sees y. We also know that z misses x , as otherwise zx would be in two distinct equivalence classes (consider the P4s zxav; and xzyv;). It follows that the edge between a and y is in , and ~ v , - ~ x a ) . both N ( v j - l v i ) and N(vivj+J (consider the P4s Z V ~ + ~ Z Uxayz However, this contradicts the fact that vi is a switch. So, we have shown that at least one of the edges we are concerned with is not the rib of any P4. Now, if vi is a switch and is the wing of some P4 then the other vertices of the P4 must see viP1irregardless of which vertex of the edge v , ~ , +is ~a tip (the easy case analysis is left to the reader). Similarly, if viP1viis the wing of a P4 then the other tip of the P4 sees vi+l. Thus if both edges are wings of P4s then we have a vertex x which sees viP1and misses vi and vi+l as well as a vertex y which sees vi+] and misses v, and vi-l (x and y are the tips of the P4s). Since the edge between vi-] and v , + ~does not extend into a P4, we know x sees y. Now, considering the P4 xyvi,,vj we see that the edge xy is in the same equivalence class as the edge uivi+l.Similarly, since yxv,-,v, is a P4,it follows that xy is in the same equivalence class as vi-lvi. However, this contradicts the fact that v, is a

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switch. This shows that at most one of the edges we are concerned with is the wing of a P4. From the preceding remarks we can deduce that one of the edges is a wing while the other is a rib. To complete Step 2, it remains only to show that vimust be the tip of this wing. By symmetry, we need only consider the case where ~ , v , + ~ is a wing. Assume that vi is the joint of this wing. Label the vertices of this P4x and y so that the induced path is ~ y v ~ v ~We + ~know . that vi--lviis the rib of some P4 which we call label zvi-luia. As mentioned previously, we know that both x and y see vi-]. If a missed vi+l then it would also miss x or x a v , ~ would ~ + ~ be an improper P4. Also, u would miss y as otherwise ayvi-lvi+lwould be a P4 with vi-lvi+l as a wing. However, this would imply that xycia was an improper P4,so a must see v i + l .Then a must also see x or uvi+lvi-lxwould be a P4 with an inactive edge. For similar reasons a sees y. Now z must see vi+l as otherwise zvi-lui+lais a P4 with an inactive edge. Now if z saw x then, because of the P4s zxav, and xz would be in the equivalence classes of both U,V,+~and U , - ~ U , . However this contradicts the fact that vi is a switch, so z misses x. Now a, together with x y z ~ , +(if~ z sees y) or yvivi_,z the following P i s ~ v , - ~ xxavi+lz, (if z misses y), imply that N ( U , - ~ ~=,N) ( ~ , V , - ~This ) . contradicts the fact that ui is a switch so this case cannot occur.

Step 3. Consider a small circuit with few switches. Previously, we have considered an arbitrary circuit of minimal length. We now restrict our attention to circuits which also have few switches. That is, we consider K which is not only a circuit of minimal length, but also has the minimal number of switches of any such circuit. We are going to show that using K, we can construct a circuit K' of minimal length with a switch that does not fulfill the requirements of Step 2. Since I/ is an interesting orientation, clearly K has at least one switch. As before, let vi be a switch of K. From Step 2, we know that one of the two edges of K adjacent to vi is a rib while the other is a wing with tip vi. We consider the case where uivi-lis a wing with joint vi-l and V , - ~ V ,is a rib (the other case is symmetric). We label the other vertices of the P4s so that x y ~ ~ + ~ v , and zvi-luia are induced paths. Recall that both x and y must be adjacent to viPl. We note that a is adjacent to u , + ~(to see this note that if a misses v i + l ,then it must miss x and see y because of the P4s xuvivi+land yv,+,v,a. But then xyav, is an improper P4). It follows that z sees vi+las otherwise avi+lvi-lz would be a P4 with an inactive edge. We also know that x sees a as otherwise x ~ , - , v , + ~would a be a P4with an inactive edge. On the other hand x misses z because otherwise we contradict the fact that vi is a switch (consider the two P4s vivi+lzxand zxav,). We note that a sees y , again because vi is a switch (consider the four vertices u,axy). In fact, for the same reason, z sees y (consider the possible P4s xyv,+,z, ar~,-~ and z xuvi+lz).Now, because both the P4s Z V , - ~ M and zvi+,xa must be is in I/ and in N(vi-lvi).Thus we can construct a circuit proper, we see that K' from K by replacing vi with z. Furthermore, z is not a switch in this cycle.

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Also, since N ( Z ~ , -=~N(viv,-,), ) u , - ~is a switch in K' precisely if it is a switch in K. Thus, since K has the minimal number of switches for cycles of its length, we know that vi-l is a switch in K'. However, zv,,, is the wing of the P4 ZV,-~UXand in this P4, v , - ~is the joint. This contradicts the facts we established about the P4s at switches in Step 2. This final contradiction establishes the lemma. 0 To prove Lemma 3.6, we shall need the following lemma.

Lemma 3.7. If an interesting orientation of the P4 edge set of a graph G contains a directed triangle, then it contains a pyramid oriented as in Fig. 5 .

Proof of Lemma 3.7. Assume vo, v , , v 2 are vertices of a triangle in some interesting orientation, U , of G. Since U is interesting, this triangle contains a switch; we label the triangle such that v 1 is the switch. Furthermore, we can label the triangle so that vovl, vlvz and vzvo are arcs of U . To prove the lemma, we will examine the ways in which vovl and v1v2 extend into P4s. Case 1: both vovl and v1v2are the ribs of some P4. Let av,vlb and cv,v,d be the two P4s. Case 1.1: c misses vo and a misses v,. In this case, a must see c or av0v2c would be improper. But then acvzvl and cav,vl are both P4s. Thus, ac is in N(vovl) and N(v,vl) which contradicts the fact that v 1 is a switch. Case 1.2: c misses vo and a sees u,. In this case b must see v, or a v 2 v l bwould be an improper P4. Furthermore, d misses v0 or dv,v,c would be improper. Now d sees a as otherwise dv,v,a would be improper. In addition, d sees b as otherwise adv,b would be improper. But now, { d , c, b, v,, v l , v,} induces a pyramid oriented as required (Note that b sees c, otherwise one of the P4s v,v2bd, cv,bd is improperly directed.) Case 1.3. c sees vo and a misses v,. This case is symmetrical to the previous one. Case 1.4. c sees vo and a sees v2. For this case, first note that d sees vo, otherwise cvovld is improperly directed. Similarly, b sees v,, otherwise bv,v,a is improperly directed. Now, if b missed c and d then { b , c, d , v,, v l , v2} would induce a pyramid oriented as required. Furthermore, if b missed d and saw c then cbv,d would be an improper P4. Thus b sees d. Now, if b missed c then all of the following P4s would exist: cvovlb, cvodb, cv,bd. But then bd would be in both N(v,vo)and N(vlv2). This would contradict the fact that v1 is a switch. Thus, b sees c and using symmetrical arguments we can show that d sees a. Also, ac cannot be an edge of G otherwise it would be in two distinct equivalence classes

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N(vov,)and N ( v l v 2 )(consider the P4s acbv, and cadv,). At this point, all of the following are P4s: avocb, cv2ad, adhc. However, it follows that v l is not a switch. Case 2. Both vovl and v2v2are wings of P4s. Case 2.1. v1 is the tip of both these P4s.

In this case we can label the vertices so that abvov, and cdv2v1are the two P4s. It is easy to see that both a and b see v 2 (we leave the case analysis to the reader). Similarly, both c and d we uo. Now, if c saw a then acvovl would be a P4 as would cav2uI.But then ca would be in both N(vOv,)and N ( v 2 v , )contradicting the fact that v1 is a switch. Thus, a misses c and uv2vocis a P4so that cv,, must be in U. Now, c sees b or abvoc would be improper. It follows that {a, b , c, vo, v,, v2} induces a pyramid as required. Case 2.2. v o v l is a wing with tip v l , while v1v2is u wing with tip v2. In this case we label the P4s as abvov, and cdvlv2.First, it is easy to see that v 2 must see both a and b and vo must see c. Now, if a sees c, then N ( v o v , ) = N ( v l v , ) (consider the P4s v1v2ac,vlvoca), and we have a contradiction. If b misses c then one of the P4s abvoc, av2v0c is improperly directed. Now, {a, b , c, v,,, vl,v2} induces a pyramid as required. Case 2.3. v2v1 is a wing with tip vl, while v l v o is a wing with tip v,,. This case is symmetric to Case 2.2. Case 2.4. Both wings have v1 as a joint. In this case we can label the two P4s abvlvo and cdv,v2. As before, vo sees c while u2 sees a. Also as before, a does not see c as otherwise we contradict the fact that v 1 is a switch. Now, c must see b as otherwise either cvovlb is improper or cvov2ais. However, now both vocba and cuou2aare P4s and at least one of them must be improper. This contradiction implies that this case cannot occur. Case 3. vovl is a wing and vlv2 is a rib. Case 3.1. v1 is the tip of the wing vovl. We can label the two P4s as abvov, and cv2v1d.Now, we note that v2 sees both a and b. If d misses vo then it must see b as otherwise bvovld would be improper. Furthermore, it would miss a as otherwise adv,vo would be improper. However, this would mean that v,dba is an improper P4 so d must see uo. It follows that c sees v o as otherwise cvavod would be improper. Now, d must also see a as otherwise either av2vld or av2vod is improper. Furthermore, since v1 is a switch, d sees b (consider the P4s badv,, abuoul, cv2vld). If c saw a then acvOvland cadv, would both be P4s. However, this would mean that ad was in both N(vovl)and N(v,v,). Since this would

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contradict the fact that v1 is a switch, it follows that a misses c. Now if c missed b then all of the following would be P4s: abvoc, cvoda, cv2ad. However, these P4s would force N(vlvo)to be the same as N(vzvl). This would contradict the fact that v1 is a switch so c sees b. Finally, we see that {a, b, vo, v l , d , c } induces a pyramid as required. Case 3.2. The tip of the wing vovl is 210. In this case we can label the two P4s as vov,ab and c v z v l d . We shall first prove that a must see v2. If a misses u2 then b must see v, as otherwise bavlu, would be an improper P4. It follows that vov,ba is a P4 and, to ensure this P4 is proper, we must have v26 in U . Furthermore, b sees d as otherwise bv,v,d would be improper. In addition, b sees c as otherwise cv,bd would be improper. Now, d sees vo as otherwise v o v l d b would be improper. Also, c sees uo as otherwise dv,)v,c would be improper. But this leads to a contradiction: the P4 v,dbc implies that the bc is in N(v1v2)but the P4 bcvov, implies that bc is in N(vovl). From the preceding remarks we can infer that a sees u2. We know that b also sees v, as otherwise, bav2vowould be an improper P4. If d misses vo then it also misses b as otherwise vov,bd would be improper. In this case, d must see a or dv,ab would be improper. But then { a , b, d , vo, v l , v 2 } is a pyramid which is directed as required. Thus, the lemma holds when d misses vo. Consider the situation when d sees vo. If d missed b then b v , v , d and bv2vod would both be P4s and they could not both be proper. Thus d sees 6. Also, c must see vo as otherwise cv2vod would be improper. Now, we show that c sees a. Assume the contrary, so c misses a. If a missed d then {vo,u , , v2, a, c, d } induces a pyramid: if a sees d then cvOda, cv,ad, are P4s, since cvovla is also a P4,cuo belongs to both N(vovl) and N ( v , u 2 ) a contradiction. Thus, a sees c. Now, c sees b as otherwise the P4s vocab, cv2bd and bdv,c imply N ( v o v l ) = N(v,v,), a contradiction. Finally a sees d as otherwise cavld would be improper. But then v,uocb and v,dbc would both be P4s; this would imply that bc was in N(vovl)and N ( v l u 2 )contradicting the fact that v1 is a switch. Thus the lemma holds in this case. Case 4. vovl is the rib of a P4, v l v z is the wing of a P4. This case is symmetrical to Case 3.

Since both uovl and v,uo extend into P4s, one of the above cases occurs and therefore the lemma holds in general. 0

Lemma 3.6. If an interesting graph G admits a proper orientation which contains a directed triangle, then G contains a homogeneous set or a good partition.

Proof of .Lemma 3.6. By Lemma 3.7, we know that G contains a pyramid directed as in Fig. 4. We assume G contains no homogeneous set and show that it admits a good partition.

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Step 1. Given a directed subgraph D of I/ isomorphic to the graph shown in Fig. 4, we can partition G into sets A l , A2, K1, K 2 , E, F, Q in such a way that: (i) Ai is homogeneous to si on D - si for i = 1 , 2 . (ii) Ki is homogeneous to ci on D - ci for i = 1, 2. (iii) E is homogeneous with p on (sl, s2, cl, c2). (iv) F is homogeneous with r on (s,, s2, cl, c2). (v) No element of F sees both p and r. (vi) No element of Q sees any element of D - p .

Proof. To show that this is true we simply need to consider every possible neighborhood of a vertex with respect to the pyramid. We use the fact that our interesting orientation contains no improper P4to show that all vertices must be of one of the above types. Obviously s1 E A , , s2 E A2, c1E K , , c2 E K 2 , p E E , r E F. Let x be an arbitrary vertex of G - D. If x sees no vertex of D - p then put it in Q, otherwise we have one of the following cases: Case 1. x misses p . x cannot see s1 and miss s 2 , because then x s l p s 2 would be an improper P4.x cannot see s2 and miss sI, because then xs2psI would be an improper P4.x cannot see s1 and s2 and miss c1 and c2 because then these five vertices would induce a Cs.x cannot see s l , s2, c1 and miss c2 because then x would see r (as otherwise x s2 c2 r would be improper) and this would mean that one of the P4sxrczp o r rxs,p was improper. Similarly, x cannot see c2, sl, s2, and miss cl. If x sees sl,s 2 , cl, c2 then put it in E. x cannot miss sl,s 2 , c2 and see c, as then one of the P4s xc,ps2 or.xcIcZs2would be improper. Similarly, x cannot miss sl, s 2 , c , and see c2. If x misses s l , s2 and sees cl, c2 then put it in F. x cannot see r and miss D - p - r for then both xrclsland xrc2s2 are improperly directed.

Case 2. x sees p and misses r. x cannot miss both c 1 and c2 as otherwise either xpc,r or xpc2r would be an improper P4.x cannot miss c I and see s, as otherwise xslclrwould be improper. Similarly, x cannot see s2 and miss c2. If x sees c I and misses c2 then it misses s2 so put it in A , . Similarly, if x sees c2 and misses c1 put it in A2. If x sees c I and c2 and misses s1 and s2 then put it in F. If x sees c1 c2 s I and s2 then put it in E. x cannot see sl, cl, c2 and miss s2 because then one of rc2xs1 or s2c2xsIwould be improper. Similarly x cannot see s2, c I , c2 and miss sI. Case 3. x sees p and r. x cannot miss both s I and s2 as then either s l p x r or s2pxr would be an improper P4. x cannot miss both sI and c1 as then x r c l s l would be improper. Similarly, x cannot miss s2 and c2. If x misses s 1 then it sees c, and s2 so put it in K 2 . If x misses s2 then it sees c2 and s 1 so put it in K,. x cannot see s,, s2 and miss c I for if it did then either clslxszor cIrxs2would be improper. Similarly x cannot see s l , s2 and miss c2. If x sees sI,s 2 , c1 and c2 then put it in E.

C.T. Hobng, B.A. Reed

partition described in Step 1 the following properties hold. K , U K 2 is a clique. A , UA2 is a stable set. If x E K i , y E Ai (i = 1, 2) then x is adjacent to y. Neither r nor p is adjacent to any element of Q. There are no edges between F and Q. If x is an element of Q then x is adjacent only to elements of E U Q. If x E F and y E A, UA2 then x is not adjacent to y. r does not see any element of F. If x E F then x sees all K , or all K 2 . If x E E then x sees all K , U K 2 . If x E E then x sees all A l or all A2.

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Proof. First, with sl, s2, cl, c2, p , r serving as distinguished vertices, we note that the following edges are in U : alcl for each a , E A , % for each a 2 e A 2 s , k , for each k , E K , k2s2 for each k 2 E K 2 p x for each a , € A 1 @ for each a 2 € A 2 k , r for each k , E K, rk; for each k2 E K 2 + for each k , E K1 pk2 for each k2 E K 2 . Now, we use the facts that G has no homogenous set and that U is an interesting orientation to show that (i) K,U K 2 is a clique. We - first show that K1 is a clique. Assume not, then let J be a big component of K , . Since J is not a homogeneous set, there exists a vertex x which disagrees on J . Now, since J induces a connected graph in G, x must disagree on two vertices, of J which are not adjacent in G. We let k , and k2 be two non-adjacent elements of J such that x sees k , and misses k2. Now x cannot be in K , because J is homogeneous with respect to K , - J. Also, x cannot be in D , because k , and k2 both have the same neighborhood as c , on D - c,. If x were in A, then one of xklrk2 or xklc2s2would be improper, so x cannot be in A , . Also, x cannot be in K 2 as then s2xrk2 would be an improper P4. In addition, x cannot be in A 2 for if it were, one of the P4s x k l r k 2 and x k l s , k 2 would be improper. Furthermore, x cannot be in F. Assume x is in F, then it misses p or r. If x missed p , then either xc2psI is improper or slk2c2xis. If x missed r, then either xklrk2 is improper or x k l s l k 2is. These contradictions show that x cannot be in F.

k,d

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We show that x cannot be in E in two steps. First, if x were in E and saw r then either kZslxs2is improper or k2rxs2 is. On the other hand, if x were in E and missed r then rkzsIx would be improper. These two contradictions show that x cannot be in E. Finally, x cannot be in Q. If x were in Q then one of x k l r k 2 or xklslk2would be improper. Since all possibilities have been examined, we see that x cannot exist. It follows that K , is a clique. Using symmetrical arguments, we can show that K 2 is a clique. It remains only to show that each vertex of K , is adjacent to every vertex of K,. Assume some k l in K1 misses kz in K 2 . Then slklrk2is an improper P4. This contradiction allows us to state that K , U K 2 is a clique. (ii) A , U A , is a stable set. We show first that A , is a stable set. Assume not, then let J be a big component of A,. Since J is not a homogeneous set, there exists a vertex x which disagrees on J. Now, since J induces a connected graph in C , x must disagree on two vertices of J which are adjacent in C. We let a , and a, be two adjacent elements of J such that x sees a, and misses a2. Now x cannot be in A , because J is homogeneous with respect to A, -J. Also, x cannot be in D, because a , and a2 both have the same neighborhood as s, on D - sl. In addition, x cannot be in K , otherwise a2alxr or aIxc2s2would be improper. If x were in K 2 then a2alxs2or azalxrwould be an improper P4, so x cannot be in K,. Also, x cannot be in A 2 for then alxc2r would be improper. Furthermore, x cannot be in F. Assume x is in F, then it misses p or r. If x missed p , then x u , ps, is improper. If x missed r, then either a,xcZr is improper or alxc2sZis. These contradictions show that x cannot be in F. We show that x cannot be in E in three steps. First, if x were in E and missed r then either rcIxs2is improper or a2cIxs2is. Second, if x were in E and missed p then either a2pczx is improper or a2ps2x is. Third, if x were in E and saw both p and r then one of the following four P4s would be improper: a2c,xs2,a2alxs2, a2alxr, or rxpa,. These contradictions show that x cannot be in E. Finally, x cannot be in Q, as then x a l c l r would be improper. Since all possibilities have been examined, we see that x cannot exist. It follows that A , is a stable set. Using symmetrical arguments, we can show that A, is a stable set. It remains only to show that each elements of A 2 misses every element of A , . Assume some a , in A , saw an a2 in A,. Then ala2c2ris an improper P4. This contradiction implies the desired result. (iii) I f x is in A ; , y in K i then x sees y (for i = 1, 2). If there is an x in A which misses some y in K , , then xpyr is improper. This contradiction shows that (iii) holds for i = 1. Using symmetrical arguments, we can show that (iii) also holds when i = 2. (iv) Every element of Q misses both p and r. If x is in Q then x misses r by definition. But then x must also miss p as otherwise either xpc,r is improper or xpc,r is. (v) Every element of Q misses every element of F.

,

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Consider an element f in F. Recall that f must miss one of p and r. Iff misses r it cannot see any x in Q as otherwise either xfclr is improper or xfc,r is. Similarly iff misses p it cannot see any x in Q (consider the P4s xfc,p, xfc2p). The desired result follows. (vi) l f x is in Q then x is adjacent only to elements of E U Q. Consider x a vertex of Q. We have already seen that x misses all of D U F. Now, x misses all of A l , for, if x saw some y in A l then x y c l r woul be improper. Similarly x misses all of A2. Furthermore, x misses all of K , . If x saw some y in K 1 then xyps2 or xyc2sz would be improper. Finally, using symmetrical arguments, we see that x misses all the vertices of K,. (vii) No element of F sees a vertex in Al U A2. Assume that some f in F sees an a in A , . If f misses p then either afc2s2 is improper or fc2psI is. Iff sees p then it misses r and then either afc2s2is improper or afc2r is. These contradictions imply that there are no edges between F and A I . By symmetry no f in F sees any a in A2 and we are done. (viii) no vertex in F sees r. Let F, be the set of vertices in F which miss p . Clearly r is in Fl. Note first that no vertex in F, is adjacent to a vertex in F - Fl (if some fi in F, saw f2 in F - F, then either flf2ps1 is improper or fifzps2 is). It remains only to show that no vertex of F, sees r. We actually demonstrate that F, is a stable set. Assume 4 is not a stable set and choose a big component J of F,. Since G has no homogeneous set there must be a vertex x which disagrees on J . As usual we choose f l and f 2 adjacent vertices of J such that x sees f l and misses f 2 . Since J is a component of F , , x is not in F,. Furthermore we have just seen that there are no edges between FI and F - F, so x cannot be in F. By (v) and (vii), we know x is not in Q, A l , A2. Now, x cannot be in K 1 as then either fzfixsl or fixpsz would be improper. Similarly, x cannot be in K 2 . Finally, x cannot be in E. If there were such an x in E then one of the following P4s would be improper: f 2 f l x s l ,fzfixs,, f2c2xs1, fzc,xsz, f2c2psI,fzclps2. Thus x cannot exist and since G contains no homogeneous set, Fl must be independent. (ix) l f x is in F then x sees either all K , or all KZ. Assume some x in F misses some vertex k , in K , and a vertex k2 in K , . If x is in 4 then either xc,k,s, is improper or xc2psl is. On the other hand, if x is in F - F, then either xpk,r is improper or xpk2r is. It follows that no such x can exist. (x) l f x is in E then x sees all Ki for i = 1, 2. Assume some x in E misses a vertex k l of K , . Now, x sees r as otherwise rk,s,x would be improper. However, it follows that either k l s l x s 2is improper or k,rxs2 is improper. Using symmetrical arguments we can show that each element of E is adjacent to every element of K 2 . (xi) l f x is in E then x sees all A , or all of A2. Assume some x in E misses a vertex a, in A l as well as a vertex a, in A2. In this case, x must see p as otherwise a , p s , x or a l p s 2 x is improper. Moreover x must see r as otherwise either rclxs2is improper or aIcIxs2is. But if x sees both p and r then either rxpal is improper or rxpa, is.

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Step 3. The following algorithm, given a partition as described in Step 1, will find a good partition of G = { K , U K 2 , A l U A 2 ,E, Q, F } . Algorithm 3.1. Creating a good partition. (Al) While there exists an element x of E s.t. x does not see all of A ,

move x

into K 2 . (A2) While there exists an element x of E s.t. x does not see all of A2 move x into K , . (A3) If there exists an element x of F s.t. x does not see all of K1 move x into A2 and return to Step A l . (A4) If there exists an element x of F s.t. x does not see all of K1 move x into A , and return to Step A l . The following properties hold throughout the application of Algorithm 3.1, ensuring that it will eventually terminate with a good partition of G: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv) (xv) (xvi)

A, U A2 is independent. K , U K 2 is a clique.

Each vertex in F misses every vertex in A l U A2. Each vertex in E sees every vertex in K1 U K 2 . Each vertex in Al sees every vertex in K1. Each vertex in u2 sees every vertex in K 2 . No vertex in Q sees any vertex in A , U A2 U K1 U K 2 U F. r sees every vertex in K1 U K 2 . p sees every vertex in A , U A 2U K , U K 2 . Each vertex in F sees either all K1 or K 2 . Each vertex in E sees either all Al or A2. Each vertex in Al misses some vertex in K 2 . Each vertex in A2 misses some vertex in K 1 . Each vertex in K1 misses some vertex in A2. Each vertex in K 2 misses some vertex in A l . The following edges are in U : ale; for each a l E A l for each a2 E A2 z sfor each k lE K1 k2s2for each k2 E K 2 pa; for each a, E A l u 2 p for each a2E A2 for each klE K1 rk2 for each k2 E K 2 + kip for each k, E K , pk; for each k2 E K 2 . (xvii) p is in E. (xviii) r is in F.

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Step 3.1. The eighteen properties hold before the algorithm is applied. Properties (i), (ii), (iii), (iv), (vii), (x), (xi) are equivalent to the following properties (respectively) of Step 2: (ii), (i), (vii), (x), (vi), (ix), (xi). Properties (v) and (vi) follow property (iii) of Step 2. Properties (viii), (ix), (xvii), (xviii) are trivially true. Properties (xii) through (xv) are also trivial, with sl, s2, cl, c2 serving as the necessary distinguished vertices. Property (xvi) has been previously established in Step 2. Step 3.2. N o property k first violated in Step A1 or Step A2. We concern ourselves first with Step A l . Assume that all the properties hold before some vertex, x , is moved from E to K 2 in Step A l . We shall show that all the properties still hold after the relabeling. The only sets affected by this relabeling are E and K 2 , thus properties (i), (iii), (v), (xi), (xii), (xiii), (xiv), and (xviii) still hold after the relabeling. Furthermore, (ii) will continue to hold because of (iv) while (vi) still holds because of (xi). Moreover, (xvii) will remain true because of (ix). It is also obvious that ( x v ) remains true. We shall now examine the vertex moved in more detail. We shall label the vertex of A l which x misses a. Now, x sees r as otherwise either rclxs2 is improper or aclxs2 is, Thus, (viii) still holds after the relabeling. Furthermore, if x misses p then aps2x is improper. Thus x sees p and (ix) remains true. At this point, we see that (xvi) will still hold: simply consider the P4s rxpa and aclxs2. Also if some vertex, q, in Q saw x then one of qxcla or qxpa would be improper (recall from (iv) of Step 2 that no element of Q sees p). Thus x misses every element of Q and (vii) is still valid. The proof that (x) remains true is analogous to the proof of property (ix) of Step 2 and is left to the reader. Now we shall prove that property (iv) holds. Note that p c E l . Consider the situation before we move a vertex x of E into K z . Let E' be the set of vertices of E that miss a. Let El be the set of vertices of E that miss r. We shall first prove the following. For each x in E', y in E l , x sees y . For each x in E', q in Q, x misses q. For each x in E', f i n F,, x seesf.

(3.3)

Note that the P4s rc2ya and rclys2 imply that ya, s z y E U for all y in E l . Now, if (3.1) fails, then either ayslx or ays2x is an improperly directed P4. To see that (3.2) holds, first note that q misses y whenever y E E l , for otherwise either qyclr or qyc2r is an improperly directed P4. Now, (3.2) holds, or else either qxya, or qxc,a is an improperly directed P4. Recall from (vii) of Step 2 that Fl consists of those vertices of F that miss p . Now, if (3.3) fails then either aclxs2 orfclxsz is an improperly directed P4 (the P4 s 2 p c l f implies that c1f E U). Now, we claim that for each y in El'= E - E l - E', we have x seeing y for any x in E'.

(3.4)

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Note that each y in E" sees both a and r. If (3.4) fails then one of the following P4 is improperly directed: uclxs2, ays2x, ayrx (note that by (3.3) x sees r ) , apxr. Next, we shall prove

E' is a clique.

(3.5)

Suppose that (3.5) fails. Let C be a big component of the subgraph of G induced by E'. We may assume that C is not homogeneous in G , or else we have a contradiction. Thus, there are vertices e l , e2 in E' and y not in E' such that y sees el but misses e 2 , and e , misses e2. By properties (iv), (xi) we know that y does not belong to A 2 U K2U K,. By (3.1) and (3.4) we know that y does not belong to E. By (3.2) and (3.3) we know that y does not belong to Q U F,. So y belongs to ( F -4) U A , . Now (3.1) and (3.4) imply that: for a , in A , , x , z in E, if a , misses x and sees z ( x E E' and z E E - E'), then x sees z. This implies that y is not a member of A l . Now, we only have to settle the case where y E F - F,. In this case, one of the following P4s is improperly directed: a c , e l s 2 ,y e l s 2 e 2 ,yelre2 (note that by (3.3) r sees e l and e 2 , and from (viii) of Step 2, r misses all vertices of F), re,pa (by (3.1) and (3.4) p sees e l ) . Now, since E' is a clique and every vertex of E - E' sees every vertex of El, we can move a vertex x of E' to K2 while satisfying property (iv). In summary, we have shown that the relabellings performed in Step A1 of the algorithm do not affect the validity of any of our eighteen properties. A symmetrical argument gives the same result for Step A2. Step 3.3. N o property is first violated in Step A3 or Step A4. We concern ourselves first with Step A3. Assume that all the properties hold before some vertex x , is moved from F to A2 in Step A3. We shall show that all the properties still hold ater the relabeling. The only sets affected by this relabelling are F and A2, thus properties (ii), (iv), (v), (viii), (x), (xii), (xiv), (xv) and (xvii) still hold after the relabeling. Furthermore, (i) will continue to hold because of (iii) while (xviii) still holds because of (viii). Moreover, (vi) will remain true because of ( x ) . It is also obvious that (vii) and (xiii) remain true. We shall now examine the vertex moved in more detail. We shall label the vertex of K1 which x misses k. Now, x sees p as otherwise either x c 2 p s I is improper or xc2ksl is. Thus, (ix) still holds after the relabelling. Furthermore, we know x misses r from property (viii) of Step 2. At this point, we see that (xvi) will still hold: simply consider the P4s xpkr and x c 2 k s l . The proofs that (iii) and (xi) remain true are analogous to the proofs of praperties (vii) and (xi) of Step 2 and are left to the reader. In summary, we have shown that the relabellings performed in Step A 3 of the algorithm do not affect the validity of any of our eighteen properties. A symmetrical argument gives the same result for Step A4. We have shown that these properties remain true throughout the algorithm,

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it follows that we terminate with a good partition { K , U K 2 , A l U A 2 , E, Q, F } of G. As mentioned before, Lemmas 3.2, 3.3, 3.4, 3.5, 3.6 imply Theorem 3.1: if for each edge a b of G, the equivalence class M ( a b ) is acyclic, then G is P,-comparable. In the next section, we shall use this characterization to design a polynomial-time algorithm for recognizing a P4-comparability graph.

4. Algorithms

In this section, we show that the problem of recognizing P4-comparability graphs is polynomial. In fact, Theorem 3.1 suggests that P,-comparability graphs can be recognized in 0(n4) time. (We shall let n denote the number of vertices of a graph.) The following algorithm RECOGNIZE determines whether a given graph G is a P,-comparability graph; if G is, then an interesting orientation of G is returned.

Procedure RECOGNIZE(G) Input: a graph G. Output: a message “G is not a P,-comparability graph”, or an interesting orientation U of G. Complexity: 0(n4) Step 0. Set all edges of G to be unmarked, set U = 0, CEC = 0. (CEC stands for Current Equivalent Class.) Step 1. If there is no unmarked edge, then return U and stop.

-

Step 2. Pick an unmarked edge vlvL Set U t U U PI, v2, CEC t v l v2. Step 3. If there is no unmarked edge in CEC then to to Step 7 else pick an unmarked directed edge %of CEC. Step 4. For each vertex v 3 that sees v 2 and misses v 1 d o begin (do loop) if { v l , v2, v3} extends into a P4, then begin (if) if v2v3 E U , then return “G is not a P4 comparability graph”, and stop if 213212 $ U , then set U +U U CEC t CEC U end (if) end (do loop)

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Step 5 . For each vertex v3 that sees v1 and misses v2, do begin (do loop) if { v l , v2,v3} extends into a P4then begin (if) if v3v1E U then return “G is not a P,-comparability graph” and stop, if v ,v3 $ U then set U + U U%+CEC t CEC U end (if) end (do loop)

-

z.

Step 6. Mark the edge v1v2and go to Step 3. Step 7 . If the graph formed by the edges in CEC contains a cycle then return “G is not a P4-comparability graph” and stop. Else set C E C t O and go to Step 1. It is easy to see that Procedure RECOGNIZE runs in O(n4) time. Steps 3 and 4 can be executed in O(n’) time: testing if {vl, v 2 ,v3} extends into a P4 is equivalent to finding a vertex v4 that (i) sees vl, misses v2, v3, or (ii) sees u 3 , misses vl, v2. We assume that G is represented by its adjacency matrix. Thus we can test the presence of an edge in constant time. Since there at most O(n’) edges, Steps 2, 3 and 4 are executed at most O(n’) time. So Procedure RECOGNIZE runs in 0(n4) time. We have seen that P,-comparability graphs can be recognized in O(n4) time. However, if we want to construct a P,-transitive orientation of a P4-comparable graph, then we have to do more work. We are going to describe the procedure ORIENT(C, U) which given a P4-comparable graph C , returns a P4-transitive orientation U of G in O(n’) time. The Procedure ORIENT first determines whether G contains a homogeneous set by calling the procedure TESTHOMOGENEITY. If G contains a homogeneous set H, then ORIENT is called on H and G - (H - x ) , for an arbitrary vertex x of H, and then ORIENT combines the two P4-transitive orientations of H and G - (H - x ) into a P,-transitive orientation of G (as shown in Lemma 3.3). If no homogeneous set is returned by TEST-HOMOGENEITY, then ORIENT calls the procedure TESTGOOD-PARTITION to test for the presence of a good partition of G. If such a partition is returned, then G is decomposed into two smaller graphs GI and G2; ORIENT is then called on Gl and G2, and then combines the two P4-transitive orientations of G1 and G2 into a P,-transitive orientation U of G (as shown in Lemma 3.4). If no good partition is returned, then the Procedure RECOGNIZE is called on G and returns an interesting orientation U of G. By Theorem 3.1, U is acyclic on EPF. Thus, we can find a P4-transitive orientation U’ of G by extending the restriction of U .

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Procedure ORIENT(G, U) Input: a P,-comparability graph. Output: a P4-transitiveorientation U of G. Complexity: ~ ( n ’ ) . Step 0. Call TEST-HOMOGENEITY(G). - if a homogeneous set H is returned, then begin (if) - select a vertex x of H. . call ORIENT(G - (H - x), Ul), call O R E I , U,). combine Ul and U, into a P,-transitive orientation U , return U and stop. end (if) Step 1. Call TEST-GOOD-PARTITION(G). . if a good partition C , S, P, Q, R is returned, then begin (if) . select nonadjacent vertices c of C , s of S. call ORIENT(G - (C - c ) - (S - s), Ul). . call ORIENT(C U S, U,). * combine Ul and U, into a P4-transitiveorientation U , return U and stop. end (if) Step 2. (Now, G has no homogeneous set and no good partition.) .Call RECOGNIZE(G), let U be the interesting orientation of G that is returned. We can test the presence of a homogeneous set in a graph G by determining whether each pair of vertices {x, y} of G is contained in some homogeneous set H of G. Initially we set H = {x, y } , A = {z: z sees all of H}, B = {z: z misses all of H}. If A U B is nonempty and A U B U H = V ( C ) then H is indeed a homogeneous set of G. Otherwise, the set H’ = V ( G )- H - A - B is nonempty. Now we set H + H U H‘ and repeat the process again until H becomes a homogeneous set of G, or H = V ( G ) .In the latter case, we know that {x, y } does not belong to any homogeneous set of G. This process of testing can be executed in O(n’) time, and so the Procedure TEST-HOMOGENEITY runs in O(n4) time. Actually, there are O(n3) algorithms ([2]) to test the presence of a homogeneous set in a graph. However, for our purpose, the upper bound of 4 n 4 ) is sufficient (since the procedure TEST-GOOD-PARTITION runs in O(n4) time).

Procedure TEST-HOMOGENEITY(G) Input: a graph G. Output: a homogeneous set of H of G , or the message “ G has no homogeneous set”.

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Complexity: 0(n4) Comment: initially all pairs { x , y} of V ( G ) are unmarked.

Step 0. - Pick an unmarked pair { x , y} of V ( G ) , and mark {x,y}. - set H t { x ,y} A t { z : zseesallofH}, B t {z: z misses all of H}, H’+V(G)-A-B-H. Step 1. If H ’ is empty but A U B is not, then return H and stop. Step 2. While H ’ is nonempty do begin (while) . pick a vertex z of H‘. .setH-HU{z},H’+H‘{z}. for each a in A, if a misses z then set H’ +HI U { a } , A + A - { a } . mforeachbin B , i f b s e e s z t h e n s e t H ’ + H ’ U { b } , B t B - { b } . . if H U H’ = V ( G )then go to Step 3. end (while) Step 3. If all pairs of vertices of G are marked then return “C has no homogeneous set”, and stop. Else go to Step 0. To see that the Procedure TEST-HOMOGENEITY runs in O(n4) time, it suffices to note that: (i) it costs O(n) to move a vertex in H’ into H and adjust the sets A , B, (ii) once a vertex is moved from H‘ into H, it remains in H. (iii) once a vertex in A U B is moved into H ’ , it stays in H U H’ It follows that Step 2 takes O(n2) time, and so the Procedure TESTHOMOGENEITY runs in 0(n4) time. We use the same trick for Procedure TEST-GOOD-PARTITION: for each edge xy, we try to determine whether (*) xy belongs to the set C of a good partition C , S, P, Q, R . We are about to show that (*) can be tested in O(n’) time, and so we can test the presence of a good partition in 0(n4) time. Now we assume that the graph G contains no homogeneous set. Let C = { x , y } , S = {z: z sees some but not all vertices of C}. Note that S is nonempty, for otherwise C is a homogeneous set of G (if G has at least three vertices), a contradiction. We can also assume that S is a stable set, otherwise (*) does not hold for xy. Next, define

P = {z: z sees all of C U S } , R = {z: z sees all of C , none of S}, Q = {z: z misses all of C U S}.

C.T. Hobng, B.A. Reed

198

Now, if C U S U P U Q U R = V ( G ) and if there is no edge between Q and R, then (*) holds for xy. As in Procedure TEST-HOMOGENEITY, we define the sets C’ and S’ to be the vertices outside C U S that must be moved into C and S, respectively. So let C’ = {c : c sees all of C, and some but not all of S}, S’ = {s:s misses all of S and s sees some but not all of C}. At this stage, if C U S U P U Q U R U C’U S’ # V ( G ) , then (*) fails for xy. Next, if some vertex q in Q sees some vertex r in R, then we move q into S’ and r into C’. Then we move the vertices in C’(S’) into C(S) while making sure that C, S, P , Q, R satisfy the properties described in Definition 3.1. We repeat this process until we can determine whether (*) holds for { x , y}. The details of our algorithm are given below.

Procedure TEST-GOOD-PARTITION(G) Input: a graph G containing no homogeneous set. Output: a good partition C, S, P , Q, R of G or the message “G has not good partition”. Complexity: O(n4). Comment: initially all edges are unmarked. Step 0. Pick an unmarked edge xy of G. mark xy. Set C t { x , Y 1, S t {s: s sees some but not all vertices of C}. Step 1. If S contains an edge then go to Step 8. Step 2. Set P t { z: z sees all of C U S} , R t { z : z sees all of C and none of S}, Q t {z: z misses all of C U S}. C‘t {z: z sees all of C, and some but not all of S}. S’ +{z: t misses all of S, and some not all of C}. Step 3. If C U S U P U Q U R U C’U S ‘ # V ( G ) then go to Step 8. Step 4. For each vertex q in Q, r in R do if qr E E ( G ) then set C ’ t C ’ U { r } ,S ‘ + S ’ U { q } ,

R + R - { r ) , Q +Q

- (4).

Step 5 . While S‘ is nonempty, do begin (while) . pick a vertex s of S’.

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- if s sees some vertex of S then go to Step 8.

set S’ t S ’- {s}, S t S U {s}. . if s sees some vertex Q then go to Step 8. .for eachp in P that missess, set C ’ t C ’ U {p}, P t P - {p}. for each r in R that sees s, set C’ c C’ U { r } , R t R - {r}. if C U S U C ’ U S ’ = V ( G )then go to Step 8. end (while)

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Step 6. If C’ is nonempty then do begin (if) . Pick a vertex c of C’. . If c misses some vertex of C then go to Step 8. Set C t C U {c}, C’ t C’ - {c}. If c misses some vertex of P - C’ then go to Step 8. * For each r in R that misses c, set S ‘ t S ’ U {r}, R t R - {r}. . For each q in Q that sees r, set S’ t S ’U { q } , Q t Q - { q } . end (if)

-

Step 7. If S‘ is nonempty then go to Step 5. If C ‘ is nonempty then go to Step 6. ‘ If C U S = V ( G )then go to Step 8. If some vertex in Q sees some vertex R then go to Step 4 Else return the good partition C, S, P, Q, R and stop.

-

Step 8. If all edges are marked then return “G has no good partition” and stop. Else go to Step 0. Note that (i) it costs O(n ) time to move a vertex into the set S ’ U C ‘ or the set PUQUR. (ii) It costs O(n) time to move a vertex from S ’ ( C ‘ )to S ( C ) and update the sets C , S, P, Q, R, C ’ , S‘, (iii) once a vertex is moved into C U S, it stays there, (iv) once a vertex is moved into C‘ U S’, it stays in C U S U C’ U S’. From the above remarks, it is easy to see that Steps 5 and 6 can be executed in O(n’) time. So our algorithm runs in O ( E ( G )* n’) time, which is O(n4) in the worst case. As mentioned in the introduction, if a perfect order of a graph G is given, then an optimal coloring of G can be found by the greedy algorithm, in linear time. Let us assume that the greedy algorithm uses k colors on the graph G. Chvital [l] proved the following: if C is a clique consisting of vertices with colors j , j - 1, , . . , k then there is a vertex with color j - 1 that is adjacent to all vertices of C. This clearly suggests an O(n2) algorithm to find a largest clique of G ;

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starting with the clique C consisting of one vertex with color k, we can enlarge C until it has k vertices. So combining the findings in this section, we have the following.

Theorem 4.1. There exists an O(n4) algorithm to recognize a P,-comparability graph ; and there exists an O(n5)algorithm which given a P,-comparability graph G finds an optimal coloring of G , and a largest clique of G.

References [ 11 V. ChvBtal, Perfectly ordered graphs, in Topics on Perfect graphs (North-Holland, Amsterdam, 1984). [2] W.H. Cunningham, Decomposition of directed graphs, SIAM J. of Alg. Disc. Math. 3 (3) 214-228. [3] A. Ghouila-Houri, Catact6risation des graphes non orient& dont on peut orienter les ar&tes de mani2re a obtenir le graphe d’une relation d’ordre, C.R. Acad. Sci. Paris 254 (1962) 1370-1371. [4] P.C. Gilmore and A.J. Hoffman, A characterization of comparability graphs and of interval graphs, Canad. J. Math. 16 (1964) 539-548. [5] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, 1980). [6] A. Hertz, Bipolarizable Graphs, to appear in Discrete Mathematics. [7] C.T. Hoang and B.A. Reed, Some classes of perfectly orderable graphs, submitted.

20 1

Discrete Mathematics 74 (1989) 201-226 North-Holland

ON CONSTRUCTIVE METHODS IN THE THEORY OF COLOUR-CRITICAL GRAPHS Dedicated to Dr. Tibor Gallai on the occasion of his 75th birthday. Horst SACHS and Michael STIEBITZ Technische Hochschule Ilmenau, DDR-6300 Ilmeanau, DDR

Some 30 years ago, G.A. Dirac, T. Gallai and G. Hajds founded and developed the theory of colour-critical graphs as an important method in graph colouring theory. Since then, about 65 papers have been written on this subject containing many ideas how to construct colour-critical graphs with some specified properties. The authors survey the use of constructive methods and, tentatively, discuss their power as well as their limitations.

1. Introduction About 1950 G.A. Dirac introduced the concept of criticality as a methodological means in the theory of graph colouring. He himself, G. Haj6s and T. Gallai were the first to develop special constructions for creating colour-critical graphs and establishing theorems on their properties. Since then about 65 papers on colour-critical graphs have been written (see the list of references). In this paper a survey of constructive methods in the theory of colour-critical graphs is given. 1 . 1 . Concepts and notation Concepts not defined in this paper can be found in [29, pp. 528-5401 or in any textbook on graph theory. Though the main objects of our investigations are graphs, it proves convenient to define the central concepts for the class of hypergraphs. A hypergraph G = (V, E) consists of a finite set V = V ( G ) of vertices and a finite set E = E ( G ) of distinct subsets of V each having cardinality at least two. The elements of E ( G ) are called the edges of G: an edge e E E ( G ) with l e l z 3 is a hyperedge and an edge e E E ( G ) with [el = 2 is an ordinary edge. If e = {x, y } is an ordinary edge of G then e is said to join x and y in C. For x E V ( G )put

N ( x : G) = { y I y

E

V ( C )& y f x & y , x

Ee

for some e E E ( G ) } .

The valency of a vertex x of a hypergraph G, denoted by val(x :G), is the number of edges of G in which x is contained. Let A(G) and 6(G) denote the maximum valency and the minimum valency of the vertices of G, respectively. 0012-365X/89/$3.50 01989, Elsevier Science Publishers B.V. (North-Holland)

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A graph is a hypergraph in which each edge is ordinary. If H and G are hypergraphs and V ( H )c V ( C )and E ( H ) c E ( G ) then we shall briefly write H c G and call H a subhypergraph of G. If G is a hypergraph and X s V ( G ) then C [ X ] denotes the subhypergraph of G induced by X , i.e. V ( G [ X ]= ) X and precisely those edges of G which are contained in X are the edges of G [ X ] .Further, for X c V ( G )and F c E ( G ) , let G - X = G [ V ( G )- XI and G - F = ( V ( G ) , E ( G ) - F ) . For two hypergraphs H and G let H U G denote the hypergraph ( V ( G )U V ( H ) , E ( G ) U E ( H ) ) . A k-colouring of a hypergraph G is a mapping c of V ( G ) into the (colour-) set {1,2, . . . , k} (k 2 1) such that Ic(e)l> 2 for every edge e E E(G). A hypergraph G which admits a k-colouring is called k-colourable. The chromatic number x(C) of a non-empty hypergraph G is the smallest integer k for which G is k-colourable. If x ( C ) = k then G is said to be k-chromatic. The empty hypergraph (0,O) is 0-chromatic. Note that H E C implies x ( H ) S x ( G ) . A hypergraph G is called colour-critical or, briefly, critical if x ( H )< x(G) for every proper subhypergraph H of G ; it is called edge-critical if x(G - {e}) < x ( G ) for every edge e of G; it is called k-(edge-)critical if it is (edge-)critical and k-chromatic. Obviously, a hypergraph G without isolated vertices (i.e. vertices of valency zero) is k-critical if and only if it is k-edge-critical. Let G be a graph and X c V ( G ) .The set X is called a clique (independent set) of G if G [ X ]is complete graph (graph without edges).

1.2. Some basic properties of critical graphs The classes of all k-colourable, k-chromatic, or k-critical graphs are denoted by Col(k), Chr(k), and Cri(k), respectively. Note that for k > 1, Chr(k) = Col(k) - Col(k - l), Cri(k) c Chr(k) c Col(k) c Col(k

+ 1).

Proposition 1.1 (immediate). Every k-chromatic graph contains a k-critical subgraph. A set S of vertices and edges of G whose removal increases the number of components of C is called a cut set; if S = {x} where x E V ( G ) then x is a cut vertex.

Proposition 1.2 (immediate, see [ 4 ] ) . If G E Cri(k) and Q is a clique where Q # V ( C ) then G - Q is connected. In particular this means that G is connected and has no cut vertex. Proposition 1.3 (immediate; see [ 3 ] ) . If G E Cri(k) then b ( C ) 2 k

- 1.

The vertices of a k-critical graph G whose valencies are equal to k - 1 are

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called the low vertices, and the others are called the high vertices, of G ; the subgraphs of G induced by the set of low vertices or the set of high vertices are called the low-vertex subgraph L ( G ) and the high-vertex subgraph H (G), respectively. Let K,, and C,, denote the complete graph and the circuit on n vertices, respectively. An even ( o d d ) circuit is a circuit having an even (odd) number of vertices. In 1941 Brooks [l] proved that if G is a connected graph then x ( C )S A(G) + 1 where equality holds if and only if C is a complete graph or an odd circuit. (For easy proofs see [26, 291 or [54]; see also Section 3.1). It is easy to see that, in terms of critical graphs, Brook’s theorem may equivalently be formulated as

Proposition 1.4. A k-critical graph G has no high vertices if and only if either G = Kk (k > 1) or k = 3 and C = CZq+,( q S 1). The graphs mentioned in Proposition 1.4 act as the “modules” of the low-vertex subgraphs of k-critical graphs (see Section 3. l ) , therefore, we shall call them bricks: B GZ Kk is a k-brick, B = C%+, (q 1) is a 3-brick. If B is a k-brick then we shall say that B has size k and write s ( B ) = k. For k s 3 the k-bricks are the only k-critical graphs.

2. Constructions I Let G and H be graphs, let x E V ( G ) , and let X = { x , , x 2 , . . . , x,,} E V ( H ) (n > 2) be an independent set of H . If the equalities

G - { x } = H - x,

(2.1)

u N ( x , :H ) n

N ( x :G ) =

i= I

hold then G is said to have been obtained from H by identifying x , , x 2 , . . . ,x, (or amalgamating X) to x ; if, in addition to (2.1) and (2.2), also

N ( x i : H )n N ( x j : H )= 0 for 1 s i < j s n (2.3) is satisfied then H is said to have been obtained from G by splitting x into the set { x , , x 2 , . . . , x,}. If a vertex of a k-critical graph is split then the resulting graph is either (k - 1)-chromatic or k-critical, where the latter case may occur if and only if k 3 4 (see [29, Problem 9.201). If e = { x , y } is an edge of C then C l e denotes the graph obtained from G - {e} by identifying x and y ; G l e is said to have been obtained from G by contracting e.

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Theorem 2.1 (Dirac’s construction). Let G1E Chr(kl) and G2E Chr(k2) be disjoint graphs and G = G1V G2. Then (a) G E Chr(kl + k2), (b) G E Cri(k, + k2) if and only if G1E Cri(kl) and G2E Cri(k2). The proof of Theorem 2.1 is straightforward. Letting G1and G2be odd circuits of the same length n, using this theorem in 1952 Dirac [4] obtained 6-critical graphs on 2n vertices and n2 + 2n edges. As easy consequences of Theorem 2.1, we formulate the following two propositions.

Proposition 2.2. For every integer k 3 6 there exist a positive constant ak and infinitely many k-critical graphs G with more than ak IV(G)12 edges. Proposition 2.3. For every integer k 3 6 there exist a positive constant bk and infinitely many k-critical graphs G with 6 ( G )3 bk I V(G)l. That Proposition 2.2 is true also for k = 4, 5 was shown in 1970 by Toft [45] (see also Section 2.4, Example A). The best values of ak are not known; a4= and a6 = are the best values presently known. Whether Proposition 2.3 holds for k = 4, 5, too, is an open problem (see also Toft [47]).

a

2.2. The Dirac-Hajbs construction Let q be a positive integer and let G1 and G2 be two disjoint graphs. For i = 1, 2 let xi’, x:, . . . ,xp, y, be q 1 vertices of G, satisfying (a) Q, := ( x i , xf, . . . , xp) is a clique in G,, (b) e , : = { x ; , y i } E E(Gi), and (c) q E (1, 2) or [q 3 3 and for j = 2, 3, . . . , q either { x i , yl) E E(G,) or (xi, Y 2 ) E E(G2)I (or both). Let G denote the graph obtained from C1U G2 by applying the following operations: (1) delete the edges e l , e 2 , (2) for j = 1,2, . . . , q, successively identify x i and x i to x J (where xic# x j 2 if j l # j 2 ) , and (3) join y , and y2 by an additional edge e*.

+

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Then we shall say that G has been obtained from G, and G2 by the construction C,. Note that Q := {XI, x2, . . . ,x q } is a clique in G. The graphs C ( i )of Fig. 1 are obtained from a copy of C5V K1 (the 5-wheel) and a copy of K4 by the construction Ci, i = 1, 2. The construction C1, usually called Hajds’ construction, was invented before 1959 by Hajds [19]. A graph G is called k-constructible if either G = Kk,or G is obtained from two disjoint k-constructible graphs by the construction C1, or G is obtained from a k-constructible graph by identifying two of its non-adjacent vertices. The following fundamental theorem due to Hajds [19] characterizes the class of graphs which are not (k - 1)-colourable in constructive terms.

Theorem 2.4. Let G be a graph. Then x ( G ) > - k if and only if G has a k-constructible subgraph. In particular, if C is k-critical then G is k-constructible. In 1955 Dirac [7, 101 observed that a construction which is a little bit more special than C, can be used to produce critical graphs.

Theorem 2.5 (the Dirac-Haj6s construction). Let k 2 4 and 1 G q 6 k - 2 and let G denote a graph obtained from two disjoint graphs GI and G2 by the construction C,. Then the following two propositions hold. (I) Zf GI, G2 E Cri(k) then G E Cri(k). (11) Zf q = 1 then G E Cri(k) if and only if GI, G2 E Cri(k). A proof of the essential part of (I) as well as a proof of (11) can be found in Gallai’s paper [16, Satz (2.7), Satz (2.9) and Satz (2.12), respectively]. Since Theorem 2.5 is fundamental and the way of reasoning is typical for dealing with questions treated in this paper, the proof of (I) shall be given here in all details. Proof of Theorem 2.5 (I). Let Hl, H2 denote the graphs uniquely determined by the conditions G - { e * } = HI U H2, Hl = G1- { e l } and H2 = G2 - { e 2 } . Then V(Hl) n V(H2)= Q (see Fig. 2).

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206

V

G

Fig. 2.

Let Gj E Cri(k) (i = 1, 2) and suppose that G has a (k - 1)-colouring c ; then the restriction ci of c to V ( H i ) is a (k - 1)-colouring of Hi = Gi - { e i } satisfying c i ( x l )= c i ( y j ) (i = 1, 2); this implies c(yl) = cl(yl) = cl(xl) = c2(x1) = c2(y2)= c(y2), contradicting {yl, y2} = e* E E ( G ) . Thus x ( G )2 k. Therefore, to prove (I) it is sufficient to show that G - { a } E Col(k - 1) for every edge of G. Let e E E(G).

Case 1. e = e*. Clearly, there are (k - 1)-colourings c1 and c2 of HI and H,, respectively, which coincide on Q (note that Q is a clique of H I as well as of H2). Then c1 U c2 is a (k - 1)-colouring of G - { e * } . Case 2. e E Q , say, e = {d', x'"} ( j ' , j " ~(1, 2, . . . , q}, j ' Zj").Put ei = {x;', x r } ; note that ei E E(GJ (i = 1, 2). G1 and G2 being k-critical, there are (k - 1)-colourings c1 and c2 of C1- {el} and G2- {e'}, respectively, with ci(xi')= c j ( x T ) (i = 1, 2) and we may assume c,(x/;) = c2(x4) for j = 1, 2, . . . , q. Let c: and c: be the corresponding (k - 1)-colourings of Hl - {e} = G1- {el, el}

and H2 - {e} = G2- { e 2 , e 2 } , respectively; clearly, c: and c; coincide on Q. If cT(yl) # c : ( y 2 ) then c ; U c: is a (k - 1)-colouring of G - {e}. If c;(yl) = c;(y2) then, because of hypothesis (c) of C,, c;(yl) 4 c : ( Q ) = c ; ( Q ) where lc:(Q)l= q - 1. Hence, keeping the q - 1 colours of Q fixed, we can choose the colour of y , among the (k - 1) - (q - 1) = k - q 2 2 remaining colours. Therefore, by suitably interchanging the colours of c:, we obtain a new colouring El of H I - {e} with E,(yl) #c:(yl) = c;(y2) such that El U c: is a (k - 1)-colouring of G - {e}. Case 3. e

& Q, e # e * .

W.1.o.g. we may assume e E E ( H l ) . Let e' be the edge

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207

of G1- { e l }=HI corresponding to e. Then GI - { e ' } has a ( k - 1)-colouring c ; with c ; ( y l )# c ; ( x ; ) , therefore, HI - { e } = G I - { e l ,e ' } has a ( k - 1)-colouring c1 with c l ( y l )# cl(xl). Choose any (k - 1)-colouring c2 of H2 = G2- { e 2 } which coincides with cl on Q: then c2(y2)= c2(x1)= cI(xl)f c l ( y l ) , hence c I U c2 is a (k - I)-colouring of G - { e } . This completes the proof of Theorem 2.5(1). 0 Remark 2.6. Let G denote a k-critical graph obtained from two disjoint k-critical graphs G, G2 by the construction C,, q E { 1 , 2 , . . . , k - 2). If GI and G2 have no cut sets consisting of q or fewer vertices then the same holds for G. However, G has a cut set consisting of q vertices and one edge. Remark 2.7. Let C denote a k-critical graph (k 2 4) obtained from two disjoint k-critical graphs GI, G2 by means of the construction C I . Then val(x' :C ) 2 k and v a ~ ( xG) : = vaI(x: C1u G2) for x E V ( G )- { x ' } . The following theorem due to Dirac [12] (for a proof constult [16, Satz (2.7)]) provides a characterization of k-critical graphs having a cut set consisting of two vertices.

Theorem 2.8. Let k a 4 , let G , and G2 be disjoint graphs, and suppose ei = {xi, y i } E E(G,), i = 1, 2. Let G denote the graph obtained from GI U G2 by deleting e l and e2 and identifying x I with x 2 as well as y , with y2. Then G E Cri(k) i f and only if either (a) G1E Cri(k), G2/e2E Cri(k), G2- { e 2 }E Chr(k - l), and e2 is not contained in a triangle of G2 or (b) G2E Cri(k), G l / e lE Cri(k), GI - { e l }E Chr(k - l), and e l is hot contained in a triangle of GI (see Fig. 3).

Fig. 3.

H.Sachs, M. Stiebitz

208

2.3. The Gallai construction

The following generalization of the construction C , is due to Gallai [16, Satz (2.12)].

Theorem 2.9. Let k 2 4 , l S p S q s k - 1 - p . Let G 1 ,G 2 , .. . , G,,,, bepairwise disjoint k-critical graphs which, for i = 1, 2, . . . , p + 1, satisfy: (A) there is a clique A j = { x f , x:, . . . ,x:} of q vertices in Gi; ( B ) there is a set B i z V ( G j ) , where IBil = bj* 1, B j n A j= 0 , and Gi[Aj]V Gi[Bj]c Gi; (C) bl bz+ * * + bP+1S k - 4. Put A ; = {xj, x f , . . . , x f } (i = 1, 2, . . . ,p 1). From G, U G2- . U G,+, construct a graph G in the following way: (i) delete all edges between Ai and Bj in Gi (i = 1, 2, . . . ,p + 1 ) ; (ii) for j = 1, 2, . . . , q , identify the vertices x:,. xi,. . . . ,x,+~; i

+

-

+

(iii) for 1 G i <j S p + 1, join every vertex from Bi with every vertex from Bi. Then the resulting graph G is k-critical. 2.4. The Toft construction

For constructing critical graphs it may sometimes be helpful to operate on critical hypergraphs because one can operate free of the constraint always present in graph theory that all edges have to be of cardinality 2. In 1974 Toft [48] developed some techniques that consist in reducing k-critical hypergraphs to k-critical graphs (for k 2 4 ) . Here we mention the following result; for a proof consult [48].

Theorem 2.10. Let H be a k-chromatic hypergraph (k 2 4) and let e c_ V ( H )be an edge of H. Let G be a k-chromatic graph disjoint from H which has a vertex x of valency at least lei. Obtain the graph G' from G by splitting x into the set e. Define H' ro be the hypergraph with V ( H ' )= V ( H )U V ( G ' ) and E ( H ' ) = ( E ( H ){ e } )U E ( G ' ) . Then (a) x(H')* k. (b) I f H and G are k-critical, then H' is k-critical if one of the following conditions is satisfied. (i) val(x : G ) G 2k - 4. (ii) le)= 2 and G' E Col(k - 1). (iii) G is the join of an odd circuit and a complete graph on k - 3 vertices containing x. Starting from a k-critical hypergraph H ( k 3 4) and using Theorem 2.lO(b)(iii), one can create a k-critical graph G * containing all vertices and all ordinary edge of H such that IV(G*)l G IV(H)I + CucE(N) max(k - 1, lei + 1). Let us mention just two applications.

Consstructive methods in colour-critical graphs

209

Fig. 4

Example A. Let XIand X 2 be disjoint sets of n vertices each, n 3 3 being an odd integer. Join XIto X , by all possible (ordinary) edges. Add the two hyperedges e , = X ,and e z = X , . Then it is easy to check that the resulting hypergraph is 4-critical. Reducing H to a 4-critical graph by means of Theorem 2.lO(b)(iii), we obtain a 4-critical graph G,, with 4n vertices and n 2 + 4 n edges (yielding a 4 = & , see Section 2.1, Proposition 2.2). These 4-critical graphs G,, which have many edges (see Fig. 4) are due to Toft [45, 481. Example B. Let XI, X 2 , X 3 , X4, X 5 be disjoint sets of II vertices each, n 2 3 . Join Xito X i + l (i = 1, 2,3, 4) and X s to X, by all possible (ordinary) edges. Add the five hyperedges X1UX3, X 3 U X s , X s U X 2 , X2UX4, and X 4 U X I .The hypergraph H obtained this way is 4-critical. Reducing H to a 4-critical graph by means of Theorem 2.10(b)(iii), we obtain a 4-critical graph G" on 15n + 5 vertices which has the remarkable property that the removal of at least n2 edges is required in order to reduce the chromatic number from 4 to 2. These 4-critical graphs G" are due to Stiebitz [44] and answer a question of Toft [52]. Using similar techniques Stiebitz (441 proved

Theorem 2.11. For k 2 4 there exist a positive constant b k and infinitely many k-critical graphs G such that it requires the removal of more than bk IV(C)12edges from G in order to reduce the chromatic number from k to k - 2. Concerning the removal of edges from critical graphs, Dirac [5] proved that a k-critical graph (k 2 4) does not have a cut set that consists of fewer than k - 1 edges (see also Toft [49, 541 or LovAsz [29, Problem 9.211).

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Let Hl denote a graph obtained from a k-critical graph G by splitting a vertex of valency k - 1 in G into a set of vertices {xl, x 2 , . . . ,x,,}, p 2 2. Let H2 denote a k-critical hypergraph with at most one hyperedge e = { y l , y z , . . . ,y,,}. Assume that Hl and H2 are disjoint. The graph H obtained from H I and H2 - { e } by identifying the vertices xi and y j for i = 1, 2, . . . ,p is k-critical, by Theorem 2.10(b)(i). Moreover, H has the property that it contains a cut set consisting of k - 1 edges which are not all incident with the same vertex. Conversely, in 1973 Gallai (unpublished) and Toft [48] proved that any k-critical graph with this property may be obtained by the above construction.

2.5. The Zykov-Schauble-Lou&

construction

Before 1949 A.A. Zykov' invented a special procedure which allows a triangle-free (k + 1)-chromatic graph to be constructed from a set of k copies of a triangle-free k-chromatic graph and a (large) set of additional vertices. In 1969 this construction was refined by Schauble [38] in order to generate a sequence of triangle-free k-critical graphs (k = 1, 2, . . .). We shall describe this construction for hypergraphs. Let HI, H,, . . . , Hk be a set of pairwise disjoint hypergraphs. Let

@ = ( A I A ~ uV ( H , ) & I A ~ I / ( H J ~ = forI k

i=1,2, ...,

j=1

k1

and associate with each set A E @ a (new) vertex X , r$U;=,V(H,); put V = {x, I A E Let H = [ H , , H2, . . . , Hk] denote the hypergraph obtained from H I U H2U * * U HkU (V, 0) by joining each vertex X, (A E @) to all vertices of A by an additional (ordinary) edge.

a}. -

Theorem 2.12. (i) If H, is i-critical for i = 1, 2, . . . , k (k 3 2) then H (k + 1)-critical. (ii) If the Hi are all triangle-free then so is H .

= [ H , , H,,

. . . , Hk] is

Proof. (ii) being obvious, we shall prove (i). It is sufficient to show that H is not k-colourable but H - { e } is k-colourable for every edge e of H . Suppose that there is a k-colouring c of H. Clearly, there is a set A E @ with Ic(A)I = k. Hence, there is a vertex y E A with c ( y ) = c(x,), contradicting {y, x,} E E ( H ) . thus H r$ Col(k). Let e be an edge of H. If e E E ( H i ) for some i , then there is a k-colouring of H , U H , U . . . U H , - ( e } such that c(V(H,))={l,2, . . . , j } for j # i and

'

A. A. Zykov, 0 nekotoryh svojstvah linejnyh kompleksov. Mat. Sbornik N.S. 24 (66) (1949) 163-188. A.A. Zykov, On some properties of linear complexes, Amer. Math. SOC.Translation no. 79

(1952) 33 pp.

21 1

Constructive methods in colour-critical graphs

c ( V ( H i ) )= (1, 2, . . . , i - l}. Hence, by a simple argument, Ic(A)I s k - 1 for all A E @, thus c can be extended to a k-colouring of H - {e}. If e is not of this type then e is an ordinary edge joining a vertex xA., for some A* E @, to a vertex y € A * = {y:, y ; , . . . ,y z } where y : E V ( H i ) ;let y = y:. By the i-criticality of Hi there is an i-colouring ci of Hi such that c i ( y t ) = i and c i ( V ( H i ) - { y ’ } ) = (1, 2, . . . , i - l} (i = 1, 2, . . . , k). It is not difficult to see that the k-colouring c = c1 U c2 U . . U c k of H I U H2 U * - U Hk satisfies Ic(A)I 6 k - 1 for every A € @ , A # A * . therefore, c can be extended to a k-colouring c’ (with c’(xA.) = io) of H - {e}. 0 The above construction was used in 1973 by LovSsz [25] to obtain critical graphs having a large independent set. For a graph G let a(C) be the stability (or independence) number of G, i.e. the cardinality of a largest independent set of G. For i = 3 , 4 , . . . , k - 1 ( k a 4 ) let Hi be an i-critical graph on N * k + 1 vertices, N odd. (Such graphs exist, by Dirac’s and Hajbs’ construction). Further, let HI = K1 and H 2 = (X,{X})with = N; note that H2 is a 2-critical hypergraph on N vertices. Then H = [HI,H2, . . . , Hk-l] is a k-critical hypergraph. Use Theorem 2.10(b)(iii) to reduce H to a k-critical graph G. If G has n vertices then n 2 a ( G )2 NkV2and n S 1 + (k - 2)N NkP2 N (k - 3) 1=s Nk-2 kN S a ( G ) kN, hence

1x1

+

+

+

n - a ( G )
+ +

+

(*I

In 1973 LovSsz [25] proved that the order of magnitude in (*) is best possible; more precisely, he proved

Theorem 2.13. Let ak(n)= max a ( G ) over the set of all k-critical graphs G on n vertices. Then (k/6)n1’(k-2)
---

212

H. Sachs, M. Stiebitz

Z

Theorem 2.14. Let k 2 2. I f G E Cri(k) then M,(G) E Cri(k + 1). In 1985 Tuza and Rod1 [57] observed that M , ( K k ) E Cri(k + 1) for all r 2 1; thus they obtained infinitely many k-critical graphs ( k 2 3 ) which can be made bipartite (i.e. 2-chromatic) by the omission of only k(k - 1)/2 edges. They also proved in [57] that this result is best possible. The proof of the following proposition is straightforward.

Proposition 2.15. Let G E Cri(k), where k a 2, and G' = M,(G) for some r 2 3. Then for every edge e of G' we have G ' - { e }E Chr(k). In particular, this means that either G' E Chr(k) or C' E Cri(k 1).

+

In general it is not true that x(M,(G))= x ( C )+ 1. However, in 1985 Stiebitz [42] proved

Theorem 2.16. Let M(3) denote the set of all 3-critical graphs (i.e. odd circuits) and for k 2 3 let M(k + 1 ) = { M , ( G ) I G E M(k) & r 2 1). Then M(k) G Cri(k) for all k 3 3. For the proof we need a result of LovBsz. The neighbourhood complex of a graph G , denoted by N ( G ) , is a simplicia1 complex whose vertices are vertices of C and whose simplicia are those subsets X of V ( G ) for which there is a vertex x E V ( G )with X E N ( x : G). In 1978 L O V ~ S[28] Z proved

Theorem 2.17. I f N ( G ) is a k-connected topological space then x ( G )2 k + 3.

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We shall only outline the proof of Theorem 2.16 which uses induction over k; for the details consult [42]. Because of Proposition 2.15 it suffices to show that M ( k ) G Chr(k). We shall prove that for every C E M(k) the topological space N ( G ) is homotopy equivalent to the (k - 2)-dimensional sphere Sk-2 which is a (k - 3)-connected space. Clearly, this is true for k = 3. If we assume that N ( G ) is homotopy equivalent to SkP2 for C E M ( k ) then it is not too difficult to verify that N ( M , ( G ) ) is homotopy equivalent to the suspension of N ( G ) , and, therefore, to Because M ( k ) E Col(k) it now follows that M ( k ) c Chr(k) for all k 2 3. 0 For a graph G containing an odd circuit let odd(G) denote the minimum length of an odd circuit in G. Let G be a given graph containing an odd circuit and put G’ = M,(G) for some r 3 1. The notation is the same as at the beginning of this section. Consider an odd circuit C of length 1 contained in G’. If z E V(C) then C contains at least two vertices of each of the sets Xi,1< i < r (note that G’ - E ( G ’ [ X , ] )is bipartite). Thus we have 1 3 2r 1. If z $ V(G) then put X‘ = {xi’ I xi’ E XI & xj E V(C) for some i 6 r}. Clearly, ( X ‘ (S 1 and C is contained in the graph M,(G’[X’]) - {z}, thus G’[X’] contains an odd circuit (which is contained in the “projection” of C into C’[X,]). Hence, 12 1X’I 3 odd(G.). This proves that odd(M,(G)) = min{odd(G), 2r l}. For r 3 1 let G(3, r) be an odd circuit of length 2r 1 and, for k 3 3, put C(k 1, r) = M,(C(k, r)). Then, by Theorem 2.16, G ( k , r) E Cri(k) where odd ( G ( k ,r)) = 2r + 1. An easy calculation shows that (V(G(k, r))l S (lDkP3)(2r+ l ) k - 2 (k 3 3). Thus for, all k 3 4 we have obtained infinitely many k-critical graphs C with odd (G) 2 21-1’1’(k-2)I V(C) I l ’ ( k - 2 ) . For k = 4 such graphs were first constructed in 1963 by Gallail [16] (with odd(C) 3 lV(G)11’2). Other examples can be found in a paper of Kierstead, Szemerbdi and Trotter [22]; in that paper it is also proved that for G E Cri(k) we have odd(G) < C, (V(G)11’(k-2)for k 3 4.

+

+

+

+

2.7. Concluding remarks (1) Starting from G2= K 2 , by the above construction (G,,, = M2(G,) for k 3 2) Mycielski [33] obtained a triangle-free k-chromatic graph for every k 3 3. These graphs are all critical and G3 and G4 are the unique smallest 3- and 4-chromatic graphs without triangles2. Gk has 3 2k-2 - 1 vertices. However, using probabilistic (non-constructive) arguments, in 1959 Erdos3 proved that for

-

*This was proved in 1973 by V. ChvBtal: The minimality of the Mycielski graph. In; Ruth A. Bari et al., ed., Graphs and Combinatorics (Proc. Capital Conf. Graph Theory and Comb., G . Washington Univ. June 1973), Lecture Notes in Mathematics 406 (Springer, Berlin, 1974) 243-246. 3 P . Erdos, Graph theory and probability. Canadian J . Math. 11 (1959) 34-38; Graph theory and probability, 11. Canadian J . Math. 13 (1961) 346-352.

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every E > O there is a k0(c) such that for all k > k o ( & ) there is a k-chromatic triangle-free graph on no more than k2+&vertices. Erdos3 also proved that for all positive integers k and 1 there are k-chromatic graphs without circuits of length smaller then 1. Extending the domain of operations to hypergraphs, in 1967 LovBsz4 gave a constructive proof of this result. Moreover, recently a purely graph-theoretic construction for such graphs has been obtained by K X 4 . (2) Using the Dirac construction it is easy to obtain, for k 2 6, k-critical graphs without low vertices. But all 4-critical graphs which can be obtained by one of the constructions described in this section do have low vertices. The first example of a 4-critical graph without low vertices was given in 1963 by Gallai [16]. By an ingenious construction he found an infinite collection of 4-critical graphs which are all 4-regular, non-planar and embeddable in the Klein bottle. Fig. 6 shows such a graph (the smallest in this collection) which has 12 vertices. A planar 4-critical and 4-regular graph was constructed in 1985 by Koester [23].

Fig. 6.

(3) It is easy to construct a sequence of k-critical graphs G,, where G, = Kk and G,+, is obtained from G,, and a copy of Kk by means of a Dirac-Hajbs construction C [ k / 2 ] , such that A(G,,) = [(3k - 1)/2] for n = 2, 3, . . . (k 3 4). Does there exist a constant c such that for all k 3 3 there are infinitely many k-critical graphs G satisfying A(G) 6 k c?

+

4 L . Loviisz, Graph and set systems, in: H. Sachs et al., ed.. Beitrage zur Graphentheorie. (Proc. Internat. Coll. Manebach (DDR), May 1967) (Teubner, Leipzig, 1968) 99-106; On chromatic number of finite set systems, Acta Math. Acad. Sci. Hungar. 19 (1968) 59-67. Also: I. Klit, A hypergraph-free construction of highly chromatic graphs without short cycles, submitted to Combinatorica.

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3. The theorems of Gallai and Greenwell/Lovasz Gallai [16] and Greenwell and L O V ~ S[18] Z characterized the class of all low-vertex subgraphs and the class of all proper subgraph of k-critical graphs, respectively.

3.1. The theorems of Gallai A maximal subgraph B of a graph G such that any two edges of B are contained in a circuit of G is called a block of G. The set of all blocks of G is denoted by B ( G ) . Clearly, a vertex of C is a cut vertex if and only if it is contained in more than one block of G. An end block of G is a block which contains at most one cut vertex of G. Two blocks which have a vertex in common (they cannot have more than one vertex in common) are called adjacent.

Definition 3.1. A connected graph all of whose blocks are bricks is called a Gallai tree; a Gallai forest is a graph all of whose components are Gallai trees. A k-Galiai-tree (-forest) is a Gallai tree (forest) whose vertices have valencies at most k - 1. The graph T depicted in Fig. 7 is a k-Gallai-tree for each k 3 8. The following theorems-both due to Gallai [161-are fundamental.

10

Fig. 7.

Theorem 3.2. If the graph G b k-critical then L(G) is a k-Gallai-forest (possibly empty)Note that if G is k-critical where k 3 4 and G is not a complete graph o n k vertices then, evidently, L ( G ) cannot contain a complete graph on k vertices as a componerit.

Theorem 3.3. Let L' be a k-Gallai forest ( k a 4 ) which does not contain a

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complete graph on k vertices as a component (L' is allowed to be empty). Then there is a k-critical graph G with L ( G ) L'.

We shall here give a proof, different from Gallai's, only for Theorem 3.2 which needs some preparation. For the proof of Theorem 3.3 the reader is referred to [16]; for the case L' # O an alternative proof can also be based on the results of Section 4.1 (Theorems 4.5, 4.6 using HajBs' construction C,to abolish undesired lower vertices). So also Remark 4.11. In what follows, let C denote an arbitrary k-critical graph having low vertices = m 3 1. Put G' = G - X and H = G [ X ] . and let X G V ( L ( G ) )with If c is any (k - 1)-colouring of G' (recall that c ( V ( G ' ) )G {1,2, . . . , k - l}), then, for any sequence 1 = ( x l , x 2 , . . . ,x,) of the m vertices of H I construct a colouring c' of G , depending on 1 and extending c from G' to G , in the following way. (i) for x E v(G') put c'(x) = c ( x ) . (ii) Suppose that i = 1 or i > 1 (i =sm ) and c'(x,) has already been defined for j = 1, 2, . . . . , i - 1. Let Xi = N ( x , : G ) - { x i + l ,xi+2, . . . ,x,} and put c'(x,) = r where r is the smallest among all integers s 3 1 with s $ c'(Xi). Then, clearly, c' is a k-colouring of G. Call the sequence 1 a feasible labelling of H if for i = 1, 2, . . . m , H [ { x i ,x i + l , . . . , x,}] is connected (this means, in particular, that for i = 1,2, . . . , m the vertex xi is adjacent to a vertex xi in H where j > i ) . If 1 is a feasible labelling of H then Ic'(N(x,: G))I = k - 1 since, otherwise, c' were a (k - 1)-colouring of G, contradicting G E Cri(k). It is easy to see that for any connected graph r with m vertices and for every x E V ( T ) there is a feasible labelling ( x , , x 2 , . . . , x,) of r s u c h that x, = x . For x E X,put M(x :G') := N ( x :G ) f l V ( G ' )= N ( x :G) - X. Assume that H is connected and has not cut vertex and let c be an ay (k - 1)-colouring of G'.

1x1

Claim 2. H is s-regular for some s 3 1 (i.e. S ( H ) = A ( H ) = s) and H 1).

E Chr(s

+

Proof of the claims. Suppose that Claim 1 is not true. Then there are two adjacent vertices x , y E X such that c ( M ( x : G')) # c ( M ( y : G ' ) ) . W.l.0.g. let i E c ( M ( x : G ' ) )- c ( M ( y : G')) and put Y = X - { y } . Clearly, there is a (k - 1)colouring h of G - Y such that c c h and h ( y ) = i. G[Y]being connected, there is a feasible labelling 1 = ( x , , x 2 , . . . , xmPl) of G[Y]with xmP1= x . From the fact that h ( z ) = i for at least two distinct vertices z E N ( x :G) we conclude that h' is a ( k - 1)-colouring of G , contradicting G E Cri(k). This contradiction proves Claim 1.

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By Claim 1, there is a set C c (1, 2, . . . , k - l} such that c ( M ( x : G ’ )= C for all x E X. Moreover, IM(x : G’)I = ICI for all x E X since, if there were an x * E X such that IM(x* :G’)l> IC( then, by the same arguments as above, from a feasible labelling I* = (xl, xz, . . . ,x,) of G [ X ] with x * = x, we would obtain a (k - 1)-colouring c‘* of G. Thus, H is s-regular where s = k - 1 - ICJ and, clearly, H E Chr(s + 1). This proves Claim 2. 0 A graph which is connected and has not cut vertex is called 2-connected.

Proof of Theorem 3.2. It suffices to show that every 2-connected induced subgraph H of L ( C ) is a brick. This can be done by induction over the number m of vertices of H. For m = 1, 2 the assertion trivially holds. Let m 2 3 and suppose that every 2-connected induced subgraph of L ( G ) with fewer than m vertices is a brick. Consider any 2-connected induced subgraph H of L ( G ) with m vertices. Case 1. There is a vertex x in H such that H - { x } is 2-connected. Then, by the induction hypothesis, H - {x} is a brick. Because of Claim 2, H is regular and we conclude that H - { x } as well as H are complete graphs. Case 2. For every X E V ( H ) , the graph H - { x } has a cut vertex. Let T = H - { x } for an arbitrary x E V ( H ) . Then, by the induction hypotheses, every block of T is a brick, hence T is a Gallai tree. H being regular (by virtue of Claim 2) we easily convince ourselves that T is a path and H is a circuit. Because of Claim 2, H is not 2-colourable, therefore, H is an odd circuit. Theorem 3.2 is now proved. 0 Clearly, a k-Gallai-forest F (k 2 4) is k-chromatic if and only if F contains a complete graph on k vertices as a component. This observation makes Brooks’ theorem (see Section 1.2) an easy consequence of Theorem 3.2. The above proof of Theorem 3.2 based on sequential colouring is a refinement of Lovisz’ proof of Brook’s theorem (see [26]). The original proof as given by Gallai is based on sequential colouring and colour interchange (see also [54]). 3.2. The theorem of Greenwell and L o v h z A graph G is called contractibly r-colourable if G itself and all graphs Gle, e E E(G), are r-colourable. Put, for r 2 1, Con(r):= {H I H E Col(r) & H l e E Col(r) for every edge e E E ( H ) } . Note that Col(r - 1) c Con(r) c Col(r) (r 2 2). Let G be a k-critical graph and e = { x , y} an arbitrary edge of G . Then G - {e} E Col(k - 1). From C 4 Col(k - 1) we infer that c ( x ) = c(y) for every (k - 1)-colouring c of G - {e}, hence C l e E Col(k - 1). This implies that if H is a proper subgraph of a k-critical graph then H E Con(k - 1). That the converse of

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this statement is also true was proved in 1974 by Greenwell and LOV~SZ [18] (see also [29, Problem 191):

Theorem 3.4. A graph H is a proper subgraph of some k-critical graph if and only if H E Con(k - 1) (k 3 2). For some extension of Theorem 3.4 the reader is referred to the papers of Muller [31, 321. Note that this characterization of the set of all proper subgraphs of k-critical graphs is not of such a simple form as Gallai’s characterization of the class of all low-vertex subgraphs since all graphs H and H l e have to be tested for (k - 1)-colourability (for k 2 4, this problem is NP-complete). However, note also that testing for the membership in Con(k - 1) requires only testing for colourability whereas testing for the membership in Chr(k) or Cri(k) requires testing also for non-colourability.

4. Constructions I1 In this chapter a general method for constructing k-critical graphs is described, applied, and some conclusions are drawn. The domain of operation of this method consists of k-critical graphs all being isomorphic to Kk or Kk-3V C2q+l. All these graphs have low vertices, and from a given set of such graphs a new k-critical graph having low vertices, too, is created. 4.1. A general procedure for constructing k-critical graphs which have low vertices

In this section a procedure is described which, starting from a suitable Gallai tree T and a vertex set Nq disjoint from V ( T ) , enables many k-critical graphs to be constructed (see the authors’s papers [35, 36, 37, 41, 421. We need some preparation. Definition 4.1. Let T be a Gallai tree. A subset B, of B ( T ) is called a matching of T if every vertex of T is contained in at most one block of B, . A matching B, of T is called perfect if every vertex of T is contained in exactly one block of B 1 . Let T be a Gallai tree having a perfect matching B1. Note that, because of the tree-like structure of T , B1 is uniquely determined, therefore we shall write B, = B,(T). For the Gallai tree of Fig. 7, B l ( T )= { B l , B4, B,, B6, B,, Blo} is its unique perfect matching. Definition 4.2. For k 3 4 and q 2 1 let Tk(q) denote the set of all k-Gallai-trees T having a perfect matching Bl( T ) such that (1) for each B E B , ( T ) , s ( B ) B k - q ; (2) for each B E B ( T ) - B l ( T ) , s ( B ) S q + 1.

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Note that K k - l and Kk belong to all sets T,(q) ( q 3 1). For q 2 1 let Nq = {1,2, . . . , q } .

Definition 4.3. Let k 3 4 and q 3 1 be given integers, put r = k - q. Denote by Uk(q)the set of all pairs ( T , u ) satisfying (1) T E Tk(q), (2) u is a mapping of B ( T ) into the power set of N, such that (2.1) for each B E B,(T), lu(B)I = s ( B ) - r ; (2.2) for each B E B ( T ) - B , ( T ) , lu(B)I = s ( B ) - 1; (2.3) for any pair of adjacent blocks B’, B ” EB ( T ) , u(B’)n

u(B“)= 0.

Let T E Tk(q)be any Gallai tree: note that, because of the tree-like structure of T, there are many mappings u such that (T, u ) E U,(q).

Definition 4.4. For k 3 4 let U ; denote the set of all pairs (T, u ) satisfying (1’) T is a k-Gallai-tree, (2’) u is a mapping of B ( T ) into the power set of Nk-, such that (2’.1) for each B E B ( T ) , l u ( B ) I = s ( B ) - l ; (2’.2) for any pair of adjacent blocks B’, B” E B ( T ) , u ( B ‘ ) n u(B”) = 0. The construction K . Let q 3 1 and k 2 4 be given integers and let (T, u ) E Cl,(q) where V ( T )n Nq = 0. The aim of the construction to be described (and which we call “the construction K ” ) is to obtain a k-critical graph G = G ( T , u , q ) containing T as an induced subgraph. Let B , ( T ) denote the unique perfect matching of T and put B,(T) = B ( T ) B , ( T ) . For every B E B , ( T ) , let C ( B ) denote the complete graph with the vertex set Nq - u(B) and put G ( B ):= B V C ( B ) . By property (2.1) of u (see Definition 4.3), C ( B ) is a complete graph on k - s ( B ) vertices, hence G ( B ) is k-critical. Let B E B,(T), x E V ( B ) , i E u ( B ) , x is also contained in exactly one brick B* E B , ( T ) hence, by property (2.3) of u (see Definition 4.3), i $ u(B*)implying that { x , i } is an edge of C ( B * ) .Put

F = { { x , i } 1 there is a B E B,(T) such that x

E

V ( B ) and i E u(B)}

and

Clearly, F E E ( G , ) . Put

and denote the resulting graph GR by C ( T , u, q ) . It can easily be shown that G ( T, u, q ) is connected.

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Let H(T, u, q ) := UBEB,(T) C(B). Clearly, G ( T , u, q ) is a subgraph of T V H ( T , u, q ) , and T as well as H ( T , u, q ) are induced subgraphs of G ( T , u, q ) . Note that V ( H ( T ,u, q ) ) G Nq (possibly, V ( H ( T ,u, q ) )# N q ; see Remark 4.8). In 1981 the authors [36] proved

Theorem 4.5. Let ( T , u ) E u k ( q ) . If H ( T , u, q ) E Con(k - 1) (in particular, if q 6 k - 1) then G ( T , u , q ) E Cri(k) and every vertex of T is a low vertex of G ( T , u, 9). The interested reader finds a detailed discussion of the construction K in [37].

The construction K ’ . The construction K has the disadvantage that Gallai trees without a perfect matching are excluded. In what follows we shall describe a possible extension of K that can be applied to arbitrary Gallai trees (see Stiebitz [41] or the authors’ paper [371). Let k 2 4 and ( T , u ) E U ; where V ( T )f l Nk-l= 0. Let H be the complete graph with vertex set N k P l and put

G’=TVH,

F’ = { { x , i} there is a B E B ( T ) such that x

E

V ( B ) and i

E u(B)},

G ( T , U ) = G’ - F ’ . In 1984 Stiebitz [41] proved

Theorem 4.6. Let ( T , u ) E U;. If T does not have a perfect matching then G( T, u ) is a k-critical graph where every vertex of T is a low vertex of C ( T , u ) . Another way of dealing with a k-Gallai-tree (or-forest) which does not have a perfect matching is to embed it in a k-Gallai-tree which does have a perfect matching - a simple device which works also the other way round. 4.2. Some remarks

Remark 4.7. Let ( T , u ) E uk(q).If T = B is a brick then G(T, u, q ) is a k-critical graph of the form B V C ( B ) . Remark 4.8. For ( T , u ) E I / k ( q ) , put I = I ( T , u ) = n B E B , ( T )u ( B ) . Clearly, I c Nq and V ( H ( T ,u, q ) ) = Nq - I . Suppose T Kk and (I1 = q’ 2 1. Then I is a proper subset of Nq and we may assume w.1.o.g. that I = { q - q ‘ + l , q - q ’ + 2 , . . . , q } , i.e. N q - I = N q - q , . For B E B ( T ) , put u ’ ( B )= u ( B ) - I. Then ( T , u ’ ) E Uk(q - 4’) and G ( T , u ’ , q ’ )= G ( T , u, 4 ) . (see ~361.1

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Remark 4.9. Let T be a k-Gallai-tree having a perfect matching. Then the following statements hold. (a) There is a positive integer qo such that T E T,(q) for every q 3 qo. (b) There is a positive integer qo(T,k) such that II(T, u)l > 0 for q 2 qo(T,k) and ( T , u ) E W q ) . As a consequence, we obtain Remark 4.10. For a given integer k 3 4 and a given k-Gallai-tree T the set of k-critical graphs obtainable from T by means of the construction K or K' is finite. Remark 4.11. The construction K' may be used to prove Theorem 3.3 for the case L' # (0,0): Let F be any k-Gallai-forest which does not contain a complete graph on k vertices as a component. There is a k-Gallai-tree T without a perfect matching which contains F as an induced subgraph. Clearly, there is a mapping u such that ( T , u ) E Uk. The graph G* = C ( T , u ) is k-critical and F E T E L(G*). In order to abolish undesired low vertices we may repeatedly apply Hajbs' construction (C,) using k-critical graphs without low vertices (which do exist: see Section 2.7 (2)). Eventually, we obtain a k-critical graph G where L ( C )= F. 0 4.3. Embedding given graphs in critical graphs

In 1981 the authors [36] proved by constructive means that every connected graph H can be represented in the form H = H(T, u, q ) for some ( T , u ) E U,(q) and used this result to establish the fact that every proper subgraph of a k-critical graph (k 2 4) can be made the high-vertex subgraph of some k-critical graph. In particular, in [37] the authors proved

Proposition 4.12. Let k 3 4 be an integer and let m 3 be an odd integer satisfying m(k - 3) 3 4. Let H E Con(k - 1) be a connected graph with q > 1 vertices. Then there is a pair ( T , u ) E Uk(q) satisfying (i) H ( T , u, q ) = H ; (ii) val(i: G ( T , u, q ) ) 3 m(k - 3) for every i E N,; (iii) IV(T)I = m(k - 2) IE(H)I. From Proposition 4.12 we obtain (see [36])

Theorem 4.13. Let k 3 4 and d 3 1 be given integers. Then for every non-empty graph H E Con(k - 1) there is a k-critical graph G satisfying (1) H ( G )= H ; (2) L ( G ) is non-empty and has the same number of components as H ( G ) ; (3) val(x : G) 2 d for every x E V ( H ( G ) ) .

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As a consequence of Theorem 4.13 we obtain the following characterization of the class of high vertex subgraphs of k-critical graphs which have low vertices. Corollary 4.14. Let k 2 4 . For any non-empty graph H the following three statements are equivalent. (a) H E Con(k - 1). (b) H is the high-vertex subgraph of some k-critical graph which has low vertices. (c) H is a proper subgraph of some k-critical graph. Note that, in particular, Corollary 4.14. implies Theorem 3.4. From Proposition 4.12 and Theorems 4.5, 4.13 we obtain

Corollary 4.15. For every k b 4 there is a constant c = ck such that any connected graph H E Con(k - 1 ) with at least one edge is contained as an induced subgraph in a k-critical graph G where IV(G)l 6 c IE(H)I c lV(H)12. In 1974 Toft [48]proved that for every integer k 2 4 there exists a constant k f k such that any graph H ~ C o l ( k - 2 ) is contained as a subgraph (not necessarily induced) in a k-critical graph of at most 2 IV(H)I + k f k vertices. Whether or not this remains true if Col(k - 2) is replaced by Con(k - 1 ) (where the factor 2 may be replaced by a constant depending on k) is an open problem.

4.4. k-critical graphs whose high-vertex subgraph is (k - 2)-colourable, or is a complete graph In 1963 Gallai [16]proved the following theorem characterizing the set of all k-critical graphs which have exactly one high vertex; this result is implicitly contained also in a paper of Dirac [lo].

Theorem 4.16. (a) Let k 2 4 and let G be a k-critical graph which has exactly one high vertex. Then L ( G ) E Tk(l). (b) Let k 3 4, let T E Tk(l ) , and assume that T has at least two blocks. Connect every vertex of T which is not a cut vertex with an additional vertex z. Then the resulting graph G is k-critical, L ( G ) = T , and H ( G ) consists of the single vertex z. The construction described in statement (b) of Theorem 4.16 is a special case of the construction K: this is just the way how to obtain G ( T , u, 1 ) for ( T , u ) E uk(1). In 1984 Stiebitz [41, 421 characterized a considerably larger class of critical graphs which can all be obtained by means of the constructions K and K’: see the three following theorems; for proofs the reader is referred to [37]and [41].

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Theorem 4.17. Let G E Cri(k) where k 3 4 and suppose that L ( G ) is connected and H ( G ) E Col(k - 2). Put IV(H(G))I= q. Then the following statements hold. (1) T = L ( G ) E Tk(q). (2) There is a mapping u such that ( T , u ) E uk(q)and G = G ( T , u, q ) . Theorem 4.18. Let G E Cri(k) where k a 4 and suppose that H ( G ) is a complete graph on q vertices where 1S q S k - 2. Then T = L ( G ) E Tk(q) and there is a mapping u such that ( T , u ) E uk(q)and G G ( T , u , q ) . Theorem 4.19. Let G E Cri(k) where k 3 4 and suppose that H ( G ) is a complete graph on k - 1 vertices. Then T = L ( G ) is a k-Gallai-tree and there is a mapping u such that ( T , u ) E U ; and G = G( T, u). As a consequence of these results and Remark 4.10, we obtain

Theorem 4.20. Let k 3 4 and let T be any k-Gallai-tree. Then the set of k-critical graphs G satisfying L ( G ) = T and H ( C ) E Col(k - 2), or H ( G ) = Kk-,, is finite. 4.5 Concluding remarks (1) Briefly speaking, the essence of Theorems 4.17-19 is that every k-critical graph (k 3 4) whose low-vertex subgraph is connected and whose high-vertex subgraph is (k - 2)-colourable, or a complete graph on k - 1 vertices, is K-constructible, or K’-constructible, respectively. Thus, from the viewpoint of constructibility, the cases H ( G ) E Col(k - 2) and H ( G ) = K k - , are essentially mastered whereas the cases H ( G ) E Chr(k - 1) ( H ( G )p Kk-,) and H ( G )E Chr(k) (i.e. L ( G ) = 0) are far from being solved. (2) Evidently, there are close relations between the constructions K and K’ on the one hand and the constructions of Gallai and Dirac-Hajds on the other. The rules of these constructions which all start from a set of k-critical graphs can be put in terms of deleting some edges, identifying vertices of certain disjoint cliques, and adding some edges. In particular, applying the construction K, the number of edges to be deleted is just twice the number of edges to be added: exactly the same happens in applying the Dirac-Hajds construction. (3) The concept of criticality-invented for simplifying graph colouring theory-has given rise to numerous investigations and beautiful theorems. However, at the same time it turned out that criticality is not as incisive a restriction as it had been expected to be: there are many k-critical graphs which significantly differ from one another with respect to a variety of properties and parameters and presently it seems to be hopeless to search for satisfactory characterization theorems; in particular this applies to the class of colour-critical graphs without low vertices, a vast area which is almost unexplored. Thus graph colouring theory remains hard. . .

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References Thb list contains all papers concerning colour-critical graphs that have come to the authors’ attention. [I] H.L. Brooks, On colouring the nodes of a network, Proc. Cambridge Phil. SOC. 37 (1941) 194- 197. [2] W.G. Brown and J.W. Moon, Sur les ensembles de sommets indtpendants dans les graphes chromatiques minimaux, Canad. J. Math. 21 (1969) 274-278. [3] G.A. Dirac, Some theorems on abstract graphs, Proc. London Math. SOC.(3) 2 (1952) 69-81. [4] G.A. Dirac, A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. SOC.27 (1952) 85-92. [5] G.A. Dirac, The structure of k-chromatic graphs, Fund. Math. 40 (1953) 42-55. [6] G.A. Dirac, Theorems related to the four colour conjecture, J. London Math. SOC.29 (1954) 143-149. [7] G.A. Dirac, Circuits in critical graphs, Monatshefte fur Math. 59 (1955) 178-187. [8] G.A. Dirac, Map colour theorems related to the Heawood colour formula, J. London Math. SOC.31 (1956) 460-471. [9] G.A. Dirac, Map colour theorems related to the Heawood colour formula (11). J. London Math. SOC.32 (1957) 436-455. [lo] G. A. Dirac, A theorem of R.L. Brooks and a conjecture of H. Hadwiger, Proc. London Math. SOC.(3) 7 (1957) 161-195. [ l l ] G.A. Dirac, 4-chrome Graphen und vollstandige 4-Graphen, Math. Nachrichten 22 (1960) 51-85. [12] G.A. Dirac, On the structure of 5- and 6-chromatic abstract graphs, J. Reine u. Angew. Math. 214/215 (1964) 43-52. [13] G.A. Dirac, The number of edges in critical graphs, J. Reine u. Angew. Math. 268/269 (1974) 150-164. [14] P. Erdos, On circuits and subgraphs of chromatic graphs, Mathematika 9 (1962) 170-175. Also in: J. Spencer, ed., Paul Erdos: The Art of Counting (Selected Writings) (The MIT Press, Cambridge, Massachusetts, 1973) 97-102. [15] S. Fiorini, On the girth of graphs critical with respect to edge colourings, Bull. London Math. SOC.8 (1976) 81-86. [16] T. Gallai, Kritische Graphen I, Magyar Tud. Akad. Mat. KutatB Int. Kozl. (Publ. Math. Inst. Hungar. Acad. Sci.) 8 (1983) 165-192. [17] T. Gallai, Kritische Graphen 11, Magyar Tud. Akad. Mat. Kutat6 Int. Kozl. (Publ. Math. Inst. Hungar. Acad. Sci.) 8 (1963) 373-395. [18] D. Greenwell and L. Lovhsz, Applications of product colouring, Acta Math. Acad. Sci. Hungar. 25 (1974) 335-340. [19] G. HajBs, Uber eine Konstriktion nicht n-farbbarer Graphen. Wiss. Zeitschrift Martin-LutherUniv. Halle-Wittenberg, Math.-Hal. X/1 (1961) 113-1 14. See also: G. Ringel, Farbungsprobleme auf Flachen und Graphen (Deutscher Verlag d, Wiss., Berlin, 1959) 27-30. [19a] D. Hanson, G.C. Robinson and B. Toft, Remarks on the graph colouring theorem of HajBs. Congressus Num. 55 (1986) 69-76. [20] C. Houchenne, Sur le crit6re de chromaticitt de HajBs et Ore. Bull. SOC.Royale Sci. Litge, 37e annte, no. 1-2 (1968) 10-13. A!so in: Colloque sur la Thtorie des Graphs, Bruxelles, 26 et 27 avril 1973; Cahiers du Centre d’Etudes de recherche Optrationnelle 15 (1973) 229-230. [21] J.B. Kelly and L.N. Kelly, Paths and circuits in critical graphs, Amer. J. Math. 76 (1954) 786-792. [22] H. Kierstead, E. Szemertdi and T. Trotter, On colouring graphs with locally small chromatic number, Combinatorica 4 (1984) 183-185. (231 G. Koester, Note to a problem of T. Gallai and G.A. Dirac, Combinatorica 5 (1985) 227-228. [24] U. Krusenstjerna-Hofstrem and B. Toft, Some remarks on Hadwiger’s conjecture and its relation to a conjecture of LovBsz, in: G. Chartrand et al., ed.. The Theory and Applications of Graphs (Proc. Fourth Internat. Conf. Theory Appl. of Graphs, Kalamazoo, May 6-9, 1980) (John Wiley & Sons, New York 1981) 449-459. [25] L. Lovhsz, Independent sets in critical chromatic graphs, Studia Sci. Math. Hungar. 8 (1973) 165- 168.

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[26] L. Lovisz, Three short proofs in graph theory, J. Comb. Theory (B) 19 (1975) 269-271. [27] L. Lovisz, Chromatic number of hypergraphs and linear algebra, Studia Sci. Math. Hungar 11 (1976) 113-114. [28] L. b v i s z , Kneser’s conjecture, chromatic number, and homotopy, J. Comb. Theory (A) 25 (1978) 319-324. [29] L. Lovisz, Combinatorial Problems and Exercises (Akad. Kiad6, Budapest 1979, Problems 9, 16-24) 60-61. [30] N.N. MoZan, 0 dva2dy krititeskih grafah s hromatiteskim Eislom pjat’. (On doubly critical graphs with chromatic number five.) (Omskij politehniteskij institut, Omsk 1985.) 14 pp. [31] V. Miiller, On colorable critical and uniquely colorable critical graphs, in: M. Fiedler, ed., Recent Advances in Graph Theory (Proceedings of the Symposium held in Prague, June 1974) (Academia, Prague, 1975) 385-386. [32] V. Miiller, On colourings of graphs without short cycles, Discrete Mathematics 26 (1979) 165-179. [33] J. Hycielski, Sur le coloriage des graphes, Colloqu. Math. 3 (1955) 161-162. (33a] V. Neumann-Lara, The dichromatic number of a digraph. J. Comb. Theory (B) 33 (1982) 265 270. (In this paper the following reference is given P. Erdos and V. Neumann-Lara, On the dichromatic number of a graph, in preparation.) [33b] F. Nielsen and B. Toft, On a class of planar 4-chromatic graphs due to T. Gallai, in: M. Fieldler, ed., Recent Advances in Graph Theory (Proceedings of the Symposium held in Prague, June 1974), (Symposia &AV, Academia Praha 1975) 425-430. [34] R.C. Read, Maximal circuits in critical graphs, J. London Math. SOC.32 (1957) 456-462. (35) H. Sachs, On colour critical graphs, in: L. Budach, ed., Fundamentals of Computation Theory (FCT ’85, Cottbus, GDR, Sept. 1985). Lecture Notes in Computer Science 199 (Springer-Verlag, Berlin, 1985) 390-401. [36] H. Sachs and M. Stiebitz, Construction of colour-critical graphs with given major-vertex subgraph, in: C. Berge et al., ed., Combinatorial Mathematics (Annals of Discrete Mathematics 17) (North-Holland, Amsterdam, 1983) 581-598. (371 H. Sachs and M. Stiebitz, Colour-critical graphs with vertices of low degree, in; L.D. Andersen et al., ed., Graph Theory in Memory of G.A. Dirac. Annals of Discrete Mathematics (North-Holland, Amsterdam), to appear. [38] M. Schauble, Bemerkungen zur Konstruktion dreikresifreier k-chromatischer Graphen, Wiss. Zeitschrift TH Ilmenau 15 (1969) Heft 2, 59-63. [39] N. Simonovits, On colour-critical graphs, Studia Sci. Math. Hungar. 7 (1972) 67-81. [40] M. Stiebitz, Proof of a conjecture of T. Gallai concerning connectivity properties of colourcritical graphs, Combinatorica 2 (1982) 315-323. [41] M. Stiebitz, Colour-critical graphs with complete major-vertex subgraph, in; H. Sachs, ed., Graphs, Hypergraphs and Applications (Proc. Conf. Graph Theory, Eyba, GDR, Oct. 1984) Teubner-Texte zur Mathematik 73 (B.G. Teubner, Leipzig, 1985) 169-181. (42) M. Stiebitz, Beitrage zur Theorie der farbungskritischen Graphen, Dissertationsschrift (B) TH Ilmenau (1985). [43] M. Stiebitz, K S is the only double-critical 5-chromatic graph. Discrete Mathematics 64 (1987) 91-93. [44] M. Stiebitz, Subgraphs of colour-critical graphs, Combinatorica 7 (1987) (to appear). [45] B. Toft, On the maximal number of edges of critical k-chromatic graphs, Studia Sci. Math. Hungar. 5 (1970) 461-470. [46] B. Toft, Some contributions to the theory of colour-critical graphs, Ph. D.-thesis Univ. of London (1970) published as No. 14 in: Various Publication Series (Matematisk Institut, Aarhus Universitet). [47] B. Toft, Two theorems on critical 4-chromatic graphs, Studia Sci. Math. Hungar. 7 (1972) 83-89. [48] B. Toft, Color-critical graphs and hypergraphs, J. Comb. Theory (B) 16 (1974) 145-161. [49] B. Toft, On critical subgraphs of colour-critical graphs, Discrete Mathematics 7 (1974) 377-392. [50] B. Toft, On colour-critical hypergraphs, in: A. Hajnal et al., ed., Infinite and Finite Sets, Vol. 111 (Coll. Math. SOC.J. Bolyai 10) (North-Holland, Amsterdam, 1975) 1445-1457. [51] B. Toft, An investigation of colour-critical graphs with complements of low connectivity. In; B. Bollobis, ed., Advance in Graph Theory (Annals of Discrete Mathematics 3) (North-Holland, Amsterdam, 1978) 279-287.

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[52] B. Toft, Problem posed at the 6th Hungarian colloquium on Combinatorics in Eger (Hungary, July 1981) in: A. Hajnal et al., ed., Finite and Infinite Sets, Vol. I1 (Coll. Math. SOC.J. Bolyai 37) (North-Holland, Amsterdam, 1984) 888. [53] B. Toft, Some problems and results related to subgraphs of colour critical graphs. In: R . Bodendiek et al., ed., Graphen in Forschung und Unterricht (B. Franzbecker, Salzdetfurth, 1985) 178-186. [54] B. Toft, Graph Colouring Theory (Matematisk Institut Odense Universitet, Preprints 1986, Nr. 2; Odense 1986). Also to appear in: Handbook of Combinatorics (ed.: R. Graham et al.). [55] D. Toft, Graph Colouring Problems, Part I (Institut for Matematik og Datalogi, Odense Universitet, Preprints 1987, Nr. 2; Odense 1987). [56] B. Toft, Graph colouring Problems, Part 11. (Odense Universitet, Preprints, to appear). [57] Zs.Tuza and V. Rodl, On colour critical graphs, J. Comb. Theory (B) (1985) 204-213. [58] H.-J. Voss, Graphs with prescribed maximal subgraphs and critical chromatic graphs, Comment. Math. Univ. Carolinae 18 (1977) 129-142. [59] J. Veinstein, Excess in critical graphs, J. Comb. Theory (B) 18 (1975) 24-31. [ a ] W. Wessel, Critical lines, critical graphs, and odd cycles. Preprint P-Math.-01/81 (Akademie der Wissenschaften der DDR, Institut fur Mathematik, Berlin 1981). [ma] W. Wessel, Eigenschaften, Operationen, Kritizitat von Graphen, Dissertationsschrift (B) T.H. Ilmenau (1982). [61] B. Zeidl, Uber 4- und 5-chrome Graphen, Monatshefte fur Math. 62 (1958) 212-218.

Discrete Mathematics 74 (1989) 227-239 North-Holland

227

CHROMATIC PARTITIONS OF A GRAPH E. SAMPATHKUMAR and C.V. VENKATACHALAM Karnatak University, Dharwad-580 003, India

Let z(G) be the chromatic number of a graph C = (V, E), and k 3 1 be an integer. The general chromatic number z k ( G )of G is the minimum order of a partition P of V such that each set in P induces a subgraph H with ~ ( H ) c k This . paper initiates a study of z k ( G ) and generalizes various known results on ~ ( c ) .

Let G = (V, E) be a graph. For a property P, let n(P) be the minimum number of sets into which V can be partitioned so that each set induces a subgraph H with property P. The number n(P) has been studied for various properties P. For example suppose PI, P' and P" be the properties defined as follows: P, :H is totally disconnected or trivial, P' :H is a forest, and P" :H is k-degenerate. Then n(PJ is the chromatic number of C, n(P') is the point arboricity of G studied in [6] and n(P") is the point partition number discussed in [13]. The number n(P) has been studied for some other properties in [ l , 2, 4-7, 10-15, 171 and [MI. Let k 3 1 be an integer, and x(C) denote the chromatic number of G. Here we initiate the study of the number n(Pk), where P k is the property: x ( H )S k. A set S c V is a Pk-set if x( (S)) s k, where ( S ) is the subgraph of G induced by S. A partition {V,, V,, . . . , V,} of V is a Pk-partition if each V , is a Pk-set. A P,-coloring of G is a coloring of the vertices of G such that the set of all vertices receiving the same color is a Pk-set. A Pk-coloring which uses r colors is called a (k, r)-coloring. If there exists a (k, r)-coloring of G for some r S n, then G is said to be ( k , n)-colorable. The chromatic partition number xk(G) of G is the minimum number of colors needed in a Pk-coloringof G. If xk(G) = n, then G is said to be (k, n)-chromatic. Clearly, x , ( G ) = x ( G ) and x k ( G ) =1 for all k > x ( C ) . Thus, if G is any bipartite graph, x,(G) = 1 for all n 3 2, and for an odd cycle C,,, xz(C,,) = 2 and x,(C,,) = 1, for all n 2 3. For any graph G, xk(G) s &(G) when ] s k. For a real number r, let [r] and {r} respectively denote the greatest integer not exceeding r, and the least integer not less than r. We start with elementary observations.

Proposition 1. For the complete graph Kp

0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

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E. Sampathkumar, C. V . Venkatachalam

Corollary 1.1. For any graph G of order p

Proposition 2. For any graph G

Proof. We establish only the first half in (3), the second half being obvious. Let {Vl, V,, . . . , V,} be a Xk(G)-partition of V and X ( = ti. Since tis k , we have

(v.))

Let P o = P,(G) be the independence number of G, and Mk be the maximum number of points in a Pk-set of G.

Proposition 3. For any graph G of order p

Proof. Let {Vl, V,, . . . , V,} be a minimum Pk-partition of V ( G ) . Then n = Xk(G)and l&l s M k , for 1 6 i 6 n. Therefore,

and the lower bound in (4)holds. To establish the upper bound in (4),let S c V be a Pk-set with IS1 = Mk. Clearly, X k ( G - S) > x k ( G )- 1. Since G - S has p - Mk points, we have

Therefore, Xk(G)s

+['}

+ 1, and (4)follows. To establish ( 5 ) , let

Gi, and IV,( = p i . It is well known that

pi

BdGi)

s x ( C j ) (see [9, p. 1281).

(V,) =

Chromatic partitions of a graph

229

and this establishes the lower bound in (5). Similarly, we can establish the upper bound in (5). 0

Results from previous studies Whenever a result follows from previous studies, we write P.S., and indicate the reference. The property P k is hereditary in the sense that a subset of a Pk-set is a &-set. We infer a number of results which are corollaries of earlier results.

P.S.l [4]. For any induced subgraph H of G , Xk(H)

Xk(G).

P.S.2 [4]. For every n 2 1, there is a graph G with Xk(G) = n. P.S.3 [15]. For every n 2 1, there is a K,-free graph G with Xk(G) = n.

P.S.4 [4]. For every two integers m, n 2 2, there exists a (k, n)-chromatic graph whose girth exceeds m.

P.S.5 [4]. For every c with 2 s c 6 n , there is a ( k , n)-chromatic graph G with clique number c. A Pk-partition { V l , V,, . . . , V,} of V is complete if V , U V, is not a Pk-set for all i, j , i # j . If V has a complete Pk-partition of order r, we write { V , Pk, r } .

P.S.6 [8] Interpolation theorem. If { V , Pk, m } and { V , Pk, M } , where m < M , then { V , P k , n }foralln, m < n < M . We now relate xk(G)with other known parameters of G. The point arboricity p ( G ) of G is the minimum order of a partition { V l , V2, . . . , V,} of V such that the subgraph ( V , ) induced by V , is a forest for each i. Clearly, for any graph G,

X2(G) zz P(G) S X(G)* Thus, in view of (3), we have for all k 2 2

(6)

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Chartrand, Kronk and Wall [7]have proved

where A = A(G) is the maximum degree of G. Hence by ( 6 ) , we have

We generalize this result for all k 3 2 in (23). For any graph G of order p , Nordhaus and Gaddum [16]have shown that

where G is the complement of G. The following analogue for p ( G ) was obtained by Mitchem [14]:

6s P(G) + P ( G ) s t ( P + 3)

(10)

E4 G p ( G ) p ( G )s [a(p + 3)]*.

(11)

Using Nordhaus-Gaddum results, ( 3 ) , (7), (10) and ( l l ) ,we have the following: If k 3 2, then for any graph G of order p , L

-

k

6s X k ( G ) + X

k m s t ( P + 3)

s X k ( ~ ) ~ k c( W ~ P) + 3 ~ .

(12) (13)

We strongly feel that the upper bounds in (12) and (13) can be improved.

Conjecture 1. Let t = { p / k } and k Xk(G)

3 1.

Then for any graph G of order p

+X k ( a t + 1

Chartrand and Kronk [6] have shown that p ( G ) 6 3 for any planar graph G , and if G is outerplanar, p ( C ) S 2. Thus, as a consequence of (7) we have the following: Let k 3 2. Then for any planar graph G , x k ( G )G 3. If G is outerplanar, then X k ( G ) s 2. Sampathkumar [18] has defined a generalization of chromatic number as follows. Let k 3 2 be an integer. For a graph C , the number x ( ~ ) ( Gis) the minimum order of a partition {V,,V,, . . . , V,} of V such that for each i, every

Chromatic partitions of a graph

component in the subgraph for any graph G Xk(G)

(K)

231

induced by V; has order at most (k - 1). Clearly, (14)

X(k+l)(G)-

Further generalizations We shall now generalize some more well known results on chromatic number. It is well known that for every positive integer n, there exists an n-chromatic triangle free graph. We generalize this result.

Proposition 4. Given a positive integer n, there exists a Kk+z-freegraph G with Xk(G) = n* Proof. We prove this result by induction on n. When n = 1, this is true. Assume that H is a Kk+2-freegraph with xk(H)= n. Let V ( H )= { v l ,v 2 , . . . , v,,} be the point set of H. We show that there exists a Kk+,-free graph G with xk(G)= n + 1. We construct a graph G from H by adding p k + 1 points { u } U Sj, where Si = {u;,,u j 2 ,. . . , U j k } , 1S i G p . The point u is joined to each point in S,, 1G i s p , and in addition, each point in Si is joined to each point to which ui is adjacent, 1 S i S p . To see that G is Kk+2-freerfirst observe that u does not belong to any Kk+2. Since no two points in Urz1 Si are adjacent, any Kk+2consists of a uii, 1 s i s p , 1S j S k, and ( k + 1) points, say v i l ,v j 2 ,. . . , u i ( k + lof ) H. But by construction, this would imply that the points v;,v i l ,ui2, . . . , V ; ( k + l ) induce a Kk+2in H , which is not true. Let a ( k , n)-coloring of H be given. Now assign to all points in Sj the same color assigned to q, and assign a (n 1)th color to u. This produces a ( k , n 1)-cooling of C. Hence, xk(C)s n + 1. Suppose xk(G)G n, and let there be given a ( k , n)-coloring of G, with colors 1, 2, . . . , n , say. Necessarily, the point u is colored differently from the colors of the points in Si, 1 ~ i s p . Suppose u is assigned the color n. Since xk(H) = n, the color n is assigned to some points in H. Recolor each vi assigned the color n with the color assigned to the points in S,. This produces a (k, n - 1)-coloring of H and a contradiction. Thus, X k ( G ) = n 1 and the proof is complete. 0

+

+

+

Note that when k

= 1,

we have a known result.

Critical and minimal graphs Let G = (V, E) be an n-chromatic graph and e E V U E. Then e is critical if x(G - e ) = n - 1. If each point (line) of G is critical, then G is n-critical

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E. Sampathkumar, C.V . Venkatachalam

(minimal). We generalize these concepts as follows: Let G be (k, n)-chromatic and e E V U E. Then e is (k, n)-critical if zk(G- e) = n - 1. If every point (line) of G is (k, n)-critical, then G is (k, n)-critical (minimal). Clearly, an n-critical (minimal) graph C is (1, n)-critical (minimal). One can easily verify the following:

Proposition 5. The complete graph Kp is (k, n)-critical o r minimal if, and only if

We now generalize many well known results on n-critical graphs. To begin with we state a result which follows directly from earlier studies.

P.S.7 [4]. A (k, n)-critical graph contains an induced (k, r)-critical subgraph for all r, 2 S r S n. The following two observations, though trivial, are quite useful later

Proposition 6. If G is (k, n)-minimal and has no isolated points, then G is (k, n)-critical. Proof. Each point v in G is incident to at least one line e. Now, X k ( G zk(G - e) = n - 1, so that xk(C- v) = n - 1, and G is (k, n)-critical.

- v) S

Proposition 7. If G is (k, n)-critical, and B is a Pk-set of G, then G - B is (k, n - 1)-chromatic. Proof. Since G is (k, n)-critical, we have r = zk(G- B) S n - 1. Let {Vl, V,, . . . , V,} be a P,-partition of V - B. Then {Vl, V,, . . . , V,, B} is a P,-partition of V. This implies zk(G- B) + 1 2 n, and the result follows. 0 In his work on n-critical graphs (those whose chromatic number decreases for any proper subgraph), Toft [20] has characterized n-critical graphs in terms of their (n - 1)-critical subgraphs. We generalize these concepts to (k, n)-critical graphs. The proof is similar to the corresponding generalization theorem obtained by Brown and Corneil [4].

Proposition 8. Let n 2 2. Then the following are equivalent: (a) G is (k, n)-critical (b) For u E V ( G ) ,let Sy, S,U, . . . , Sx, be all maximal Pk-sets containing v. Then G - Sy contains a (k, n - 1)-critical subgraph. If FY is any such one, then V ( G )- {v} =

0 V(Fy)

i= 1

Chromatic partitions of a graph

233

(c) Let S,, S,, . . . , S, be all the maximal Pk-sets of C. Then G - Si contains a (k, n - 1)-critical subgraph. If l$ is any such one, then V ( G )= ljV ( 6 ) i=l

Proof. (a)+(b). By P.S.7 and Proposition 7, C - Sy contains a (k, n - 1)critical subgraph. Let w be any point of C different from v, {V,, V,, . . . , V n - , } be any P,-partition of G - w, and v E V , . Clearly, xk(G- w - V , ) = n - 2. Since V, is a Pk-set, v E V;., V , c Sy and w $ Sy, since otherwise xk(G- w - V , ) 3 %,(G - S y ) = n - 1, a contradiction. In fact, w is in any (k, n - 1)-critical subgraph of G - V , (hence of G - Sy) as xk(G- w - V , ) = n - 2 and xk(GV , ) = n - 1. Thus, w E l$ and (15) follows. E &, . . . , S,}, so (b)J(c). For any v E V ( G ) and 1 S i S r u , we have S ~ {S,, that

and (16) follows. (c)+(a). If {V,, V,, . . . , Vn-.i} is a P,-partition of V ( C ) ,then xk(G - V;.) s n - 2, and if V, c S,, then xk(G- Si) S xk(G- V , ) 6 n - 2, a contradiction. Thus, xk(G)2 n. We now show that xk(G- w ) S n - 1 for all w E V ( G ) . Let w E V ( G ) . Then for some i E (1, 2, . . . , r}, w is in every (k, n - 1)-critical subgraph of G - S i , since otherwise we could choose 6's such that w$U:=,l$ Thus, xk(G- Si- w ) = n - 2, which implies that zk(G- w ) S n - 1. This completes the proof. 0 It is well known that every critically n-chromatic graph, n connected. We now generalize this result.

Proposition 9. Every (k, n)-critical graph C, n 3 2, is n

3 2,

is (n - 1)-line-

+ (k - 2)-line-connected.

Proof. The result is true when k = 1. Assume that k 2 2 , and C is not (n + k - 2)-line-connected. Then there exists a partition of V ( C )into sets V, and V, such that the set E' of lines joining V, and V, contains fewer than (n k - 2) elements. Since G is (k, n)-critical, the subgraphs Gi = ( V , ) , i = 1, 2 are (k, n - 1)-colorable. Let each of C , and G2 be P,-colored with at most n - 1 colors, using the same set of n - 1 colors. If each line in E' is incident with the points of different colors, then C is (k, n - 1)-colorable. This contradicts the fact that x k ( G )= n. Hence there are lines in E' incident with points assigned the same color. We show that the colors assigned to the points in V, may be permuted so that each line in E' joins points assigned different colors. Again this will imply that xk(G)s n - 1, produce a contradiction and complete the proof.

+

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E. Sampathkumar, C. V . Venkatachalam

In the Pk-coloring of G1, let U1, U2, . . . , U,,, be those color classes of G1 such that for each i, 1S i s m S n - 1, there is at least one line joining U, and G2.For i = 1, 2, . . . , m, assume that there are ni lines joining Ui and G2.Hence for each i, l s i s m , we haveni>Oand C c l n i ~ n + k - 3 . If no point u1 of Ul is adjacent to a point of G2 having the same color of ul, then the assignment of colors to the points of GI is not altered. If not, in Gl we may permute the n - 1 colors, so that in the new assignment of colors to the points of GI, no point of U1 is adjacent to a point of G2 having the same color. This is possible since the points of Ul may be assigned any one of at least (n - 1) - n, colors and (n - 1) - n, > 0. If in the new assignment of colors to the points of GI, no point u2 of U2 is adjacent to any point of G2 having the same color as u2, then no (additional) permutation of colors in GI is needed. If not, in GI we may permute the n - 1 colors, leaving the color assigned to Ul fixed, so that no point of Ul U U2 is adjacent to a point of G2 having the same color. This can be done since the points of U2 can be assigned any one of (n - 1) - (n2 1) colors, and (n - 1) - (n2 + 1) 2 (n + 1) - (nl nz) > 0. Continuing this process, we arrive at a (k, n - 1)coloring of G, which is a contradiction. 0 Since every connected (k, n)-minimal graph is (k, n)-critical, we have

+

+

Corollary 9.1. If G is a connected (k, n)-minimal graph, n 2 2, then G is (n + k - 2)-line-connected.

+

The above two results imply that the line connectivity h(G) 5 (n k - 2) for every (k, n)-critical graph, or connected (k, n)-minimal graph G. Since h(G)=s 6(G), the minimum degree of G, we have

Corollary 9.2. If G is (k, n)-critical or connected (k, n)-minimal, then

d(G) 2 (n + k - 2).

(17)

When k = 1, this gives a known result. If H is an induced subgraph of C , we write H < G. We shall now obtain three generalizations of the following known results.

+ A(G) x ( G )s 1 + max 6 ( H ) x ( G )S 1

(Szekeres and Wilf [19])

H
Proposition 10. For any graph G with Xk(G) = n 2 2, &(G) s max 6 ( G ' )- k

+2

where the maximum is taken over all induced subgraphs G' of G.

Chromatic partitions of a graph

235

Proof. By P.S.7, there exists an induced (k, n)-critical subgraph H of G. Clearly, S ( H ) s maxG.
G’
By (17), S ( H ) 3 n

G’
+ k - 2, so that

max S ( G ’ )3 n G’
+ k - 2,

and (20) follows.

0

Corollary 10.1. For any graph G with xk(G) 3 2,

The results (21) and (20) generalize the results (18) and (19) respectively. We now characterize (k, n)-critical points.

Proposition 11. For a point v in a graph G , the following statements are equivalent. (i) v is (k, n)-critical (ii) There exists a minimum Pk-partition of V such that {v} E Pk, and for each % E Pk, V; # {v}, v is adjacent to at least k points in V;. (A) Proof. (i) j (ii). Clearly, {v} E P k for some minimum Pk-partition of V . Suppose {{v}, V., V,, . . . , Vn} is such a partition. If for some i, 2 s i = s n , v is not adjacent to k points in V;, then since x(( V)) S k, we have x(( V; U {v})) G k, and this implies xk(G) < n, a contradiction. Clearly, (ii) j (i). 0 Corollary 11.1. Zf v is (k, n)-critical, then deg v

3 k(n

- 1).

Corollary 11.2. If G is (k, n)-critical, then (i) S(G)3 k(n - 1) (ii) G should have at least k(n - 1) + 1 points. The next result characterizes (k,n)-critical graphs.

Corollary 11.3. The following statements are equivalent. (i) G is (k, n)-critical. (ii) The statement (A) holds for each point v in G. It is not hard to see that if &(G) = n, then G should have at least k(n - 1)+ 1 points. Hence, any (k, n)-critical (or minimal) graph should have at least k(n - 1) 1 points. The following result shows that the unique graph G with z k ( G )= n and k(n - 1) + 1 points is Kk(,,-l)+l.

+

236

E. Sampathkumar, C.V . Venkatachalam

Proposition 12. Zf G is a graph with t = k(n - 1) + 1 points, then xk(G)= n and only if, C = Kt-

if,

Proof. If G = K,, then x k ( G ) = n by (1). Conversely, let G have t points and x,(G) = n. If n = 1, then G = K,. Otherwise, let u and v be two distinct points of G. If they are not adjacent, then we may partition V into (n - 2) sets, say V,, V,, . . . , Vn-* each containing k points and one set Vn-, containing u and v together with (k - 1) other points. Clearly, x(( 6 k for i = 1,2, . . . , n - 1, and hence xk(G)S n - 1, a contradiction. Therefore, u and v are adjacent and G=K,.

v,))

The second generalization of the results (18) and (19) is given by the next result and its corollary.

Proposition 13. For any graph G and k 2 1

where the maximum is taken over all induced subgraphs H of G.

Proof. Clearly, the result is true when k = 1. Also, this is true when k > x ( G ) , since in that case xk(G)= 1. Let 1 < k < x(G) = n. By P.S.7, G has an induced subgraph H which is (k, n)-critical. Clearly, S ( H ) S maxC.
G’
C’iG

Since H is (k, n)-critical, S ( H ) 3 k(n - 1) by Corollary 11.2, and hence max S(G‘)2 k(n - 1) = kxk(G)- k, and (22) holds. 0 G‘
Since S(G’)=sA(G) for any induced subgraph G’ of G , we have the following.

Corollary 13.1. For any graph G,

We shall now adopt a method described in [4] to obtain another generalization of the results (18) and (19). First we define new terms using hypergraph theory. Recall that a hypergraph H consists of a set V of vertices and a collection of subsets of V of size at least two, called edges. The degree of a vertex v of hypergraph H, deg(v), is the maximum number of edges of H that intersect pairwise only in {v}. The chromatic number x ( H ) of H is the minimum number

Chromatic partitiom of a graph

237

of colors needed to color V ( H ) so that no edge is monochromatic. Given k L 1 and a graph G, form a hypergraph H k with vertex set V ( G ) , where E c V ( H ) is an edge of Hk if, and only if, the subgraph ( E ) G of G induced by E is (k,2)-critical.

Proposition 14. x(Hk)= xk(C). Proof. Let r = x(Hk) and s = xk(G). Suppose { V , , U,, . . . , U,} is a 2(Hk)partition of V ( H k ) , and { V , , V,, . . . , VT} is a Xk(G)-partition of V ( G ) . The result follows if we show that each Uj is a Pk-set and no edge of Hk is contained in any V,. Consider any Uj. Since no edge of H k is contained in Ui,we have xk(( Q ) )= 1 and this implies that Uj is a Pk-set. Suppose now, some edge E of H k is contained in a V,. Then xk( 2, and this implies that V;. is not a Pk-set, a contradiction. This completes the proof. 0

(v))

The k-degree dk(v) of v in G is defined as the degree of v in Hk. Let A'(G) and 6"(C) denote respectively, the maximum and minimum k-degree of a point in G. Using known results in hypergraph theory (see [3, p. 431]), we deduce the following for any graph G.

Proposition 15. (i) xk(G)S 1 + A k ( G ) (ii) xk(G)=s 1 max hk(F)

+

(iii)

F
Zf G is (k, n)-critical, then a k ( G )2 n - 1

(26)

We conclude this paper by obtaining an upper bound for t ( G ) in terms of the largest eigenvalue of the adjacency matrix of the graph G. For a graph G of order p, let A = A[G] denote the p X p adjacency matrix of G. Let m = n ( A ) = m(G) be the largest eigenvalue of A . If d* is the average degree of G, it is well known that d* G m(G)G A(G). Wilf [21] has proved that xI(G) = x(G) < 1 m ( G ) . We now generalize this result.

+

Proposition 16. For any graph G and any positive integer k 3 1

and this inequality is the best possible.

Proof. If xk(G)= 1, then the inequality (27) holds since m ( G ) 0. Let xk(G)= n L 2 and F be a (k, n)-critical subgraph of G. Let A[G] be a p X p matrix and A[F] be a p' x p' matrix. Suppose A ' is the p X p matrix obtained from A[G] by

E. Sampathkumar, C.V . Venkatachalam

238

replacing by zero’s those rows and columns that correspond to the points of G which are deleted in forming F. Then

m(F) = max m(A‘)S m(G). The equality above holds since the eigenvalues of A’ are those of A [ F ] plus an additional p - p ’ zeros, and the inequality follows from the entry-by-entry domination of A [ G ] over A’. Since F is a (k,n)-critical graph, it follows from Corollary 11.2 that 6 ( F ) 3 k(n - 1). Since m ( G ) m ( F ) 3 6 ( F ) , we have

m ( G )3 k(n - l), and the result follows. 0 That this result is the best possible follows from the fact that

Xk(Kp)=

[;}

=1

+ [p+]

and m ( K p )= p - 1.

References [l] M.O. Albertson, R.E. Jamison, S.T. Hedetniemi and S.C. Locke, The subchromatic number of a graph, unpublished manuscript. [2] J.A. Andrews and M.S. Jacobson, On a generalization of chromatic number, Congr. Numer. 47 (1985) 33-48. [3] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam 1973). [4] J.I. Brown and D.G. Corneil, On generalized graph colorings, J. Graph Theory 11, 1 (1987) 87-99. [5] G. Chartrand, D. Geller and S. Hedetniemi, A generalization of chromatic number, Proc. Cambridge Philos. SOC.64 (1968) 265-271. [6] G . Chartrand and H. Kronk, The point arboricity of planar graphs, J. London Marh. SOC. 44 (1969) 612-616. [7] G . Chartrand, H. Kronk and C. Wall, The point arboricity of a graph, Israel J. Math. 6 (1968) 169-179. [8] E.J. Cockayne and G.G. Miller, An interpolation theorem for partitions which are complete with respect to hereditary properties, J. Combin. Theory Ser. B. 13 (1972) 290-296. [9] F. Harary, Graph Theory (Addison-Wesley, Reading Mass. 1969). [lo] F. Harary, Conditional colorability in graphs, Graphs and Applications, Proc. First Col. Symp. Graph Theory (Boulder, Colo., 1982), 127-136 (Wiley-Intersci. Publ., Wiley, New York, 1985). [11] S. Hedetniemi, Disconnected coloring of graphs, Combinatorial Structures and their Applications, (Proc. Calgary Int. Conf. Calgary, Aka., 1969) 163-167 (Gordon and Breach, New York, 1970). [12] D.R. Lick, A class of point partition numbers, Recent trends in Graph Theory (Proc. Conf. New York, 1970) Lecture Notes in Mathematics, Vol. 186, 185-190 (Springer-Verlag. Berlin, 1971). (13) D.R. Lick and A.R. White, k-Degenerate graphs, Cand. J. Maths, XXII, 5 (1970) 1082-1096. [14] J. Mitchem, On the point arboricity of a graph and its complement, Cand. J. Math., XXIII, 2 (1971) 287-292. [15] C.M. Mynhardt and I. Broere, Generalized colorings of graphs, Graph theory with Applications to Algorithms and Computer Science, (Kalamazoo, Mich., 1984) 583-594 (Wiley-Intersci. Publ. Wiley, New York, 1985).

Chromatic partitions of a graph

239

[16] E.A. Nordhaus and J.W. Gaddum, On complementary graphs, Amer. Math. Monthly. 63 (1956) 175- 177. [ 171 E. Sampathkumar, P.S. Neeralagi and C.V. Venkatachalam, A generalization of chromatic and line-chromatic numbers of a graph, J. Karnatak Univ. Sci. 22 (1977) 44-49. [18] E. Sampathkumar, A generalization of independence and chromatic numbers of a graph, preprint (March, 1988). [I91 G . Szekeres and H.S. Wilf, An inequality for the chromatic number of a graph, J. Combin. Theory 4 (1968) 1-3. [201 B. Toft, On critical subgraphs of color-critical graphs, Discrete. Math. 7 (1974) 377-392. [21] H.S. Wilf, The eigenvalues of a graph and its chromatic number, J. London Math. SOC.42 (1967) 330-332.

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241

Discrete Mathematics 74 (1989) 241-243 North-Holland

SEQUENTIAL COLORING VERSUS WELSH-POWELL BOUND Maciej M. SYSLO Fellow of the Alexander von Humboldt-Stiftung; on leave from the Institute of Computer Science, University of Wrochw, Przesmyckiego 20, 51 151 Wroctaw, Poland.

We comment in this note on the relations between sequential coloring and the Welsh-Powell upper bound for the chromatic number of a gcaph.

In 1967, Welsh and Powell introduced in [3] an upper bound to the chromatic number x ( G ) of a graph G that is usually formulated in the following form (in what follows, n denotes the number of vertices of a graph): If the vertices of a graph G are arranged in non-increasing order of their degrees, i.e.

d ( v J 2 d(vJ

2.

-

d( v , )

(1)

then x ( G )s us(G;vl, v2, . . . , v,)

def

=

max min{d(v,)

Isisn

+ 1, i } .

(2)

We have checked all available standard texts on the theory and applications of graphs and papers on graph coloring published after appearing [3] (including [3]), and in all of these publications the bound (2) is preceded by the assumption (1). On the other hand, the bound (2) is quite often presented in the context of the sequential coloring algorithm (algorithm S, for short) which can be described as follows: Given an ordering u l , u2, . . . , u, of the vertices of a graph G. Assign color 1 to vertex u l , and for i = 2, 3, . . . , n , assign to ui the smallest color not used for any neighbor uj of ui such that j < i. Let xs(G;u l , u 2 , . . . , u,) denote the number of colors used by algorithm S. It is clear that each vertex ui can be colored by the smaller of two colors, i and d ( q ) 1. Therefore, for any vertex ordering u , , u 2 , . . . , u, of a graph C we have

+

X(G)SXs(G;ul,Uz,...,Ufl)dU S(G;UI,Uz,...

,u,).

(3)

We encourage the reader to prove, for instance by applying an exchange argument, that u s ( G ;u , , u2, . . . , u,) is minimized by u l , u 2 , . . . , u,

arranged in non-increasing order of their degrees. 0012-365)
(4)

M.M. S y s b

242

c--’i f

-

vertex ordering

XS

b e c d g f a h

3

3

a b c d e f g h

2

4

a

Fig. 1. A bad example for the Welsh-Powell sequential coloring.

The fact (4) is probably an implicit motivation behind appearing (1) as the assumption for (2). There are however two issues that should be emphasized here and which support our claim that (1)-(2) should be rather presented in the form of two separate statements, as (3) and (4). First, it is well-known that the number of colors used by the algorithm S, xs(G; u l r u2, . . . , u,) is almost always much smaller than us(G;u l , u2, . . . , u,) the upper bound to xs (and also to x) resulting from the algorithm, and there is no relation between these two numbers. Second, the smaller bound u s ( G ;u , , uz, . . . , u,) does not necessarily guarantee the better sequential coloring. In particular, for certain graphs no ordering of vertices satisfying (1) may produce a chromatic coloring. Fig. 1 shows a bipartite (i.e. 2-chromatic) graph for which both vertices of degree 3 cannot be colored with the same color in any optimal coloring. However they will get the m- 1

m-2

m-3

1

Johnson’s g r a p h G

m

m- 1

m-2

( s e e F i g u r e 1 i n C11)

m-3 Fig. 2. Another bad example.

1

Sequential coloring us. Welsh-Powell bound

243

same color when vertices are ordered according to (1). Note that in this case we have xs(a. . . h ) < x s ( b . . . h) although us(a. . . h ) > us(b. . . h). In general, if two vertices of a bipartite graph located at odd distance from each other are forced by the algorithm S to receive the same color, then the coloring produced is not optimal. Fig. 2 shows a member of an infinite family of bipartite graphs, for which the number of colors used by the algorithm S applied to any vertex ordering satisfying (1) is of order fi. To summarize, the upper bound (2) to the chromatic number of a graph is valid for any vertex ordering and is minimized for ordering (1). However, it may happen for some graphs, that for no vertex ordering satisfying (l),the algorithm S produces an optimal coloring. Hence some other vertex orderings may be of interest while searching sequentially for a chromatic coloring. This motivates our claim that (1)-(2) should be rather presented as two separate statements, (3) and (4). We refer the reader to Section 4.1 in [2] devoted entirely to coloring algorithms, where the results of some computational experiments with sequential coloring algorithms are also presented. An extensive annotated bibliography is also included there.

References [I] D.S. Johnson, Worst case behavior of graph coloring algorithms, Proc. 5th S-E Conf. on Combinatorics, Graph Theory and Comput., Utilitas Mathematica (Winnipeg, 1979) 513-527. [2] M.M. Systo, N. Deo, and J.S. Kowalik, Discrete Optimization Algorithms with Pascal Programs (Prentice Hall, Englewood Cliffs, NJ., 1983). [3] D.J.A. Welsh and M.B.Powell, An upper bound for the chromatic number of a graph and its application to timetabling problems, Comput. J . 10 (1967) 85-86.

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245

Discrete Mathematics 74 (1989) 245-252 North-Holland

A GENERALIZATION OF ROBACKER’S THEOREM S.K. TIPNIS and L.E. TRO’ITER, Jr. Let B be the polyhedron given by B = { x E R”: Nx = 0, a x =Z b}, where N is a totally unimodular matrix and a and b are any integral vectors. For x E R” let ( x ) + denote the vector obtained from x by changing all its negative components to zeros. Let X I , . . . ,x p be the integral points in B and let B+be the convex hull of ( X I ) ’ , . . . , ( x p ) + . In this paper we derive the blocking polyhedron for {x E RT: Mx 3 1) where the rows of M are integral points in B+. We also show that the optimum objective function values of the integer programming problem max( 1 . y : yM S w , y 0 and integral} and its linear programming relaxation differ by less than one for any nonnegative, integral vector w. As a special case of this result we derive Robacker’s theorem which states that for a directed graph with two distinguished nodes s and I, the maximum value of an integral packing of forward edges in (s, r)-cuts into a nonnegative, integral weighting w of the edges is equal to the minimum w-weight of an (s, r)-path.

1. Introduction and definitions Let G = ( V , E ) be a directed graph with node set V and edge set E and with two distinguished nodes, s and t. Let X G V with s E X and t c$ X . The set of edges of G with one end in X and the other end in X = V\X is called an (s, t)-cut and is denoted by [ X , 31. The set of edges {e E E: e = (u, v), u E X , v E X } is called the 31. Define set of forward edges in the (s, t)-cut [X, % = {S E E: edges in S are forward edges in a minimal (s, t)-cut in C}

and

%’ = {S E E: edges in S are a directed, minimal (s, t)-path in G}. Let M be the cut-edge incidence matrix for % and M be the path-edge incidence matrix for %‘.It is well-known that M and A? form a “blocking” pair of matrices (defined later in this section-see Fulkerson [ 2 ] ) . Also, the following two combinatorial max-min relations are well-known where w is any nonnegative integral vector with IEl components, max{l- y : y~ s w , y

3

o integral} = min{ w - 8: ii a row of M }

(1.1)

max{ 1 .y : yA? s w , y

S

0 integral} = min{w * p : a row of M}.

(1.2)

Since M and

M are a blocking pair of matrices, (1.1) and (1.2) are equivalent to

This work was supported in part under NSF grant #ECS-8504077 to Cornell University. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

246

S . K . Tipnis, L.E. Trotter, Jr.

the assertion that the corresponding linear programming problems in (1.1) and (1.2) have integral optimum solutions for all nonnegative integral right-hand side vectors w. (1.2) is the max-flow min-cut theorem of Ford and Fulkerson and (1.1) is a theorem of Robacker [5]. Trotter and Weinberger [7] have obtained a generalization of (1.2) by treating the problem in the setting of packing into a nonnegative integral vector w , the integral solutions to the system Nx = 0, a S x S b, where N is a totally unimodular matrix and a and b are nonnegative integral vectors. In this paper we extend this approach to obtain a generalization of (1.1). We define the positive part of x E R", denoted by x + , as the vector in R" obtained from x by changing all its negative components to zeros. We consider incidence vectors of forward edges in (s, t)-cuts from the general viewpoint of positive parts of integral vectors in P = { x E R": Nx = 0, a s x S b } , where N is a totally unimodular matrix and a, b are integral vectors. Let V E R" be a linear subspace. The support of a vector x E V, denoted by S ( x ) , is defined by S ( x ) = {j:xi # O}. The positive support of a vector x E V, denoted by S + ( x ) , is defined by S + ( x ) = { j : xi > O}. The negative support of a vector x E V, denoted by S - ( x ) , is similarly defined. A vector x E V is said to be an elementary vector in V if x # 0 and has minimal support in V, i.e. provided O f y E V implies S ( y ) is not a proper subset of S ( x ) . We denote by 9 ( V ) the frame of V defined as the set of all elementary vectors of V. For a detailed development of elementary vectors the reader is referred to Fulkerson [l]. We say that the vector y conforms to the vector x if S + ( y )G S + ( x ) and S - ( y ) E S-(x). It is well-known that if V has dimension m then there exists a matrix A = (I,,,IL)where I,,, denotes the m X m identity matrix and L is an m x (n - m) matrix such that after a suitable permutation of the co-ordinates V is the row space of A. Matrix A is called a standard representative matrix of V. Let A be an m X n matrix with nonnegative, real entries and with no zero rows. Let 93 be the polyhedron 93 = { x E R;: Ax 3 l}, where 1 denotes an m-vector whose components are all equal to one. The blocking polyhedron of 93 is defined by = { b E R:: b x 3 1, for every x E 93}. The following theorem of Fulkerson [3] shows that 6 is a polyhedron and that blocking polyhedra occur in dual pairs.

Theorem 1.1. Let the rows of matrix A be the extreme points of 93 and let %? = { x E R:: Ax 3 1). Then, 6 = %? and 6 = 93. 0 We call any nonnegative matrix A such that 93 and 8 as in Theorem 1.1 form a blocking pair of polyhedra, a blocking matrix of A . If we restrict to matrices A having only rows that are essential in defining 93 (a row of A is said to be inessential for 93 if its corresponding inequality may be omitted when defining 3) we obtain unique pairs of blocking matrices. The next theorem of Fulkerson [3] shows the relationship of blocking theory to the optimal objective function values of certain linear programming problems.

Generalization of Robacker's Theorem

247

Theorem 1.2. Let A and k be nonnegative matrices, each with n columns and without zero rows. Then max{ 1 - y : y A s w , y 3 O} = min{ w - a;.: (ui some row of k)for every w E R: if and only if A and A are a blocking pair of matrices. 0 For xl, x 2 , . . . , x p E R" we denote by conv{xl, x 2 , . . . ,x " } the convex hull of the vectors xl, x 2 , . . . , xp. Let xl, x 2 , . . . ,x p be the integral points in polyhedron 9 defined above. Let 9+=conv((x1)+, ( x 2 ) + , . . . , (x")'}. In Section 2 we derive the blocking polyhedron for { x E R:: M x 3 l}, where the rows of M are integral points in 9+. In Section 3 we prove that the optimum objective function values of the integer programming problem max{ 1 y : y M s w, y 3 0 integral}, where w is any nonnegative integral vector, and its linear programming relaxation differ by at most one. Finally, we point out that a particular choice of N, a and b leads to relation (1.1) as a corollary.

-

2. A blocking relationship Let N be an m x n matrix and let [ai, bi], 1S j S n , be nonempty, closed intervals. We will be concerned with the polyhedron 9= { x E R": Nx = 0, xi E [ai, b,], 1 ~j G n } . The following theorem (see Fulkerson [ l ] ,Rockafeller [6]) will be useful.

Theorem 2.1. Suppose N , 9 and [aj,b,], 1 S j is nonempty if and only if 1

2

jeS+(k)

kibi +

kiai 3 0 for all k

S

E

n, are as defined above. Then 9

9(V),

jeS-(k)

0

where V is the row space of N.

For the rest of this paper, let N be a totally unimodular matrix and suppose a and b are integral vectors. Consider the polyhedron 9= { x E R": N x = 0 , a ~x s b } . Let x ' , . . . , x p be the integral vectors in 9. Note that since N is totally unimodular and a and b are integral vectors, the extreme points of 9 are among X I , . . . , xp. Let 9+= conv{(x')+, . . . , ( x p ) + } and let the integral vectors in 9+be the rows of the matrix M . We wish to derive the blocking polyhedron for 93 = { x E R:: M x 3 1 ) . We first establish the following two lemmas.

Lemma 2.1. If x

E

9,then x +

E

9+, for 9 and 9+as defined above.

Proof. Notice that it suffices to show that for every x in 9,there exist integral

248

S . K . Tipnb, L.E. Trotrer, Jr.

vectors X I , . . . ,x k in 9,conforming to x and scalars )L1, . . . , A k such that k

x =

k

c Ai

2 &xi,

i=l

= 1,

Ai 3 0,

1 F i F k.

i=l

This will suffice since, then, we have, +

k i=l

i=l

which in turn belongs to 9+. Consider any x E 9. Let 9'= 9 f l { y E R": yi 2 OVJ E S + ( x ) , yi F OVJ E S - ( x ) , yi = OVJ $ S ( x ) } . Then, clearly 9' G 9. Also, the constraint matrix defining 9'is totally unimodular since N is. Hence 9'has integral extreme points. By the way 9' is defined, each vector y in 9' (and in particular any extreme point of 9') conforms to x . Also, x E 9'and hence there exist integral extreme points X I , . . . , x k of 9'that conform to x and scalars A l , . . . , A& such that k

x =

2 &xi, i= 1

c Ai k

= 1,

Ai 3 0, 1 6 i 6 k.

i=l

These xi are integral vectors in 9, as observed, and so the proof is complete. 0

Lemma 2.2. There exists x E 9"such that x S w if and only if there exists y E 9 such that y F w, where w is any nonnegative vector and 9 and 9+are as in Lemma 2.1.

Proof. If x is any vector in 9+such that w 3 x then we have, x = C:='=,&(xi)+, where ,YJz1 Ai = 1, Ai 2 0, 1F i S p . Hence x 3 Cr='=, Aix' = y which in turn is in 9. Hence there exists y in 9 such that y S w. Conversely, let y be a vector in 9 such that y s w. Then, since w 3 0, y + S w. But from Lemma 2.1, y + E 9+and hence there exists x = y + in 9+such that x 6 w. We now derive the blocking polyhedron for 9 = { x E R:: M x matrix M is as defined above.

3

l}, where the

Theorem 2.2. The blocking polyhedron for 9 is given by (where V denotes the row space of N ) [ x E R::

2 kixi 3

jaJ

c

jeS-(k)

(-ki)ai -

c

jeS+ ( k) \J

kibi, k E 9 ( V ) , J

Proof. The blocking polyhedron for 9 is given by

& = { x E R::

x 2 z for some z E conv{rows of

= { x E R:: x 3 z for some z E

S'}.

M}}

E s+(k)).

249

Generalization of Robacker's Theorem

Using Lemma 2.2, we have that,

9= { x E R:: x 3 z

for some z

E

9)

= { x E R:: N z = 0, aj S zj S min{xj, b j } , 1 ~j

s n for some z } .

Using Theorem 2.1 with V as the row space of N, we have, x

E R::

c

kj min{xi, bi} +

jsS+(k)

Hence, we have

9as required.

c

kjaj3 0 , V k E 9(V)}.

jsS-(k)

0

As a special case of Theorem 2.2, we can easily show that positive parts of minimal (s, t)-cuts in a network and directed (s, t)-paths form a blocking pair of clutters. (See [8].)

3. Packing positive parts of integral vectors in 9 As above we let N be a totally unimodular matrix and suppose a and b are integral vectors. Consider the polyhedron 9 = { x E R": Nx = 0, a S x 6 b } . Let xl,. . . , x p be the integral vectors in 9 and let 9+=conv{(x')+, . . . , ( x P ) + } . Let the integral vectors in 9+be the rows of matrix M . In this section we show that the objective function values of the integer programming problem, max{ 1 .y : yM s w , y 3 0 and integral} and its linear programming relaxation differ by less than one for all nonnegative integral vectors w. Towards this end, we first prove a conformal decomposition theorem for 9. For any integer, k 3 1, let k 9 = { x : x = ky for some y E 9).We note here that the following decomposition theorem is a direct generalization of a decomposition theorem by Trotter and Weinberger [7] to the case where a, b are integral vectors as opposed to both a and b being in 2:. In the case where a, b are integral vectors we get a conformal decomposition theorem. This theorem may also be deduced from results of McDiarmid [4], where it is shown (Section 4) that the decomposition x = X I + . . + x k may be presumed equitable, i.e. x i - x j s 1 for all i , j = 1 , . . . , k.

-

Theorem 3.1. Let N be a totally unimodular matrix. Suppose a , b are integral vectors. Let 9= { x E R": Nx = 0, a S x 6 b } . Then for each positive integer k and each integral vector x in k 9 there exist integral vectors X I , . . . ,x k in 9 and conforming to x such that x = X I + * * xk.

+

Proof. We prove the theorem by induction on k. The case k = 1 is trivially true. We assume the result to be true for k = 1, 2, . . . ,p - 1. Consider an integral vector x in p 9 . Then, aj s xilp s b j , 1 6 j s n .

(3.1)

S . K . Tipnis, L.E. Trotter, Jr.

250

Multiplying both sides of (3.1) by ( p - 1) and rearranging, we have xi - ( p - l)ai 3 xi/p 3 xi - ( p - l)bi,

1 G j s n.

From (3.1) and (3.2) we have that x l p is in { x where

Ji = [max{ai, xi - ( p - l)b,},

E R":

Nx

(3.2) = 0)

n {J1x . - - x J , } ,

min{bi, xi - ( p - l)ai}]= [c,, d,]

and ci, d, are integers, 1 sj 6 n. Since N is totally unimodular, there exists an m x n totally unimodular matrix A whose rows are a basis for the subspace { x E R": Nx = 0). Thus there exists a vector y E R", such that x l p = y A . Also, c s y A s d. Since p is a positive integer, x l p conforms to x ; and xi/p s xi if xi 3 0 and xi/p a x j if xi s 0. Hence there exists y E R"' such that, csyASd xj2yA,30

ifxiso

xisyAisO

ifxjsO

(3.3)

y A j = 0 if xi = 0,

where Ai is the jth column of A . Since A is totally unimodular, so is the matrix defining the inequality system in (3.3). Hence, because c, d , x are integral, there exists an integral vector y' satisfying (3.3). Hence, x ' = y ' A is an integral vector satisfying c s x 1 S d , Nx' = 0 and conforming to x . Now, X I E J , x . x J , and so we have that ( p - l)ai s x , - x f s ( p - l)bi, l s j s n , and it is clear that we also have x - x ' ~ { z : N z = O , ( p - 1 ) a s z s ( p - l ) b } = ( p - 1)9. From (3.3), x - X I conforms to x . Hence by the induction * + x p where x 2 , . . . , x p are integral vectors in 9 hypothesis, x - x l = x 2 which conform to x. I7

--

+. -

Lemma 3.1. Let k

E Z,

and w

E Z:.

Then,

k 9 + n { x : x s w } # O j k 9 n { x : x s w } f l Z" # O .

Proof. First note that for every x f E 9+, there exists y ' since i=I

E

9 such that x f 3 ( y f ) + ,

i=l

and hence x ' 3 (CfLl &xi)+ = ( y f ) + , where y ' E 9. Now, if x E k 9 + and x s w , then x l k E !iP+ and x / k s w l k . Hence, by the argument above, there exists y ' E 9 such that ( y f ) + s x l k s w l k . Taking Z' = ky', we obtain z f E k 9 and ( z f ) + s w. Since z' s ( z ' ) + ,we have z f E k 9 and Z' G w. But since 9 # 0 is defined by a totally unimodular matrix and since w is an integral vector, there exists an integral vector z E k 9 such that z s w. 0

Generalization of Robacker's Theorem

25 1

We are now ready to prove the main theorem of this paper

Theorem 3.2. Where the rows of M are integral vectors in 9+and w is any nonnegative' integral vector, the objective function values of the integer programming problem max{ 1 * y :yM S w , y 3 0 integral} and its linear programming relaxation differ by less than one. Proof. If 0 E 9, then 0 is a row of M and both programming problems are unbounded. Also when M is vacuous (i.e. 9 = $), there is nothing to show; thus wesupposey* s o l v e s m a x { l . y : y M s w , y 2 0 ) and l . y * = r * . I f O S r * < l , the theorem holds trivially. So, let r* 2 1. Let x ' = [ r * Jy * M / r * . Then x ' E Lr*] 9+ and x ' =s w . Hence by Lemma 3.1, there exists an integral vector z E Lr*] !?? such that z =s w. Thus by Theorem 3.1 there exist integral vectors zl, . . . , zLr*l in 9 and conforming to z such that z = z1 + * * * + zLr'I S w . Hence by Lemma 2.1, there exist integral vectors ( z ' ) + ,. . . , (zLr*])+ in 9+ such that (zl)+ +. . - + (zLr'l)+ 6 w. This defines a solution to the integer programming problem max{l my: yM S w , y 2 0 and integral} of value Lr*], which completes the proof. 0 We now indicate how Robacker's theorem (see (1.1) in Section 1) can be obtained as a corollary of Theorem 3.2. Let G ' = (V, E') be a directed graph with IVI = m 1 , IE'I = n - 1 and let s and t be distinguished nodes of G'. Add the special edge from t to s to E' to get G = (V, E). Then there exists (possibly after edge permutation) a standard representative matrix A' = (Z, I L) for the row space of A , where A is the node-edge incidence matrix of G. Let N = (-L' I Zn-,J. Then, N is a totally unimodular matrix. Let 9 = { x E R": Nx = 0, -1 s x =s 1, xu = -1}. Integral vectors in 9 correspond to unions of disjoint cocycles of G, exactly one cocycle containing the edge from t to s. This follows because if x is in 9 and integral (i.e. has components in (0, +1, -1)) then x is in the null space of N and hence in the row space of A. Thus, there exist x ' , . . . , x k , nonproportional elementary vectors in the row space of A which conform to x such that x = x ' . + x k . Since elementary vectors in the row space of A correspond to incidence vectors of cocycles in G and since x, = -1, exactly one of the x's is the incidence vector of a cocyle containing the edge from t to s. Furthermore, due to conformality and nonproportionality , the x i s must be incidence vectors of disjoint cocycles. Conversely, if x is the incidence vector of a disjoint union of cocycles of G with exactly one cocycle containing the edge from t to s, then, after a sign adjustment, if necessary, x is in the row space of A and all components of x are in (0, + 1 , - 1 ) with x, = -1. Hence x is an integral vector in 9. Let the rows of M correspond to positive parts of integral vectors in 9 and the rows of M' correspond to positive parts of cocycles of G containing the edge from t to s (with negative sign). Since there exists an optimum solution y * to the

+

+- -

252

S . K . Tipnis, L.E. Trotter, Jr.

integer programming problem max{ 1 .y : y M S w ,y 2 0 and integral} with y: = 0 if row j of M is not minimal in support, and since by Theorem 3.2, the optimum objective function values of this integer programming problem and its linear programming relaxation differ by less than one, we have that for any nonnegative integral vector w , max{ 1 y : y M '

1

S

w, y 5 0 and integral}

= max{ 11 .y : y M ' = min{ Lw

-

s w, y

2

o}

p ] : p the incidence vector of a minimal (s, t)-path} (since M' and the matrix whose rows are incidence vectors of minimal (s, t)-paths form a blocking pair of matrices) = min{ w * p : ,u the incidence vector of a minimal (s, t)-path}. This is precisely Robacker's Theorem as stated in (1.1) in Section 1.

References [ l ] D.R. Fulkerson, Networks, Frames, Blocking Systems, in Mathematics of the Decision Sciences, Lectures in Applied Mathematics, Eds G. B. Dantzig and A. F. Veinott, J r . , Vol. I1 (American Mathematical Society, 1968). [2] D.R. Fulkerson, Blocking Polyhedra, in Graph Theory and its Applications, Ed. B. Harris (Academic Press, New York, 1970) 93-112. [3] D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Mathematical Programming 1 (1971) 168-194. (41 C. McDiarmid, Integral decomposition in polyhedra, Mathematical Programming 25 (1983) 183-198. [5] J.T. Robacker, Min-max Theorems on Shortest Chains and Disjunct Cuts of a Network, Rand Corporation Report RM-1660-PR (1956). [6] R.T. Rockafeller, The Elementary Vectors of a Subspace of R", in Combinatorial Mathematics and its Applications, Eds R. C. Bose and T.A. Dowling, Proc. of the North Carolina Conference, Chapel Hill, (University of North Carolina, 1969) 104-127. [7] L.E. Trotter, Jr. and D.B. Weinberger, Symmetric blocking and anti-blocking relations for generalized circulations, Mathematical Programming Study 8 (1978) 141-158. [8]S.K. Tipnis, Integer Rounding and Combinatorial Max-min Theorems, Ph.D. Thesis (School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, 1986).

Discrete Mathematics 74 (1989) 253-261 North-Holland

253

A RANDOMISED 3-COLOURING ALGORITHM A.D. PETFORD Department of Astrophysics, University of Oxford, U.K .

D.J.A. WELSH Mathematical Institute, University of Oxford,

u.K .

This paper describes a randomised algorithm for the NP-complete problem of 3-colouring the vertices of a graph. The method is based on a model of repulsion in interacting particle systems. Although it seems to work well on most random inputs there is a critical phenomenon apparent reminiscent of critical behaviour in other areas of statistical mechanics.

1. Antivoter models In the original voter and antivoter models studied by Donnelly and Welsh [2,3] the vertices of a graph G are either black or white and at random epochs of time a vertex, chosen at random, changes colour according to a specified stochastic mechanism. In the antivoter model each vertex has attached to it a clock which acts independently for each vertex. When the clock at vertex u rings, one of its neighbours, say u, is selected at random and u changes its colour so that it is different from that of u. Thus if u happened to be a different colour to u the colour of IJ would not change. The process is a Markov process with state space consisting of all possible 2-colourings of the vertex set V of G. In the antivoter model there are two possibilities depending on the graph G: (a) if G is bipartite, the system is, with probability one, absorbed in one of the proper 2-colourings, or (b) if G is not bipartite then the system continues to evolve without ever reaching an absorbing state. Prompted by noticing the speed with which the antivoter model with 2 colours seemed to settle down to equilibrium we proceeded to try to extend the model to 3 colours. The underlying combinatorial problem is now the 3-colouring problem which is known to be NP-complete. Hence the likelihood of being able to find a truly randomised algorithm analogous to the Rabin-Solovay-Strassen algorithm for primality testing is remote. The existence of such an algorithm would imply that random polynomial time (RP) equalled nondeterministic polynomial time (NP). This would be highly surprising unless, of course, NP turns out to equal p (see for example Welsh [lo]). With more than 2 colours the antivoter model has at least two natural 0012-365X/89/%3.5001989, Elsevier Science Publishers B.V. (North-Holland)

254

A . D . Petford, D.J.A. Welsh

formulations of the way in which a vertex chooses its new colour once it has seen and been repelled by the colour of a randomly chosen neighbour. We proceed to define the general antivoter model with 3 colours as follows. Consider a fixed instant of time t. For 1S i S 3 let Si(v) denote the set of neighbours of vertex v which are coloured i at t and let si(v) be the cardinality of S,(v). When the clock at v rings, instead of v changing colour to that of some randomly chosen neighbour we stipulate that the new colour of v is a random colour X where

P ( X = j ) =p(sl, s 2 , s3:j ) ,

1 s j s 3,

and where si = si(u) and where p is the transition function satisfying the conditions (1)

$2, 3

(2) C j=l

s3:j ) 3 0,

As1,s2,s3:j ) = 1.

Note. Here and throughout, where necessary, we shall identify the set of colours used with the set of integers { 1,2,3}. By suitably choosing p we get different versions of the antivoter model.

Example 1. Consider now the antivoter model having transition function given by

, 1sis3. This represents the situation where if the clock at v rings, a neighbour u is chosen at random and vertex v then chooses a colour at random from the two colours which are not the colour of u. In other words it is repelled by u’s colour but otherwise chooses independently. 2. The randomised algorithm The basic idea of our algorithm for deciding if a graph is 3-colourable is as follows: (1) Colour the vertices arbitrarily with 3-colours (2) Allow the antivoter mechanism with transition function p to operate on G for a time t ( n ) where n is the number of vertices and t is the threshold function (3) After time t ( n ) announce the graph as 3-colourable if a proper 3-colouring has been achieved and as not 3-colourable otherwise. Clearly the algorithm has the following properties. (a) If it says G is 3-colourable then it is correct (b) If it says G is not 3-colourable it may be incorrect.

Randomised 3-colouring algorithm

255

The probability of an error depends on G, and the choice of threshold function t, and on the choice of transition function p. The object of the simulation exercise is to (i) find a ‘good’ transition function p (ii) for this good transition function find a threshold function t in which we can have a fair degree of confidence. Although the physical models studied in [2] and [3] work in continuous time, there is no advantage in sticking to this in using these ideas for developing an algorithm. Accordingly we stipulate that the “clock ringing random mechanism” works as follows:

The Clock Ringing Mechanism: A vertex is bad at time t if it has a colour which is the same as any one of its neighbours. B, denotes the set of bad vertices at time t and as time progresses through t = 0, 1, 2, . . . , the clock chosen to ring at time t + 1 is a random member of B,. Initially we start with a random colouring. As soon as B, becomes empty the system stops. We measure the time taken by the process to be the number of times a random member is chosen from the set of bad vertices. We denote it by T ( G ) and it is clearly a random variable whose distribution depends on C and the transition function p. For any non-trivial p and G it is clear that T = T ( G ) has a non-zero probability of taking arbitrarily large values and the real quantities of interest are the first moment of T and the size of the tail distribution. As a test of the proposed algorithm we introduce the concept of a random benchmark for the problem, and thus evaluate its performance on ‘typical’ or ‘random’ 3-colourable graphs. We construct these graphs in the following way. Choose integers k , , k,, k3 with ( k , + k , + k 3 = n ) and take disjoint sets V,, V,, V, with = ki. Take the vertex set of the benchmark graph to be V = V, U V, U V3and for each pair of vertices x and y with x E V, and y E V,(i # j ) join x , y by an edge with probability p, 0 < p < 1, independently for each such pair of vertices x and y. We denote the class of such graphs by %(k,, k,, k 3 ; p ) . In each of the simulations described below we use T ( n ) to denote the estimate of the average time to colour an n vertex graph from the proposed class of graphs under consideration.

1x1

3. The simulation results Our pilot runs were on the class of graphs %(k, k , k ; 4) for k = 4 to 20. We took the view that any algorithm worth pursuing ought to be tested first on this easily constructed model. Preliminary results were reported in [ l l ] and can be summed up in the following observations:

Observation I. Random algorithms based on antivoter models with transition

256

A . D . Petford, D.J.A. Welsh

T(n)

n

Fig. 1. Data from 100 trials for each value of n O p = 0.5 xp=0.9 O p = 0.3

functions p of type described in Example 1 above or of the other repulsive type functions (1) p i ( s I , ~ 2 ~, 3 i :) l/si (2) p i ( s I ,s2, s3:i ) Us? do not efficiently 3-colour random 3-colourable graphs.

-

Observation 11. A random algorithm based on an antivoter model with transition function p given by p i @ , ,s2,

s3:

i)

- 4-”<

appears to achieve a 3-colouring of a random member of the class %(n/3, n / 3 , n/3; 4) in time which is linear in n. Because of the success with this transition function we ran 100 simulations for each n in the range n = 60-300, step size 60. The results as shown in Fig. 1 seem to justify Observation 11. One slight curiosity in Fig. 1 is the way in which the curve of p = 0.3 crosses the two other curves at a relatively low value of n. We should say that we have no theoretical justification for the choice of 6 = 4 in our transition function except that it seems to work at least as well as any other 0 used. The same applies to our use of an exponential form for the transition function, though intuitively we believe that the random colouring process we are constructing is somewhat akin to approaching the position of minimum energy in more classical problems of statistical mechanics, as suggested by Kirkpatrick’s method of simulated annealing [8] for the travelling salesman problem. In order to see whether the linear trend exhibited in Fig. 1 holds for larger values of n we proceeded to run the experiment for larger values of n , and for the

Randomked 3-colouring algorithm 6000-

257

/

T(n) Tn:o)O5000 o[

4000 -

3000 -

2000 -

1000O0

1 */

. 1I 0

II

II

II

400

200

II

II

II

II

600

II

800

n

II

1000

Fig. 2. p = 0.5

cases p = 0.3, 0.5 and 0.9 found this to be the case. In Fig. 2 we show the results of our simulation for the case p = 0.5. The cases p = 0.3, p = 0.9 are similar. However for the case p = 0.1 as shown in Fig. 3 we did notice a curious phenomenon, namely the existance of a ‘hump’ at about n = 80. This critical region was examined in greater detail and the results are shown in Fig. 4. For future reference note that the critical region ( n = 80) means that the average vertex degree of a vertex in the critical case is 2np/3 = 16/3. Further simulation for the cases p = 0.05 and p = 0.02 (one of which is shown in Fig. 5) suggest that the existence of a critical region is not an isolated phenomenon but appears to occur at about a value of n corresponding to the case where the average vertex degree is about 5 or 6. As a further test we ran

loo0

tf

- 0 0

200

400

Fig. 3. p

600

n

= 0.1

800

1000

A . D . Perford, D.J.A. Welsh

258

T(n)

30002000 -

n Fig. 4. The critical region in detail for p = 0.1

simulations for the case p = 0.005 (shown in Fig. 6) which tend to confirm our ideas that for this particular method of colouring the most difficult case among the class of “roughly regular” graphs is the case when the graphs have low vertex degree, say 5 or 6. We have no theoretical explanation of this curious behaviour. It is not unlike the phenomenon of phase-transition which occurs in the Ising model, Potts model and other models of statistical mechanics and to which this model bears some resemblance (though of course we are using a finite version here). We close this section by reporting briefly the results of some further experiments. (I) Changing the value of 8 did not seem to affect the position of the critical region in the ranges of n that we were able to work with. 7000Tin)

6000 -

5000LOO0 -

n

Fig. 5. p =0.05

Randomised 3-colouring algorithm

259

3000r

n Fig. 6. p =0.005

(11) There seems very little difference in the behaviour of the algorithm on the slightly “more random case” where instead of the benchmark graphs having equal colour sets, the graphs used as test graphs were constructed by the following method used by KuEera [9]; (i) Take a set V of n vertices and arbitrarily colour the members of V with 3 colours so that each vertex has, independently, probability 4 of being assigned any particular colour (ii) Do not connect a pair of vertices which are assigned the same colour (iii) For each pair of vertices in different colour sets let them be joined with probability p , independently of the presence of other edges (iv) Forget the colouring of the vertices. This procedure will give a graph which by construction is 3-colourable.

4. Conclusion

Various problems are suggested by the above results. First we should remark that the randomised algorithm proposed does seem to work well in a wide variety of cases. As far as we know the two situations in which it does not appear to 3-colour a 3-colourable graph G efficiently are: (a) when the only 3-colouring of G decomposes the vertex set V into disjoint sets A, B, C in which the size of A is much larger than that of B or C; (b) when G is approximately regular with a low vertex degree, say of the order of 5 or 6 for a loo0 vertex graph. It led us to suggest that possibly Garey Johnson and Stockmeyer [7] did the community a disservice in proving their theorem which states that 3-colouring is NP-complete, (even) for graphs of maximum vertex degree 4. It could be that these graphs are among the hardest to colour and it led us to conjecture that if for any fixed a>O Y&(n) denotes the collection of graphs with n vertices and minimum vertex degree at least an then for this class of graphs the colouring problem can be done in polynomial time. This was proved for the case of 3-colouring and a > $ by Farr [5] and then extended by Edwards [4] who proved

260

A.D. Petford, D.J.A. Welsh

that for k a 3 the conjecture is true if a > ( k -3)/(k -2) and false for O < a ~ ( -3)/(k k -2). In addition to the “random benchmark” test graphs used in the above simulation we have tried the algorithm out on some “apparently difficult” graphs sent to us as a challenge by R . Irving and K.W. Regan. These examples consisting of two Kneser type graphs of 70 and 130 vertices respectively together with a graph of 341 vertices which was the line graph of a planar graph with no small reducible configuration were correctly coloured in a matter of seconds by the programme (written in the language C) on a Perkin Elmer mini-computer. We have made a preliminary approach to extending the above methods to the k-colouring problem for general k > 3. In principle this should not present any greater problem. In practice this does not seem to be the case, preliminary investigations for the case of k = 10 suggest that the method is not as good; one reason for this may be that the amount of experimentation needed to find a good transition function in this case takes much more space and time. Another feature of the method is that preliminary work to start off with a “good colouring” in the sense that the initial bad set Bo was small did not seem to speed up the algorithm. This can be explained by regarding such a good colouring as approaching a local optimum which is a long way in the metric of exhanges from the true global optimum. Finally we remark that since the above experiments were first carried out in 1985 Zerovnik [12] has checked the algorithm by independently verifying our results using a different language and machine at the University of Ljubljana.

Acknowledgement We would like to thank R.W. Irving and K.W. Regan for communicating their “difficult” test graphs, D.E. Blackwell for allowing the use of the computing facilities in the Department of Astrophysics and J. Zerovnik for communicating his results to us.

References [l] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, 1976). [2] P.J. Donnelly and D.J.A. Welsh, Finite particle systems and infection models, Math. Proc. Camb. Phil. SOC.94 (1983) 167-182. [3] P.J. Donnelly and D.J.A. Welsh, The antivoter problem: random 2-colourings of graphs, Graph Theory and Combinatorics (ed. B. Bollobas) (Academic Press, 1984) 133-144. [4] K.J. Edwards, The complexity of colouring problems on dense graphs, Theoretical Computer Science, 43 (1986) 337-343. [5] G. Farr (private communication, 1985). [6] M.R. Garey and D.S. Johnson, Computers and Intractability (W.H. Freeman and Co., San Francisco, 1979).

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261

[7] M.R. Garey, D.S. Johnson and L. Stockmeyer, Some simplified NP-complete graph problems, Theor. Comput. Sci. 1 (1976) 237-267. [8] S. Kirkpatrick, C.D. Gelatt Jr. and M.P. Vecchi, Optimisation by simulated annealing, Science, 220, NO. 4598 (1983) 671-680. [9] L. KucCra, Expected behaviour of graph colouring algorithms, Lecture Notes in Computer Science, 56 (1977) 477-483. [lo] D.J.A. Welsh, Randomised algorithms, Discrete Applied Maths. 5 (1983) 133-145. [ 1I] D.J.A. Welsh, Correlated percolation and repulsive particle systems, Proc. Conference Heidelberg (Sept. 1984), Stochastic Spatial Processes (ed. P. Tautu) Springer Lecture Notes 1212, 300-311. [12] J. Zerovnik (private communication, 1986).