GRAPH THEORY
General Editor
Peter L. HAMMER, University of Waterloo, Ont., Canada 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 PUBLISHING COMPANY - AMSTERDAM
NEW YORK
OXFORD
NORTH-HOLLAND MATHEMATICS STUDIES
62
Annals of Discrete Mathematics (13) General Editor: Peter L. Hammer University of Waterloo, Ont., Canada
Graph Theory Proceedingsof the Conference on Graph Theory, Cambridge
Editor:
Bela BOLLOBAS Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge CB2 lSB, England
1982
NORTH-HOLLAND PUBLISHING COMPANY
- AMSTERDAM
NEW YORK
OXFORD
North-Holland Publishing Company, 1982 AII rights reserved. N o part of this publication may be reproduced, stored in a retrievalsystem or transmitted,in any otherform or by any means, electronic, mechanical, photocopying, recording or otherwise, without theprior permission of the copyright owner.
ISBN: 0 444 86449 0
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Main entry under t i t l e : Graph theory.
(Annals of d i s c r e t e mathematics ; U) (NorthHolland mathematics studies ; 62) Papers presented a t the Cambridge Graph Theory Conference, held a t T r i n i t y College 11-l.3 Mar. p81. 1. Graph theory--Congresses. I. Bollob&, Bela. 11. Cambrid e Gra h Theory Conference (1 81 : T r i n i t y Colfege, e n i v e r s i t y of Cambridge? 111. Ser i e s . I V . Series: North-Holland mathematics s t u d i e s ; 62. QA166.G718 1982 5ll'.5 82-8098 ISBN 0-444-86449-0
AACF2
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PRINTED IN THE NETHERLANDS
FOREWORD The Cambridge Graph Theory Conference, held at Trinity College from 11 to 13 March 1981, brought together top ranking workers from diverse areas of the subject. The papers presented were by invitation only. This volume contains most of the contniutions, suitably refereed and revised. For many years now, graph theory has been developing at a great pace and in many directions. In order to emphasize the variety of questions and to preserve the freshness of research, the theme of the meeting was not restricted. Consequently, the papers in this volume deal with many aspects of graph theory, including colouring, connectivity, cycles, Ramsey theory, random graphs, flows, simplicial decompositions and directed graphs. A number of other papers are concerned with related areas, including hypergraphs, designs, algorithms, games and social models. Thiswealth of topics should enhance the attractivenessof the volume. It is a pleasure to thank Mrs. J.E. Scutt and Mrs. B. Sharples for retyping most of the papers so quickly and carefully, and to acknowledge the financial assistance of the Department of Pure Mathematics and Mathematical Statistics of the University of Cambridge. Above all, warm thanks are due to the participants for the exciting lectures and lively discussions at the meeting and for the many excellent papers in this volume. €#la Bollobis 25th February 1982 Baton Rouge
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TABLE OF CONTENTS Foreword
J. AKIYAMA, K.AND0 and H. MXZUNO, Characterizationsand classificationsof bioonnected graphs
V
1
R.A. BAN, Line graphs and their chromatic polynomials
15
J.C. BERMOND, C. DELORME and G. FARHI, Large graphs with given degree and diameter 111
23
B. BOLLOBh, Distinguishing vertices of random graphs
33
B. BOLLOBh and F. HARARY, The trail number of a graph
51
J.H. CONWAY and M.J.T. GUY, Message graphs
61
D.E. DAYKIN and P.FRANKL, Sets of graph colourings
65
P.DUCHET and H. MEYNIEL, On Hadwiger’s number and the stability number
71
H.DE FRAYSSEIX and P. ROSENSTIEHL, A depth-first-search characterization of planarity
75
J.L. GROSS, Graph-theoreticalmodel of social organization
81
R. I-I&XKVIST, Odd cycles of specified length in non-bipartite graphs
89
R. HALIN, Simplicial decompositions: Some new aspects and applications
101
F. HARARY, Achievement and avoidance games for graphs
111
viii
Cbnrenrs
A J.W. HILTON, Embedding incomplete latm rectangles
121
AJ.W. HILTON and C.A. RODGER, Edgecolouring regular bipartite graphs
139
AJ. W S F I E U ) and DJ.A. WELSH, Some colouring problems and their complexity
159
A. PAPAIOANNOU,A Hamilbnian game
171
J. SHEEHAN, Finite Ramsey theory and strongly regular graphs
179
R. TINDELL, The connectivities of a graph and its complement
191
Annalsof Discrete Mathematics 13 (1982) 1-14 0 North-Holland Publishing Company
CHARACTERIZATIONS AND CLASSIFICATIONS OF BICONNECTED GRAPHS J. AKIYAMA, K. AND0 and H. MIZUNO Nippon Ika University, Kawasaki, Japan and University of Electrocommunications, Tokyo, Japan Dedicated t o Frank Harary f o r h i s 6 0 t h b i r t h d a y
b i c o n n e c t e d i f b o t h G and i t s complement 5 a r e connected. I n t h i s paper w e mainly d e a l w i t h biconnected g r a p h s and c l a s s i f y a s e t G o f biconnected g r a p h s i n t o s e v e r a l c l a s s e s i n terms of t h e number of c u t v e r t i c e s ( o r e n d v e r t i c e s ) of G and E. Furthermore w e g i v e s t r u c t u r a l charact e r i z a t i o n s of t h e s e classes, t h a t i s , c h a r a c t e r i z a t i o n s of graphs G such t h a t G h a s m c u t v e r t i c e s ( e n d v e r t i c e s ) and E h a s n c u t v e r t i c e s ( e n d v e r t i c e s ) r e s p e c t i v e l y , where m and n a r e g i v e n , 1 5 m 2 2 and 1 6 n 5 2 . A graph G i s s a i d t o be
5 1.
INTRODUCTION
graph G i s b i c o n n e c t e d i f b o t h G and i t s complement are connected. I n t h i s paper w e a r e mainly concerned w i t h biconnected graphs. W e d e n o t e by G t h e set of a l l biconnected g r a p h s and by Gp t h e set of a l l biconnected g r a p h s of o r d e r p. W e u s e t h e n o t a t i o n s and terminology of C11 o r C21. The c h a r a c t e r i z a t i o n of b i connected graphs i s w e l l known, which i s s t a t e d a s f o l l o w s : A
Theorem A . C 1 , p. 261 A graph G of o r d e r p ( p 2 2 ) i s biconnected i f and o n l y i f n e i t h e r G nor E c o n t a i n s a complete b i g r a p h K(m,n) a s a spanning subgraph f o r some m and n w i t h m + n = p. W e d e n o t e by c ( G ) , e ( G ) t h e number of c u t v e r t i c e s and e n d v e r t i c e s of a connected g r a p h G , r e s p e c t i v e l y . W e s a y t h e c u t v e h t e x - t y p e of G E G i s (m,n) i f {c(G), = cm,n). Similarly t h e c n d v e k t e x - t y p e of G E G i s (m,n) i f { e ( G ) , e ( E ) j = {m,n). Note is t h a t t h e c u t v e r t e x - t y p e ( t h e endvertex-type) of a g r a p h G E t h e same a s one of i t s complement E from t h e d e f i n i t i o n . The
c(c)}
1
J. Akiyama, K.Ando and H.Mizuno
2
c u t v e r t e x - t y p e and t h e endvertex-type f o r a l l g r a p h s o f o r d e r
p
2
5,
a s p r e s e n t e d i n Table 1.
Table 1.
The c u t v e r t e x - t y p e and t h e endvertex-type f o r g r a p h s ( E G) of o r d e r p ( 5 5 ) .
W e now p r e s e n t a l i s t of symbols which w e u s e i n t h i s paper: C(m,n) = {G E G I G i s a g r a p h w i t h c u t v e r t e x - t y p e (m,n) 1 ,
E(m,n) = ( G
E
G I
G
i s a graph w i t h endvertex-type
(m,n)l,
Cp(m,n) = C(m,n) n G P' Ep(m,n) = E(m,n) n G P' C ( G ) = { v E V ( G ) J v i s a c u t v e r t e x of GI, End(G) = ( v E V ( G ) Iv i s an e n d v e r t e x of G I , E = U E(m,n), t h a t is E is a s e t of a l l b i c o n n e c t e d g r a p h s m,n?J such t h a t b o t h G and z have a t l e a s t one e n d v e r t e x .
G
The f o l l o w i n g two w e l l known r e s u l t s a r e r e q u i r e d t o prove t h e lemmas w e w i l l mention l a t e r . P r o p o s i t i o n 1. Every n o n t r i v i a l graph G of o r d e r l e a s t two v e r t i c e s t h a t a r e n o t c u t v e r t i c e s ; c6G)
2
p -2
p
f o r every n o n t r i v i a l graph
contains a t
G.
0
P r o p o s i t i o n 2 . (Harary C31) I f v i s a c u t v e r t e x of a biconnected graph G , t h e n v i s n o t a c u t v e r t e x of E ; C(G)
n C ( c ) = Ip
f o r e v e r y graph
G
E
6.
0
Characterizations and classifications of biconnected graphs
3
W e now p r e s e n t two fundamental lemmas.
Lemma 1.
Let
C(G).
Then
(i) (ii)
c(G
- v ) 2- c ( G ) - 1, and if t h e e q u a l i t y h o l d s , t h e n
Proof. of
be a connected g r a p h and
G
Let
G , where
C ( G ) = {u,,
n = c(G).
u2,
..., u n l
Assume t h a t
v
v
be any v e r t e x n o t i n
i s an e n d v e r t e x of
G.
be t h e c o l l e c t i o n of c u t v e r t i c e s G
-
mi ( i ) Gk r
(i)
ui = k;l
where Gk
a r e components of G - ui. Without loss of g e n e r a l i t y w e may assume t h a t G i i ) c o n t a i n s v. W e d i v i d e o u r proof i n t o two c a s e s depending on t h e degree of v. (i5 k
5 m,)
Case 1.
degG v
2
2.
I n t h i s c a s e , t h e d e g r e e of
Gji)
v
in
.ai)
G1( i ) i s
a t l e a s t 1, and so - v i s n o t a n u l l graph. Thus, f o r e v e r y i ( l 5 i 5 n) , G ui - v = (G v ) - ui = ( G : ~ ) v) u u u Gm( i ) h o l d s . T h i s f a c t i m p l i e s t h a t ui i s a l s o a c u t v e r t e x
-
...
of
G
i
- v.
-
-
...
Therefore we obtain t h e i n e q u a l i t y
Case 2 . degG v = 1. I n t h i s c a s e t h e r e a r e two p o s s i b i l i t i e s depending on whether v i s a d j a c e n t t o a c u t v e r t e x of G v o r not.
-
or
By (l), (2) and ( 3 ) w e complete t h e p r o o f .
0
Remark: Lemma 1 g i v e s t h e lower bound of c ( G - v) i n t e r m s of f o r a connected graph G and a non-cutvertex v of G , howe v e r it i s i m p o s s i b l e t o e s t i m a t e t h e n o n t r i v i a l upper bound of c(G v ) i n terms of c ( G ) a s s e e n i n t h e example G = Pn + K1. Furthermore, n o t e t h a t t h e converse of s t a t e m e n t (ii)of Lemma 1 d o e s n o t hold when t h e e n d v e r t e x v of G i s a d j a c e n t t o a c u t v e r t e x of G - v, a s shown by t h e graph i n F i g u r e 1.
c(G)
-
Figure 1
J. Akiyama, K.Ando and H. Mizuno
4
I t i s e a s y t o prove t h e f o l l o w i n g r e s u l t s corresponding t h e p r e v i o u s two p r o p o s i t i o n s .
Proposition 3 .
For any biconnected g r a p h
G,
Proof.
For any connected g r a p h H of o r d e r p , t h e i n e q u a l i t y - 1 h o l d s , and i f t h e e q u a l i t y h o l d s , namely e ( H ) = p(H) 1, t h e n H must be a s t a r . However e v e r y s t a r i s not biconnected, thus we obtain e(H)
2
p(H)
-
P r o p o s i t i o n 4 . I f v i s an e n d v e r t e x of a biconnected graph t h e n v i s n o t an endvertex o f E ; End(G) n End(E) = 4
f o r every graph
G
E
6.
GI
0
Lemma 2 . L e t G E G and u , v be any v e r t i c e s w i t h degG u = 1, P' degG v = p 2, t h a t i s , u E End(G) and v E E n d ( c ) . Then, G c o n t a i n s a p a t h P4 a s an induced subgraph, which c o n t a i n s b o t h u and v .
-
Proof. W e may assume t h a t uv E E ( G ) , o t h e r w i s e w e may c o n s i d e r uv i n E i n s t e a d of G. S i n c e degG v = p 2, there is exactly one v e r t e x w which i s n o t a d j a c e n t t o v i n G . S i n c e degG u = 1 and uv E E ( G ) by t h e assumption, u i s n o t a d j a c e n t w I n . G . S i n c e G i s connected, w i s a d j a c e n t t o some v e r t e x x o t h e r t h a n u o r v . S i n c e degG v = p - 2 and degG u = 1, w e see t h a t v is a d j a c e n t t o x , and u i s n o t a d j a c e n t t o x . (See F i g u r e 2 . )
-
Figure 2. T h e r e f o r e t h e subgraph induced by in
E)
is
P4
containing both
{ u , v , w, x ) u
and
v
in
G
as r e q u i r e d .
(and a l s o 0
Characterizations and classifications of biconnected graphs
5
I f G and Pn a r e g r a p h s w i t h t h e p r o p e r t y t h a t t h e i d e n t i f i c a t i o n of any v e r t e x o f G w i t h an a r b i t r a r y e n d v e r t e x o f H r e s u l t s i n a u n i q u e g r a p h ( u p t o isomorphism), t h e n we w r i t e G Pn f o x t h i s graph.
-
W e s h a l l show t h e e x i s t e n c e theorem f o r g r a p h s w i t h c e r t a i n c u t vertex-types o r endvertex-types. Theorem 1.
For a l l p o s i t i v e i n t e g e r s
p
2
and
6
m(0
2
m
(i1
Cp(m,O)
# Izl,
(ii) and
Ep(m,O)
#
(iii)
Ep(m,l)
# 8.
Proof.
To p r o v e t h i s , it i s s u f f i c i e n t t o e x h i b i t g r a p h s
5__
5
-
p
2),
PI
G
E
G
belonging t o t h e v a r i o u s sets (see F i g u r e 3 ) :
(i)
(m = 01,
C
P
'p-m P
'm+l
(1 5 m
2
P
-
11,
( m = p - 2).
P
m = l
m = O
m = 2
m = 3
m - 4
Figure 3 . p = 6 . (ii)
Tm = T(m/2
Let
, m/2)
b e a d o u b l e s t a r and d e f i n e a g r a p h
T; as t h e g r a p h o b t a i n e d from Tm by i n s e r t i n g p - m - 2 new v e r t i c e s of d e g r e e 2 i n t o t h e edge j o i n i n g two n o n e n d v e r t i c e s o f
(see F i g u r e 4 ) . m(0
5 m 2
p
-
2).
Then t h e g r a p h
Tk
is i n
E(m,o)
for
T~
J. Akiyama, K. Ando and H. Mizuno
6
Figure 4 .
p = 8 , m = 5 , when
m = 0,
it
coincides with t h e case (ii). (iii) D e f i n e a g r a p h
(1
5 m 5
p
-
GA
of o r d e r
where
KO
Pp-m-1
u
Then w e o b t a i n
stands f o r a n u l l graph.
by adding a new v e r t e x Of
p, m
a s follows: f o r
set
21,
GA = K1 + (Pp-m-l
-
p(2 6)
x
and j o i n i n g
(see F i g u r e 5).
x
from
Gm
G;
t o one of t h e e n d v e r t i c e s
Then t h e g r a p h
Gm
.
Ep(m,l)
belongs t o
0
m = 4
m = 2
m = 3
m = l
Figure 5 . p = 6 . Before p r e s e n t i n g Theorem 2 , w e i n t r o d u c e a d e f i n i t i o n and a lemma which a r e r e q u i r e d i n t h e p r o o f s l a t e r . b e a connected graph. An edge uv of G i s said t o be a d o m i n a t i n g e d g e of G i f any v e r t e x w o t h e r t h a n u o r v i n G Let
G
is adjacent t o e i t h e r Lemma 3 .
Let
or
v.
be a connected g r a p h w i t h dominating edge
G
and w i t h c u t v e r t e x Proof.
u
x,
then
is either
x
F i r s t w e o b s e r v e t h a t f o r any
a l s o a dominating edge of
-
G
w.
w
E
u
V(G)
or
-
uv
E
E(G)
v. {u, v ) ,
uv
is
Then it f o l l o w s immediately from
t h e f a c t t h a t a g r a p h w i t h a dominating edge i s c o n n e c t e d t h a t e v e r y c u t v e r t e x of
G
is either
u
or
v.
0
7
Characterizationsand classifications of biconnected graphs
Theorem 2 . proof.
For
Let
G
2
mn
C (m,
E
c(E) 2 1 and t h a t G
where
-
n ) , mn E
2 , C(m, n)
2
1, m
,
C(G)
2
c
2
k
f o r some
v
3
k - p a r t i t e graph k = 2
bigraph
v
f o r some
G
U...U
IG1l,
K(IG1l,
E
C(G)
- -
2
3, G
2
2,
.., l G k l 1,
I,.
KIGl
and
k
If
lG21
and so
IG21),
2
c(G)
2,
then
v = G1 u G2
C(G).
E
Em Assume t h a t
2.
n'
i s t h e number of components o f
k
Case 1.
If
v
2
1, m
- G E
or
-
G
v.
k = 2
v
and
lGll,
2
IG21
2
c o n t a i n s a spanning complete
E -
and t h u s
i s a block.
v
v
c o n t a i n s a spanning complete
v
is a l s o a block.
That i s ,
c(E - v ) = 0. I t f o l l o w s from Lemma 1 ( i i )t h a t v E End(E) s i n c e c(E) = 1 by t h e h y p o t h e s i s . T h e r e f o r e v i s a d j a c e n t t o e x a c t l y one v e r t e x , s a y u , i n E, t h a t i s uv E E ( C ) . However i n t h i s c a s e , G
-
u
c o n t a i n s a spanning s t a r w i t h c e n t e r
-
c(G) 2 2
Since that
u
End(G).
E
Case 2 .
-v
G
v e r t i c e s of
v.
Thus
c(G
-
u)
2
1.
by t h e h y p o t h e s i s , it f o l l o w s from L e m a 1 ( i i )
=
Thus,
11-11
and
G
G
E
f o r any
u G2
u, u '
respectively.
Denote by
four vertices
u, u ' , v
E. v
C(G)
E
.
be e n d v e r t i c e s o f
Let G
v, v '
t h e g r a p h o b t a i n e d from
H
and
v'
(see F i g u r e 6).
be c u t -
adjacent t o G
Then
v, v ' ,
be removing dG(u, u ' ) 2 3 .
Figure 6. T h i s i m p l i e s t h a t t h e edge hypothesis, neighbourhood in
E,
i s a dominating edge o f
5 h a s a t l e a s t one c u t v e r t e x .
may assume t h a t V(H)
uu'
u NCu'l
v'
is a c u t v e r t e x of in
E
contains
E. v
must be an e n d v e r t e x of
E.
By t h e
Applying Lemma 3 , w e Since t h e closed
and a l l t h e v e r t i c e s o f
E.
0
J. Akiyama, K.Ando and H.Mizur;o
8
I2.
A CLASSIFICATION OF BICONNECTED GRAPHS
W e c l a s s i f y t h e set
of a l l b i c o n n e c t e d g r a p h s i n t o 5 p o s s i b l e
classes depending on t h e c u t v e r t e x - t y p e C ( m , n ) , and a l s o t h e e n d v e r t e x - t y p e E(m,n) r e s p e c t i v e l y , w e g i v e t h e s t r u c t u r a l c h a r a c t e r i z a t i o n s f o r t h e f o l l o w i n g 6 c l a s s e s of g r a p h s : C(2,2), C(2,1), C ( 1 , l )
E,
and
E(2,2),
E(m,l)
(m
2 1).
Before p r e s e n t i n g o u r r e s u l t s , w e i n t r o d u c e a c o n v e n i e n t n o t a t i o n i n o r d e r t o d e s c r i b e v a r i o u s k i n d s of f a m i l i e s of g r a p h s . Let
G,
H
be p o i n t - d i s j o i n t
subgraphs of a g r a p h
F.
The induced subgraph of < V ( G ) u V ( H ) > i n F c o m p l e t e b i p a r t i t e subgraph w i t h p a r t i t e sets
G - H :
and
V(G)
V(H).
There i s a p a i r o f a d j a c e n t v e r t i c e s u , v
G-H:
has a
with
and a t t h e same t i m e t h e r e i s a u E V ( G ) , v E V(H) p a i r of nonadjacent v e r t i c e s u, v w i t h u E V ( G ) , v G G
----
H : H :
E
V(H)
in
F.
N o edge j o i n s a v e r t e x of
G
t o a vertex of
H.
This n o t a t i o n s t a n d s f o r every p o s s i b l e adjacency r e l a t i o n between
G
V(G)
and
V(H)
.
"L: Figure 7.
W e e x p l a i n t h e s e n o t a t i o n s by t a k i n g g r a p h s i l l u s t r a t e d i n F i g u r e 7
a s an example. The f a m i l y of g r a p h s r e p r e s e n t e d i n F i g u r e 7 a r e t h e 4 g r a p h s i n F i g u r e 8 i f G = K3, H = K1.
F i g u r e 8.
9
Characterizations and clussifications of biconnected graphs
Note t h a t i f G i s t h e g r a p h i n F i g u r e 9 ( a ) , t h e n i t s complement is one o f t h e form i n F i g u r e 9 ( b ) , where F is an a r b i t r a r y g r a p h and F ' i s i t s complement.
E
1 (a) Figure 9. W e a r e now r e a d y t o c h a r a c t e r i z e g r a p h s i n Theorem 3 .
Both
G
and
E.
have a t l e a s t one e n d v e r t e x , i . e . G
E
E
i f and o n l y i f G is one of t h e f o l l o w i n g g r a p h s G1 o r G2 i n F i g u r e 10, where F i s an a r b i t r a r y g r a p h ( p o s s i b l y a n u l l g r a p h ) .
F i g u r e 10. P r o o f . F i r s t n o t e t h a t i f G is one of t h e g r a p h s of t h e form G1 o r G2 i n F i g u r e 10, t h e n E i s t h e o t h e r . And so i f G i s rep r e s e n t e d i n t h e form G1 o r G 2 t h e n G E E. It thus suffices t o show t h a t i f G E E t h e n G o r E i s r e p r e s e n t e d a s one of t h e graphs
G1
or
G2.
For any G E E, n e i t h e r End(G) nor E n d ( e ) i s empty. I t f o l l o w s from Lemma 2 t h a t G c o n t a i n s an induced subgraph P4 c o n t a i n i n g b o t h u and v f o r any u E End(G) and any v E End(G). (we may assume w i t h o u t loss of g e n e r a l i t y t h a t uv E(G).) L e t F be t h e g r a p h o b t a i n e d from
G
by removing
V(P4).
Then
J. Akiyama, K. Ando and H.Mizuno
10
(1) u
End(G)
E
to (2)
v
u in
i f and o n l y i f e v e r y v e r t e x o f
E(E)
uv
F
i s not adjacent
F
i s adjacent t o
G.
i s a g r a p h o f t h e form
Thus, G or
i f and o n l y i f e v e r y v e r t e x o f G.
End(e)
E
v
in
G1
or
G2
depending on
uv i E ( G )
0
respectively.
We now p r e s e n t two c h a r a c t e r i z a t i o n theorems, t h e f i r s t o n e f o l l o w s immediately from Theorems 2 and 3 , t h e second from Theorem 3.
or
E
A biconnected graph G i s i n C ( 2 , l ) i f and o n l y i f i s one of t h e g r a p h s o f t h e form G1 o r G 2 i n F i g u r e 11,
where
F
i s any g r a p h .
C o r o l l a r y 3.1. G
F
F
F i g u r e 11. C o r o l l a r y 3.2.
is i n graphs G3
E(m,l) G1,
G2
m
For any g i v e n i n t e g e r i f and o n l y i f or
G3
G
or
E
2
i s any g r a p h , and t h e induced subgraph
Figure 1 2 .
a biconnected graph
i s one o f t h e f o l l o w i n g
i n F i g u r e 1 2 , where
endvertex.
1,
F
is i n
i n
G I , G2 G3
and
h a s no
G
Characterizations and classifications of biconnected graphs Theorem 4 .
If
E
has a t l e a s t 3 endvertices, then
G
11
has a t m o s t
one e n d v e r t e x , i . e .
Proof.
Let
G
E
E
be a graph with
one of t h e g r a p h s o f t h e form
-
e(G)
or
G1
G2
2
3.
By Theorem 3 ,
is
G
i n F i g u r e 10.
i s a g r a p h o f t h e form G1 i n F i g u r e 10. S i n c e e ( G ) 2 3 , u E V(F) such t h a t u E End(G). T h i s implies t h a t uvl, uv3 L E ( G ) , i.e., uvlI uv3 E E ( C ) , and so v l , v3 u E n d ( E ) . Furthermore, s i n c e UW, v4w E E ( E ) f o r any w E V(F) - {u}, w e see t h a t w L End(E) f o r any w E V(F) {u). Thus w e o b t a i n End(E) = {v2}, i . e . e ( E ) = 1. Case 1.
G
there exists a vertex
-
i s a g r a p h of t h e form G2 i n F i g u r e 10. S i n c e e ( G ) 2 3, t h e r e e x i s t s a v e r t e x u E V(F) such t h a t u E E n d ( G ) . S i m i l a r t o Case 1, w e o b t a i n t h a t End(E) = {v3}, and t h u s e ( G ) = 1.
Case 2 .
G
By Case 1 and 2 , w e see t h a t
e(E)
= 1
if
e(G)
2
0
3.
The f o l l o w i n g r e s u l t s a r e immediately from Theorems 2 and 4 . Corollary 4.1.
m , n (mn
For any i n t e g e r
2
31,
0
C(m,n) = 0. Corollary 4.2.
The set
f u r t h e r m o r e any g r a p h 1 3 , where
F
coincides with t h e set
c(2,2)
in
G
E(2,2),
i s one of t h e graphs i n Figure
c(2,2)
i s an a r b i t r a r y g r a p h ( p o s s i b l y a n u l l g r a p h ) .
Figure 13. Theorem 5. Proof. G
L E.
C(l,l) = E u
W e sliow t h a t
Assume t h a t
G
E(1,O). E
E ( l ,O)
e(G) = 0
and
provided t h a t C ( G ) = {v},
G
E
then
c (1,l)
and
J. Akiyama, K. Ando and H.Mizuno
12
...
-
G v = G1 u G2 u u Gn, n e n t of G - v . Denote by
where Gi ( i = 1 , 2 , . . . , pi t h e o r d e r of Gi,
n ) i s a compot h e n pi 2 2
-
f o r e a c h i (1 5 i 5 n ) , s i n c e e ( G ) = 0. Thus, G - v c o n t a i n s a spanning complete n - p a r t i t e g r a p h K(pl, p 2 , . , pn) Thus E - v i s a b l o c k and so c ( z v ) = 0 by t h e assumption. I t f o l l o w s from
..
-
Lemma 1 ( i i )t h a t v Therefore, we obtain
E
End(E), s i n c e
c (z) = 1 by t h e a s s u m p t i o n .
The s t r u c t u r a l c h a r a c t e r i z a t i o n f o r g r a p h s of from Theorem 5. C o r o l l a r y 5.1.
A biconnected graph
.
G
in
c(l,l)
C(1,l)
is derived
if and o n l y i f
o r E i s one of t h e g r a p h s i n t h e form i n F i g u r e 1 4 , where F i s an a r b i t r a r y g r a p h i n F i g u r e 1 4 ( a ) , and b o t h F and H a r e g r a p h s such t h a t n e i t h e r F n o r H - (N(u) u N ( w ) ) h a s an i s o l a t e d vertex. G
U
Figure 1 4 . P r o o f . F i r s t w e o b s e r v e t h a t t h e g r a p h s o f F i g u r e 1 4 ( a ) and F i g u r e 14(b) a r e i n C ( 1 , l ) . I n o r d e r t o prove t h e n e c e s s i t y , w e d i v i d e o u r proof i n t o two c a s e s a c c o r d i n g t o Theorem 5. Case 1. G E E n C ( l r 1 Theorem 3 t h a t G o r 1 4 (a)
.
).
E
I n t h i s c a s e , it f o l l o w s immediately from must be a g r a p h o f t h e form i n F i g u r e
13
Characterizationsand classificationsof biconnected graphs Case 2 . G
G
E
w
and
Assume
has one endvertex
be t h e v e r t e x a d j a c e n t t o
u
Let
be a v e r t e x adjacent t o
u / C(G)
u
in
v
and
in
u / End(G),
Since
G.
v
by t h e assumption and P r o p o s i t i o n 2 , r e s p e c t i v e l y , w e
obtain t h a t
vertex o f
C(G
- u,
G
component of
-
G
-
u) = C(G)
-
C(G
v
by Lemma 1.
u ) = C ( G ) = {v}.
Since Let
H I
u.
containing t h e vertex
v
i s a dominating be t h e c o n n e c t e d
Set
i s n o t a n u l l g r a p h f o r d e g G ( u ) = d e g H , ( u ) 2 2 and a l s o i s n o t a n u l l g r a p h f o r v i s a cutvertex of G. Furthermore,
Then F
n C(1,l).
E(l,O)
h a s no e n d v e r t e x .
H
since
h a s no e n d v e r t e x , n e i t h e r
G
i s o l a t e d vertex.
Thus i f
g r a p h of F i g u r e 1 4 ( b ) P r o p o s i t i o n 5.1. Proof.
If
G
G
.
C(0,o)
E
E(l,O)
-
H n
N(u,w)
c(l,l)
nor
F
h a s an
then
G
must be a
0 c
E(0,o).
has an endvertex, then
G
0
has a cutvertex.
From t h e r e s u l t s o b t a i n e d above, w e c a n c l a s s i f y a s e t
G
of b i -
connected g r a p h s a s i n T a b l e 2 . REFERENCES
C11 M. Behzad, G. C h a r t r a n d and L . L e s n i a k - F o s t e r , Ghaphb and Dighaphb, P r i n d l e , Weber & Schmidt, ( 1 9 7 9 1 , Reading. ( 1 9 6 9 ) , Reading.
C21
F. Harary, Ghaph T h e o h y , Addison-Wesley
C31
F . Harary, S t h u c t u h a t d u a t i t y , B e h a v i o u r a l S c i . 2 (1957) 225-265.
x
rI
A
8 Table 2.
A classification of a set
6 of
biconnected graphs.
Annals of Discrete Mathematics 13 (1982) 15-22 0 North-Holland Publishing Company
LINE GRAPHS AND THEIR CHROMATIC POLYNOMIALS RUTH A. BAR1 George Washington University Washington, D.C.20052 U.S.A. Let G be a connected (p,q)-graph with q > 0 , and let L(G) be the line graph of G Denote the chromatic polynomial of G by P(G,X), and the line chromial of G, that is, the chromatic polynomial of L(G), by PL(G;A). Since L(G) has relatively many lines, it is most convenient to compute the line chromial of G in factorial form. A useful method for computing PL(G,A) in this form is to multiply the partition matrix of the lines of G by the By this method, certain adjacency vector of L(G) relations are derived between the points and lines of G and the coefficients of PL(G,A). An appendix gives the coefficient vectors for the factorial forms of P(G,A) and PL(G,X) for all connected (p,q)-graphs with p S 5 and O < q s 8 .
.
.
1.
Introduction The definitions and notation in this paper are based on C21. In addition, we will need the following definitions. Let G be a (p,q)-graph whose lines have been labelled with integers 1,2,...,q, The and let L(G) , the line graph of G , be a (pL,qL)-graph. adjacency v e c t o h of L(G) is the column vector lT , with entries AL = Ca12,a13,...,a l q ~ a 2 3 ~ a 2 4 ~ . . . ~ a 2 q ~ .q-lrq ..~a a for all pairs i, j such that i j s q , where a = 1 if ij ij lines i and j are adjacent, and aij is 0 otherwise. If n is a partition of the lines of G into k parts, 1 s k 5 q , the paktitio n v e c t o k of n is the vector P = Cp12,p131.. Iplqr~231p24r.. I j s q , where pij = 1 PZq' * * * lPq-lrqI , with entries pij for all i if lines i and j are in the same cell of n , and pij = 0 if A A - c o e o k i n g of L(G) is a i and j are in distinct cells of n function from the set of lines of G into the set I A = {lI2,...,A1 , wbse elements,are called c o e o k d . A A-coloring is p k o p e h if no two
.
.
15
.
R.A. Ban
16
a d j a c e n t l i n e s a r e a s s i g n e d t h e same c o l o r .
Associated with each
X-coloring of L ( G ) , t h e r e i s a p a r t i t i o n o f t h e l i n e s o f G i n t o c o l o r classes. A c o t o h ceabd i s a s e t o f i n d e p e n d e n t l i n e s o f G o r a matching. A c o t o h p a h t i t i o n of L(G) is a p a r t i t i o n of t h e l i n e s of G i n t o c o l o r c l a s s e s . I f a c o l o r p a r t i t i o n TI h a s k c o l o r classes, t h e number o f p r o p e r X-colorings a s s o c i a t e d w i t h
,
is = X(X-1) (A-2) (A-k+l) Thus, i f P L ( G , X ) denotes the Line chhomiaL 0 6 G , i . e . t h e c h r o m a t i c polynomial o f L ( G ) , t h e s e t o f a l l c o l o r p a r t i t i o n s of t h e l i n e s of G l e a d s t o t h e f a c t o r i a l form
.
...
of
PL(G,X) PL(G,A)
TI
P
= boX(q)
+
+
blX(q-l)
b2X(q-2)
+...+
bq-lX
bi i s t h e number o f p a r t i t i o n s o f t h e l i n e s o f G i n t o q - i c o l o r classes. The S t i h L i n g numbehd 0 6 t h e d i h d t kind, d e n o t e d s ( n , r ) , a r e d e f i n e d
where
.
f
= X ( X - 1 ) . ..(X-n+l) = s ( n , r ) Xr I t follows by t h e r e l a t i o n r= n from t h i s d e f i n i t i o n t h a t s ( n , n ) = 1, s ( n , n - l ? = - ( 2 ) , ‘ n-1 s ( n , l ) = (-1) (n-1) I , and s ( n , O ) = 0 The qxq S t i h t i n g mathiX 0 6 t h e d i h b t kind i s t h e m a t r i x Sl,q whose e n t r i e s a r e S t i r l i n g
.
numbers o f t h e f i r s t k i n d , where
Let
PL(G,X)
=
q-1 C
i=o
biX(q-i)
=
q-1 C
i=O
be t h e l i n e c h r o m i a l - o f
cihq-i
G
i n f a c t o r i a l and s t a n d a r d forms r e s p e c t i v e l y , and l e t
...,
and C = Cco,cl, c 1 be t h e i r r e s p e c t i v e q- 1 Then it i s c l e a r t h a t B.S1 = C *q Counting t h e c o l o r p a r t i t i o n s o f L ( G )
B = Cbo,bl,...,bq-ll
coefficient vectors. 2.
.
.
The f o l l o w i n g approach t o c o u n t i n g c o l o r p a r t i t i o n s i s a m o d i f i c a t i o n o f t h e method i n t r o d u c e d by O’Connor C31 Theorem 1. L e t n be an a r b i t r a r y p a r t i t i o n o f t h e l i n e s of G into k parts. L e t P be t h e p a r t i t i o n v e c t o r o f n , and l e t AL
.
.
be t h e a d j a c e n c y v e c t o r o f L ( G ) Then n i s a c o l o r p a r t i t i o n o f L ( G ) i f and o n l y i f t h e s c a l a r p r o d u c t P.% = 0 Proof. Suppose ’ n i s a c o l o r p a r t i t i o n o f L ( G ) Then i f l i n e s i and j a r e i n t h e same c e l l o f n , t h e y a r e i n d e p e n d e n t , so pij = 1 implies a = 0 , and a = 1 implies p = 0 for a l l
.
ij
ij
i j
.
Line graphs and their chromatic polynomials
.
17
.
p a i r s i , j of l i n e s of G Thus P.% = 0 C o n v e r s e l y , i f ‘TI i s n o t a c o l o r p a r t i t i o n of L ( G ) , t h e n some c e l l of T c o n t a i n s a n a d j a c e n t p a i r of l i n e s , i and j Thus p i j = 1 and a = 1 ,
.
.
ij
p i j a i j = 1 , and P.AL # 0 0 Suppose t h a t AL h a s e x a c t l y r 0 - e n t r i e s . Then G h a s r p a i r s o f non-adjacent l i n e s , o r 2 - l i n e matchings. L e t M = {ml~m2,...,m,~ b e t h e s e t o f 2 - l i n e matchings o f G . Then e v e r y matching of G i s e i t h e r a s i n g l e l i n e o r a u n i o n of 2 - l i n e
so
matchings. A partition
‘TI i s an M - p a h t i t i o n o f t h e l i n e s of G i f e a c h c e l l of n c o n t a i n s e i t h e r a s i n g l e l i n e o r a u n i o n of 2 - l i n e matchings of G The p a h t i t i o n m a t h i x PM of L ( G ) i s t h e m a t r i x whose rows a r e t h e p a r t i t i o n v e c t o r s o f t h o s e M - p a r t i t i o n s o f L ( G ) which have a t l e a s t A p a r t s . I n PM , a p a r t i t i o n v e c t o r i s s a i d t o b e The a t Levee k i f i t r e p r e s e n t s a k - p a r t M - p a r t i t i o n o f L ( G ) vectors a t l e v e l q , q - l , A a p p e a r i n t h a t o r d e r i n PM , b u t t h e o r d e r o f t h e v e c t o r s w i t h i n t h e same l e v e l i s a r b i t r a r y . D i f f e r e n t l e v e l s a r e s e p a r a t e d by dashed l i n e s . Example 1
.
.
...,
G =
3Q5
A = 3
1 2 1 3 1 4 15 23 2 4 25 34 35 45 AL = C 1 ,
1, 0, 0, 1, 1, 1, 1, 0, 1 1
M = [{l,4l,~l,5lI~3,5l) M-parti t i o n s 5 parts: 1;2;3;4;5 4 parts: 1,4;2;3;5 1,5;2;3;4 3 parts: 1,4,5;2;3 1,4;3,5;2 P a r t i t i o n matrix PM(G)
3,5;1;2;4 1,3,5;2;4
R.A. Ban
18
Theorem 2 .
The number o f z e r o s i n t h e v e c t o r
number o f c o l o r p a r t i t i o n s of
.
L(G)
Pw.k = D
is t h e
The number o f z e r o e s a t
i s t h e c o e f f i c i e n t bk of A ( q - k ) i n the f a c t o r i a l form o f t h e l i n e c h r o m i a l P L ( G , A ) Proof. By Theorem 1, e a c h z e r o i n D r e p r e s e n t s a c o l o r p a r t i t i o n of L ( G ) S i n c e t h e p a r t i t i o n v e c t o r s a t l e v e l q-k correspond t o ( q - k ) - p a r t p a r t i t i o n s o f t h e l i n e s o f G , t h e number o f z e r o e s i s t h e number o f ( q - k ) - p a r t c o l o r p a r t i t i o n s of L ( G ) , a t l e v e l q-k t h a t i s , t h e c o e f f i c i e n t bk o f A(q'k) i n L(G) 0 level
of
(9-k)
D
.
.
.
L e t G be a ( p , q ) - g r a p h w i t h p o i n t s Theorem 3 . and l e t L ( G ) be t h e l i n e g r a p h o f G Then i f
.
triangles,
L(G)
contains
= T
T~
.
- iP=o (:i)
...,
v1,v2, v P I G contains T
t r i a n g l e s , where
di
is t h e d e g r e e o f v i Corresponding t o each t r i p l e o f m u t u a l l y a d j a c e n t l i n e s o f G t h e r e i s a t r i p l e of m u t u a l l y a d j a c e n t p o i n t s o f L(G) Three l i n e s o f G a r e m u t u a l l y a d j a c e n t o n l y i f t h e y form a t r i a n g l e i n G o r i f t h e y a r e i n c i d e n t w i t h a common p o i n t of G 0 Proof.
.
.
Theorem 4 . and l e t A
Let
PL(G,A)
...,
be a ( p , q ) - g r a p h w i t h p o i n t s v1,v2, v P' n-1 = C biA(n-i) L e t di d e n o t e t h e d e g r e e o f vi, G
.
i=o
t h e maximum d e g r e e o f
G
, and
T
bi
of
A(n-i)
t h e number o f t r i a n g l e s o f
G.
Then
1.
The c o e f f i c i e n t
c o u n t s t h e number o f
( n - i ) - p a r t c o l o r p a r t i t i o n s of t h e l i n e s o f 3.
n = q , t h e number of l i n e s o f I f br # 0 , t h e n r s q-A
4.
bl = (q2 )
5.
b2 =
2.
Proof.
1.
-
[ti] . [ti)
.
i=l
i=l
+
G
-
bl[";')
,
G.
and
bo = 1.
.
~(q~q-2)
The number o f A-colorings a s s o c i a t e d w i t h e a c h ( n - i ) - p a r t
.
Hence bi , t h e c o e f f i c i e n t c o l o r p a r t i t i o n of L ( G ) i s A ( " - ~ ) of A ( n - i ) , c o u n t s t h e number of ( n - i ) - p a r t c o l o r p a r t i t i o n s of L ( G ) . 2 . The unique c o l o r p a r t i t i o n o f L(G) w i t h t h e g r e a t e s t number of p a r t s i s t h a t i n which e a c h l i n e i s a c o l o r c l a s s . Thus c o l o r i n g s of L ( G ) a s s o c i a t e d a = 1 0 3 . The s m a l l e s t number o f c o l o r c l a s s e s i n a c o l o r p a r t i t i o n o f L ( G ) i s x' , t h e l i n e c h r o m a t i c number o f G , so e v e r y
t h e r e a r e q c o l o r c l a s s e s and A ( q ) w i t h t h i s p a r t i t i o n , so n = q , and
.
19
Line graphs and their chromatic polynomials
color partition has at least X I color classes. Since, by Vizing's Theorem, x' = A or x' = A+l , if br # 0 , then q-r ;r A or q-r 2 A + l Hence r 5 q - A 1 s q - A , so r s q A 4. Let B = Cl,bllb21...lbr,0,0,...101 be the coefficient vector of PL(G,X), the line chromial of G expressed in factorial 1 the coefficient vector of PL(G,X) form, and C = ~1,c1,c2,...,c 9-1 expressed in standard form. Since Bas1 = C , we get 1 9 l*s(q,q-l) + bl*s(q-l,q-l) = c1 B u t since s(q,q-1) = -(;) and
-
.
-
.
.
Again from the fact that Basllq= c , we get 5. l*~(q,q-2)+ bl*s(q-l,q-2) + b2.s(q-2,q-2) = c2 . But ~ ( q - l ~ q - 2 ) =
(q;l)
Hence
,
s (q,q-2)
[ZL) - ' -
b2 =
Example 2.
= 1
+
,
~ ( q - 2 ~ q - 2=) 1
1 2(3)
-
+
bl Cq;l)
' ]
P di i=l [ 3
and it is known that
+
b2 = [ZL)
bl(q;l)
-
-
T
-
2
(3)
7 = (2)
-
+
1
2
+
4 3(2)
-
[zL]-
TL =
so
.
~(q~q-2)
3 2(3) = 1 + 0 + 0 + 2 = 3.
-
:p),
i=l
From Example 1, we know that if
+
c2 =
0
G =
Therefore
.
35 = 1
References C11 Bari, R.A., and Hall, D . W . , Chromatic Polynomials and Whitney's Broken Circuits, J. Graph Theory 1 (19771, 269-275. c21 Harary,~., Graph Theory, Addison Wesley, Reading, 1969. C31 O'Connor,M.G., A Matrix Approach to Graph Coloring, Honors Thesis , Mt .Holyoke College (1980) C41 Riordon,J., An Introduction to Combinatorial Analysis,Wiley, New York, 1967.
.
20
R.A. Ban' APPENDIX Chromials and L i n e Chromials o f Connected Graphs w i t h a t most F i v e P o i n t s and E i g h t L i n e s I n t h e following t a b l e , t h e c o e f f i c i e n t v e c t o r s of t h e f a c t o r i a l
form of
P(G,A)
(p,q)-graph with
and p
a r e given f o r each connected
PL(G,A) 5
5
and
0
q s 8
.
C1,Ol
c11
C 1 , l ,Ol
c1,01
c1,0,01
c1,0,01
C1,3,1 to1
c1,1,01
C1,3,1,01
c1,0,01
c1,2,0,01
C 1 , l ,O,Ol
c 1 , 2,1,01
c1,2,1,01
c1,1,0,01
c1,2,1,0,01
c1,0,0,01
[1,3,3,1,0,01
C1,6,7,1,01
c1,0,0,01
C1,6,7,1,01
C1,3,1,01
C1,6,7,1,01
c1,2,0,01
C1,5,4,0,01
C1,3,1,0,01
C1,5,5,4,01
C1,4,3,0,01
C1,5,4,0,01
C1,4,2,0,01
C1,5,4,0,01
c1,2,0,0,01
Line grapk and their chromatic poiynomiuls
C 1,5,1,0,01 c1.5,4 , O f 0 1 C1,4,3,0,0,01 C1,6,8,1,0,01 C1,6,9,2,0,01 ri,5,5,i,o,oi
C 1,7,12,1,0,0,01 ri,6,9,2,0,0,01 C1,6,9,4,0,0,01
C 1,0,17,8,0,0, 01 c1,9,23,17,2,0,0,01 c1,10,29,25,2,0,0,01
21
This Page Intentionally Left Blank
Annals of Discrete Mathematics 13 (1982) 23-32 0 North-Holland Publishing Company
LARGE GRAPHS WITH GIVEN DEGREE AND DIAMETER I11
J.C.BERMOND, C. DELORME and G. FARHI UniversitC de Paris Sud, Orsay, bit. 490 France
The following problem arises in the study of interconnection networks: find graphs of given maximum degree and diameter having the maximum number of vertices. In this article we give a construction which enables us to construct graphs of given maximum degree and diameter, having a great number of vertices from small ones: in particular we obtain a class of graphs of diameter 3 , maximum degree A and having about 8A 3 /27 vertices. I.
INTRODUCTION
We are interested in the (AID) graph problem, one of the problems arising in telecommunications networks (or microprocessor networks). Let G = (X,E) be an undirected graph with vertex set X and edge The d i d t a n c e between two vertices x and y I denoted by set E 6(x,y) , is the length of a shortest path between x and y The d i a m e t e h D of the graph G is defined as D = MX 6(x,y). The (XIY)EX degree d(x) of a vertex x is the number of vertices adjacent to (For a survey on x and A denotes the maximum degree of G diameters see C11.) The (AID) graph problem is that of finding the maximum number of vertices n(A,D) of a graph with given maximum degree A and diameter D This problem arises quite naturally in the study of interconnection networks: the vertices represent the stations (or processors), the degree of a vertex is the number of links incident at this vertex and the diameter represents the maximum number of links to be used to transmit a message. The problem seems to have been first set in the literature by Elspas C71. Different contributions have been made in the 70's and the known results were summarized in Storwick's article C101. These results have been recently improved by Memmi and Raillard C81 and Quisquatter C91.
.
.
.
.
A
simple bound on
n(A,D)
is given by Moore (see C3 or 4 1 ) : 23
J.C. Berrnond, C. Delorme and G. Farhi
24
n(2,D) s 2 D + 1
and f o r
A > 2
,
n(A,D)
5;
A ( AA-2 -1)D-2
s a t i s f y i n g t h e e q u a l i t y a r e c a l l e d Moohe g h a p h b .
.
The g r a p h s
I t h a s been
proved by d i f f e r e n t a u t h o r s (see Biggs C3 chap.231) t h a t Moore g r a p h s can e x i s t o n l y i f A = 2 ( t h e g r a p h s b e i n g t h e ( 2 D + 1 ) c y c l e s ) o r if D = 2 and A = 3 , 7 , 5 1 ( f o r A = 3 and A = I t h e r e e x i s t s a unique Moore graph r e s p e c t i v e l y P e t e r s e n ' s g r a p h on 10 v e r t i c e s , and Hoffman and S i n g l e t o n ' s graph on 50 v e r t i c e s , f o r i s n o t known).
A = 57 t h e answer
The aim of t h i s a r t i c l e i s t o g i v e a new c o n s t r u c t i o n , which g i v e s g r a p h s having a g r e a t number o f v e r t i c e s i n p a r t i c u l a r f o r s m a l l d i a m e t e r s ; a s a c o r o l l a r y we o b t a i n t h a t l i m i n f n ( A , 3 ) c 8 A 3 A + m. T h i s p a p e r i s t o be c o n s i d e r e d as a companion o f C2,5,61 t h e cons t r u c t i o n is based on a new p r o d u c t o f g r a p h s which i s s t u d i e d i n C21 and i n C5,61 are g i v e n i n f i n i t e f a m i l i e s o f g r a p h s o f g i v e n A D d e g r e e and d i a m e t e r . I n E51 i t was proved t h a t n(A,D) 5 (?) 3 which g i v e s o n l y i n t h e c a s e of d i a m e t e r 3. 8
,
I1 DEFINITIONS
2.1
*
The
product
L e t G = (X,E) and G' = ( X ' , E ' ) be two g r a p h s . Take an a r b i t r a r y o r i e n t a t i o n of t h e e d g e s o f G and l e t U be t h e s e t o f a r c s .
F i n a l l y , f o r each arc ( x , y ) of U
.
,
let
be a one t o one
(X,Y)
mapping from X' t o X ' W e d e f i n e t h e p r o d u c t G*G' a s f o l l o w s : The v e r t e x s e t of G*G' i s t h e C a r t e s i a n p r o d u c t vertex if
(x,x')
elthee OR.
is joined t o a vertex
x = y (x,y)
and E
U
(y,y')
(xl,yl) E El , and y ' = f ( x , y )
in
X
x
G*G'
X'
.
A
i f and o n l y
-
Remark: G * G ' c a n be viewed a s formed by 1x1 c o p i e s of G' where two c o p i e s g e n e r a t e d by t h e v e r t i c e s x and y a r e j o i n e d i f ( x , y ) i s an arc and i n t h a t c a s e t h e y a r e j o i n e d by a p e r f e c t matching depending on ( x , y ) . Examples: G' = C5
.
( a ) L e t G = K2 , ( 1 , 2 ) b e i n g t h e o r i e n t e d a r c and l e t I n F i g . 2 . l . a , we have r e p r e s e n t e d K2*C5 where
f(l,2)(x') = x' (mod.5).
and i n F i g . Z . l . b ,
K2*C5
where
f(l,2)
(XI)
= 2x'
fi
Large graphs with given degree and diameter 111
114
111
25
114
211
2 ,3
212
113
211
112
113
2.1.a
112 2.l.b
Fig. 2.1 : K2*C5 with two different
flI2 Note that the diameter of the graph of fig.2.1.a is 3 , but the diameter of Petersen's graph (fig.2.1.b) is 2. (b) If we choose for any arc (x,y) fx (x') = x' then G*G' is rY nothing else than the Cartesian sum (called also Cartesian product) of G and G' and in this case the diameter of G+G' is the sum of the diameters of G and G'
.
Degrees and diameter of G*G' If G is of maximum degree A and G' of maximum degree A ' , then G*G' has as maximum degree A + A ' If G is of diameter D and G' of diameter D' , then the diameter of G*G' is less than or equal to D + D' and a clever choice of the functions f can give a small diameter.
.
(XIY)
2.2 The property P* graph G = (X,E) is said to have the property P * if G has diameter at most 2 and if there exists an involution f of X (i.e. f2 is the identity of X) , such that for every vertexx of G we have x = (xlulf (x)IUIf (r(x)1 luIr (f(x)) l A
.
Examples: In figure 2.2 are shown three graphs satisfying property Other examples will P* with respectively 5,8 and 9 vertices. be given in section 4 . f (d)
Figure 2 . 2
26
J. C.Bermond, C. Delorrne and G. Farhi The p r o p e r t y
2.3
P
G w i l l be s a i d t o s a t i s f y p r o p e r t y P , i f any p a i r o f v e r t i c e s a t d i s t a n c e D (where D i s t h e d i a m e t e r o f G) c a n be j o i n e d by a p a t h o f l e n g t h D + l A t r i v i a l example i s g i v e n by t h e complete g r a p h Kn,n 2 3 : o t h e r examples w i l l be g i v e n i n s e c t i o n 4 . A graph
.
THE M A I N THEOREM
I11
G be a graph o f diameter D 2 1 , s a t i s f y i n g property G' be a graph s a t i s f y i n g p r o p e r t y P* ( w i t h i n v o l u t i o n Then G*G' , w i t h f ( x , y ) ( x ' ) = f ( x ' ) f o r e v e r y arc ( x , y ) f) of an a r b i t r a r y o r i e n t a t i o n of G , i s a graph of diameter a t m o s t
Theorem:
P
Let
and l e t
.
D + 1 .
Proof: Any two v e r t i c e s ( x , x ' ) and ( y , y ' ) o f G*G' c a n be connected w i t h a p a t h o f l e n g t h a t most d G ( x , y ) + 2 ( a s G ' i s of d i a m e t e r a t most 2 by p r o p e r t y P*) and t h e r e f o r e i f d G ( x , y ) < D , then
(x,x')
and
(y,y')
a r e a t d i s t a n c e a t most
D
+
1
.
Now l e t x and y be a t d i s t a n c e D and l e t z be t h e v e r t e x p r e c e d i n g y on a p a t h o f l e n g t h D between x and y Let z' = f D - 1 ( x ' ) Then t h e v e r t e x (z,z') i s a t d i s t a n c e D-1 of
.
.
(x,x') ;
therefore t h e v e r t i c e s i n
(y,f(z'))
a r e a t d i s t a n c e a t most
D
( z , r ( z l ) ) and t h e v e r t e x from ( x , x ' ) and t h e v e r t i c e s
i n ( y , f ( I ' ( Z ' ) ) ) and ( y , r ( f ( z ' ) ) ) a t d i s t a n c e a t most D + l o f (x,x') F i n a l l y by u s i n g t h e p a t h o f l e n g t h D + l between x and y (by p r o p e r t y P ) t h e v e r t e x ( y , f D + l ( x ' ) ) = ( y , z ' ) is a t
.
.
d i s t a n c e a t most D + 1 f a c t t h a t by p r o p e r t y
from ( x , x ' ) i n G*G' According t o t h e P* t h e v e r t e x set o f G' i s e q u a l t o (zi)uIf(zi)}uEf(r(zi))}u{r(f(zi))}e v e r y v e r t e x o f ( y , ~ ' ) i s a t d i s t a n c e a t most D + l from ( x , x ' )
.
IV
4.1
GRAPHS SATISFYING
P* AND P
Graphs s a t i s f y i n g P*
W e have s e e n t h r e e examples i n 2.2 of d e g r e e s 2,3,4 and r e s p e c t i v e l y
5,8,9 v e r t i c e s . Note t h a t a g r a p h o f d e g r e e A s a t i s f y i n g P* h a s a t most 2A+2 v e r t i c e s . The n e x t c o n s t r u c t i o n shows t h e e x i s t e n c e o f g r a p h s o f d e g r e e A on 2A vertices. P r o p o s i t i o n : For e v e r y i n t e g e r A , t h e r e e x i s t s a g r a p h G A s a t i s f y i n g p r o p e r t y P* , o f d e g r e e A and h a v i n g 2A v e r t i c e s . Construction: F i g . 4.1.a shows such g r a p h s w i t h A = 1 and Suppose w e have c o n s t r u c t e d a g r a p h G A , s a t i s f y i n g P* , of
A = 2.
27
Large graphs with given degree and diameter III degree
A
and such t h a t t h e
IAl = I f ( A )
I
4
vertices
x,y,f(x),f(y)
a s shown i n F i g . 4.1.b.
G
extends t h a t of
vertices are
and
A
f(A)
with
L e t GA+2 b e t h e graph o b t a i n e d from GA by
s a t i s f i e d by G I and G 2 ) . adding
2A
and s u c h t h a t r ( A ) ? f ( A ) and I ' ( f ( A ) ) z A ( t h a t i s
= A
j o i n e d between t h e m s e l v e s and t o
Then t h i s graph (where t h e i n v o l u t i o n f
has c l e a r l y degree
GA)
A+2
,
2(A+2)
vertices
and it i s n o t d i f f i c u l t t o show t h a t i t s a t i s f i e s p r o p e r t y
and
P*
t h e hypothesis of induction.
X
I,
(x) G2
G1
F i g . 4.1.a 4.2
F i g . 4.1.b
Graphs s a t i s f y i n g P
A s a l r e a d y n o t i c e d t h e complete g r a p h s
Kn,
n
h
3
,
satisfy
P
.
One can show a l s o t h a t P e t e r s e n ' s g r a p h , Hoffmann-Singleton's graph, t h e products of F i g . 2 . 2 ,
Kn*X, n
3
2
s a t i s f y property
P
,
.
where
X
i s one o f t h e g r a p h s
An i m p o r t a n t c l a s s o f g r a p h s (which w i l l be u s e d i n s e c t i o n 5 ) i s
P of diameter 2 a s s o c i a t e d t o p r o j e c t i v e q L e t u s r e c a l l , t h a t i f q = pr where p is a prime number,
formed by t h e g r a p h s planes.
th ere e x i s t s a projective plane of order
i s a numbering o f t h e p o i n t s lines, M.
3
E
Dj
Di
.
,
1
5
j
Thus
5
U
q2 D
+ j
q
+
q
with a p o l a r i t y , t h a t + 1 and o f t h e
q2 + q 1 such t h a t i f Mi
Mi,
1
5
i
5
E
D
j
then
c o n t a i n s a l l t h e p o i n t s of t h e p l a n e .
b e t h e graph whose v e r t i c e s a r e t h e p o i n t s o f t h e p r o j e c t i v e p l a n e , t h e v e r t i c e s Mi and M a r e j o i n e d i f and o n l y i f M j e D i j The p r o p e r t i e s o f t h e numbering shows t h a t P is of degree q + l ; q 2 i t h a s q +q+l v e r t i c e s and i t s d i a m e t e r i s 2 NOW, l e t Mi As be two v e r t i c e s a t d i s t a n c e 2 , t h e n M . # Di Let
P
.
9
.
M1
4
3
.
and ID nD = 1 , there exists a vertex , such t h a t % E D i and i j Mk # D j ; a s is a t d i s t a n c e 1 from Mi and 2 from M j i t i s c l e a r t h a t Mi and M j a r e j o i n e d by a p a t h o f l e n g t h 3
4
S i m i l a r l y one can prove t h a t t h e q u o t i e n t o f Levi g r a p h s of
,
.
J. C. Bermond, C. Delorme and G. Farhi
28
g e n e r a l i z e d q u a d r a n g l e s and hexagons, h a v i n g a p o l a r i t y , by t h a t p o l a r i t y have p r o p e r t y P (See C51.)
.
V
APPLICATIONS
and G I = X8 t h e g r a p h on 8 v e r t i c e s o f d e g r e e 3 (see F i g . 2.2) t h e n Kn*X8 h a s 8n Vertices, d e g r e e n+2 and diameter 2 I n p a r t i c u l a r with n = 3 ( r e s p . 4 ) we o b t a i n graphs Let
G = Kn
.
The of d i a m e t e r 2 , d e g r e e 5 ( r e s p . 6 ) on 2 4 ( r e s p . 3 2 ) v e r t i c e s . b e t t e r g r a p h o f d i a m e t e r 2 d e g r e e 6 , known b e f o r e had o n l y 31
v e r t i c e s (namely
P5)
.
By t a k i n g
K3*X8
a s graph
G
and a g a i n
X8 a s G I w e o b t a i n a g r a p h of d i a m e t e r 3, d e g r e e 8 , on 1 9 2 v e r t i c e s ( t h e b e t t e r g r a p h known b e f o r e h a s 1 1 4 v e r t i c e s ) . By t a k i n g f o r G t h e g r a p h on 585 v e r t i c e s o f d e g r e e 9 and d i a m e t e r 3 and f o r
t h e g r a p h s on
GI
5(8,9)
v e r t i c e s of F i g . 2 . l
we give
examples o f g r a p h s o f d i a m e t e r 4 of d e g r e e 11 ( r e s p . 12,13)
on
2925 ( r e s p . 4680, 5265) v e r t i c e s which a r e t h e b e s t known now. Another a p p l i c a t i o n i s t h e f o l l o w i n g r e s u l t .
l i m i n f n(A,3)
Theorem:
A+m
2
$ A3
Proof: L e t G be t h e graph P d e f i n e d i n s e c t i o n 4 , w i t h q odd, 2 q q a prime power, o f d i a m e t e r 2 , d e g r e e q + l and on q +q+l 3 ( +1) v e r t i c e s which s a t i s f i e s p r o p e r t y P Let A = and l e t
4
.
G ' = GA13 be t h e g r a p h o f d e g r e e on - v e r t i c e s s a t i s f y i n g p r o p e r t y P* c o n s t r u c t e d i n 4 . 1 . By t h e main theorem Pq*GA13 is A a graph o f d i a m e t e r 3 d e g r e e q + l + = A , h a v i n g 2 A2 - 2A + i i T2A = 2 ;A37 = (4T 4-A2 + 2-A v e r t i c e s p r o v i n g (q 3 9 3 t h e r e f o r e t h e theorem.
-
VI
6.1
GENERALIZATIONS The p r o p e r t y
A graph
i s s a i d t o have p r o p e r t y
G
mutation
Pi
Pi
of i t s v e r t i c e s , such t h a t
f
, if f'
there e x i s t s a peri s a n automorphism o f
G and t h a t t h e g r a p h G* , which h a s t h e same v e r t i c e s a s G whose e d g e s are t h e e d g e s of G o r t h e i r images by f , h a s diameter 1
and
.
6.2
Examples
The g r a p h
G
of F i g . 6 , on 4 vertices, where
f ( x ) = x + l (modulo 4 )
29
Large graphs with given degree and diameter III
has property
P
1'
G*
b e i n g t h e complete g r a p h
4L
K4
.
3 Fig. 6
The graph G , on 29 v e r t i c e s , where t h e v e r t e x x and y a r e j o i n e d if y x E 1 , -1, 7 , -7 (mod.29) and where f ( x ) E 12x (mod.29) h a s p r o p e r t y P2
-
If
.
is a prime power, c o n s i d e r t h e g r a p h whose 4 e + l v e r t i c e s a r e t h e e l e m e n t s of t h e f i e l d GFqe+l : two v e r t i c e s x and y a r e a d j a c e n t i f x-y is a s q u a r e i n GF4e+l This graph has p r o p e r t y P1 ( d i a m e t e r 2 and d e g r e e 2 e ) . (These g r a p h s a r e known a s P a l e y ' s g r a p h s and a r e examples o f s t r o n g l y r e g u l a r g r a p h s . ) 4e+l
.
6.3
Theorem
Let
G
b e a graph of d i a m e t e r
.
D z i+l and l e t
b e a graph
G'
Then G*G' , w i t h (XI) = f(x') s a t i s f y i n g p r o p e r t y Pi (x,y) f o r every a r c (x,y) o f an a r b i t r a r y o r i e n t a t i o n of G , has diameter
D+i
.
Proof:
Let
x
and
y
be two v e r t i c e s of
.
(not n e c e s s a r i l y elementary) of length j a a r c s o f G and t h e r e f o r e j - a a r c s o f
j o i n e d by a p a t h P
G
I f t h i s path contains G , l e t E ( P ) = 2a-j.
-
W e w i l l p r o v e t h a t i f t h e r e e x i s t s a p a t h of l e n g t h j ' from f E ( P ) ( x ' ) t o y' i n GI* , w i t h j ' s j , t h e n t h e r e e x i s t s a p a t h of length
j+jl
in
G*G'
between ( x , x ' ) and ( y , y ' )
.
The proof
g o e s by i n d u c t i o n on j ( t h a t is t r u e f o r j = 0 ) and f o r j by i n d u c t i o n on j ' ( j ' 5 j ) ; t h a t is t r u e f o r j ' = 0 ( y , f
.
fixed (XI))
b e i n g j o i n e d t o ( x , x ' ) by t h e o b v i o u s p a t h o f l e n g t h j Suppose j l > 0 and l e t ( z ' , y ' ) be t h e l a s t edge o f t h e p a t h between f E ( P ) ( x ' ) and y ' in GI* If (z',y') E G ' , w e o b t a i n a p a t h o f l e n g t h ( j + j ' ) from ( x , x ' ) t o ( y , y ' ) by a d d i n g t h e arc
.
( ( y , z ' ) ( y , y ' ) ) t o t h e p a t h o f l e n g t h j + j ' - l from ( x , x ' ) t o ( y , z ' ) (which e x i s t s by i n d u c t i o n h y p o t h e s i s ) . If (z',y') # G' let z be t h e v e r t e x p r e c e d i n g y on t h e p a t h P o f G : t h e n f E ( Y , Z (2') ) and f E ( Y r Z()y ' ) are j o i n e d by an edge i n G' that to
E(Y,Z)
G).
equals
1 or
-1
according
(ylz)
(recall
belongs o r not
J. C. Bermond, C. Delorme and G. Farhi
30
By i n d u c t i o n h y p o t h e s i s w i t h exists in
G*G'
( z I f E ( Y r Z( )2 j+jl
j+j'-Z
from
5 j-1)
from
( z , f E ( Y ' Z ) ( y ' ) )( y , y ' ) (x,x') to (y,y')
.
there
(x,x')
and t h e n by a d d i n g t h e p a t h of l e n g t h
' ) )
G * G ' ( z , f E ( Y r z()2 ' ) ) length
jl-1 ( j l - 1
and
j-1
a path of l e n g t h
to
2
of
w e o b t a i n a p a t h of
.
Now l e t
( x , x l ) and ( y , y ' ) be any two v e r t i c e s o f G*G' There i s a ( n o t n e c e s s a r i l y e l e m e n t a r y ) p a t h P from x t o y o f l e n g t h
.
j = D or D - 1 Since D 2 i+l , j 2 i holds. As GI* is o f d i a m e t e r i , t h e r e e x i s t s a p a t h o f l e n g t h j ' s i between f E ( P ) ( x l ) and y ' i n GI*
.
We have
jl s j
(as j 15 i
and
i
5
above t h e r e e x i s t s a p a t h of l e n g t h (YIY')
6.4
.
j )
and by t h e p r o p e r t y proved
j+jl s D + i
from
(x,x')
to
Applications
By u s i n g t h e example o f 6 . 2 , w e can b u i l d new good g r a p h s .
For
example w i t h t h e P a l e y ' s g r a p h s w e o b t a i n i n f i n i t e l y many g r a p h s o f d i a m e t e r 3 which have a c t u a l l y t h e h i g h e s t known number of v e r t i c e s , a l t h o u g h t h e a s y m p t o t i c v a l u e o b t a i n e d i n theorem of s e c t i o n 5 i s n o t improved.
By t a k i n g f o r
on 585 v e r t i c e s and f o r
GI
G
t h e graph of diameter 3 , degree 9
t h e P a l e y ' s graph on 1 3 v e r t i c e s w e
o b t a i n a graph o f d i a m e t e r 4 , d e g r e e 15 on 7605 v e r t i c e s .
(This
i s i n f a c t t h e b e s t known v a l u e ) . Remark:
The c o n d i t i o n on
e x t r a c o n d i t i o n s on GI
satisfies
P1
GI
:
G I D 2 i+l c a n be d e l e t e d i f one a d d s
f o r example i f
G
and i s o f d i a m e t e r 2 , t h e n
i s o f d i a m e t e r 1 and G*G1
has diameter 2 .
REFERENCES c11
J.C.Bermond and B.Bollobds, Diameters i n g r a p h s : a s u r v e y , P r o c . 1 2 t h S o u t h e a s t e r n Conference on C o m b i n a t o r i c s , Graph Theory and Computing, Baton Rouge 1981, t o a p p e a r .
c21
J . C . Bermond, C . Delorme and G . F a r h i , Large g r a p h s w i t h g i v e n d e g r e e and d i a m e t e r 11, t o a p p e a r .
C31
N.
C41
B . B o l l o b l s , Extremal g r a p h t h e o r y , London Math. SOC. Monographs N o . 11, Academic Press, London 1978.
C51
C.
C61
C . Delorme and G. F a r h i , Large g r a p h s w i t h g i v e n d e g r e e and diameter I , t o appear. B . E l s p a s , T o p o l o g i c a l c o n s t r a i n t s on i n t e r c o n n e c t i o n l i m i t e d l o g i c , Proc. 5 t h Symposium on s w i t c h i n g c i r c u i t t h e o r y and 5-164 ( 1 9 6 4 ) , 133-197. l o g i c a l d e s i g n , I.E.E.E.
C71
Biggs, A l g e b r a i c g r a p h t h e o r y , Cambridge U n i v e r s i t y Press, Cambridge, England, 1 9 7 4 .
Delorme, Gmnds g r a p h e s de d e g r 6 e t d i a m b t r e donngs, toappear.
Large graphs with given degree and diameter III
C81
G . Memi and Y .
C91
J.J. Q u i s q u a t t e r :
ClOl
R a i l l a r d , Some new r e s u l t s a b o u t t h e ( d , k ) T r a n s . on Computers, t o a p p e a r . graph problem, I . E . E . E . manuscript
t o be p u b l i s h e d .
R.M. S t o r w i c k , Improved c o n s t r u c t i o n s t e c h n i q u e s f o r (d,k) T r a n s . Computers, 1 9 , ( 1 9 7 0 ) 1 2 1 4 - 1 2 1 6 . graphs, I.E.E.E.
31
This Page Intentionally Left Blank
Annals of Discrete Mathematics 13 (1 982) 33-50 0 North-Holland Publishing Company
DISTINGUISHING VERTICES OF RANDOM GRAPHS
BELA B O L L O B ~ University of Cambridge England
The d i b t a n c e b e q u e n c e of a v e r t e x x of a graph i s (di(x)): d i ( x ) i s t h e number o f v e r t i c e s a t d i s t a n c e i from x The paper i n v e s t i g a t e s under what c o n d i t i o n it i s t r u e t h a t almost e v e r y graph of a p r o b a b i l i t y space i s such t h a t i t s v e r t i c e s a r e u n i q u e l y determined by a n i n i t i a l segment of t h e d i s t a n c e sequence. I n p a r t i c u l a r , it is shown t h a t f o r r 2 3 and E > 0 a l m o s t e v e r y l a b e l l e d r - r e g u l a r graph i s such t h a t e v e r y v e r t e x x i s u n i q u e l y determined by (di ( x ) ) U , where u = L ( # + e l l o n J
.
where
,
log7r-l) .
Furthermore, t h e paper c o n t a i n s an e n t i r e l y c o m b i n a t o r i a l proof of a theorem of Wright C101 a b o u t t h e number o f unl a b e l l e d graphs of a given s i z e . 50.
INTRODUCTION
I n t h i s n o t e w e s h a l l s t u d y two models of random g r a p h s : G(n,M) and G(n,r-reg) The f i r s t c o n s i s t s of a l l g r a p h s of s i z e M w i t h vertex set V = {1,2, n1 and t h e second is t h e s e t of a l l r - r e g u l a r graphs w i t h v e r t e x s e t V I n b o t h models a l l g r a p h s a r e given t h e same p r o b a b i l i t y . When d e a l i n g w i t h t h e model G(n,r-reg) w e s h a l l assume t h a t r n is e v e n , so t h e r e a r e r - r e g u l a r graphs of order n i f n 2 r+l For t h e b a s i c f a c t s c o n c e r n i n g random graphs w e r e f e r t h e r e a d e r t o C2, Ch.VII1. I n p a r t i c u l a r , we s a y t h a t atrnobt e v e h y (a.e.1 graph i n a model h a s a c e r t a i n p r o p e r t y i f t h e p r o b a b i l i t y o f having t h e p r o p e r t y t e n d s t o 1 as t h e number of v e r t i c e s tends t o m L e t G be G(n,M) o r G ( n , r - r e g ) W e a r e interested i n the exist e n c e of i n d i s t i n g u i s h a b l e v e r t i c e s of a graph GEG f o r two d i f F i r s t w e c o n s i d e r two f e r e n t d e f i n i t i o n s of being d i s t i n g u i s h a b l e . v e r t i c e s o f G d i b t i n g u i b h a b l e i f no automorphism of G maps one of t h e v e r t i c e s i n t o t h e o t h e r one. Thus G E G c o n s i s t s of d i s t i n g u i s h a b l e v e r t i c e s i f f t h e automorphism group o f G i s t r i v i a l . To g i v e t h e second d e f i n i t i o n o f being d i s t i n g u i s h a b l e , g i v e n a v e r t e x x6G w r i t e d i ( x ) f o r t h e number o f v e r t i c e s a t d i s t a n c e i from x Call (di(x)): t h e d i b t a n c e 6eQUenCs 0 6 x and d e f i n e two v e r t i c e s t o be d i b t i n g u i b h a b l e i f t h e y have d i f f e r e n t
.
...,
.
.
.
.
.
33
B. Bollobas
34
sequences.
C l e a r l y two v e r t i c e s d i s t i n g u i s h a b l e i n t h i s s e n s e a r e
a l s o dis tinguis hable i n t h e f i r s t sense.
A fundamental theorem o f Wright C l O I i m p l i e s t h a t i f such t h a t a l m o s t no g r a p h
in
GM
o r two v e r t i c e s o f d e g r e e n-1,
G(n,M)
is
M = M(n)
h a s two i s o l a t e d v e r t i c e s
t h e n a l m o s t e v e r y random g r a p h
h a s t r i v i a l automorphism group.
more namely t h a t t h e number o f l a b e l l e d g r a p h s o f o r d e r
n
and
size
M
d i v i d e d by t h e number of u n l a b e l l e d g r a p h s o f o r d e r
size
M
i s asymptotic t o
.
n!
GM
I n f a c t t h e theorem s t a t e s much n
and
I n 5 1 w e g i v e a n e n t i r e l y combina-
t o r i a l proof o f t h i s r e s u l t . The rest o f t h e p a p e r i s concerned w i t h t h e problem o f d i s t i n g u i s h i n g I n 52 w e t r e a t
v e r t i c e s w i t h t h e a i d o f t h e i r d i s t a n c e sequences. t h e random g r a p h s
and i n 53 we s t u d y random r e g u l a r
GN E G(n,M) graphs of a f i x e d degree r
.
I n b o t h cases o u r aim i s t o u s e
s m a l l p o r t i o n s o f t h e d i s t a n c e sequences t o i d e n t i f y t h e v e r t i c e s . 51.
THE NUMBER OF UNLABELLED GRAPHS
Denote by and s i z e graphs.
t h e number o f u n l a b e l l e d g r a p h s o f o r d e r
UM = U n I M
M = M(n) and w r i t e LM = Ln , M W e s h a l l p u t N = (n2 ) so t h a t
show t h a t under s u i t a b l e c o n d i t i o n s on
-
UM
N
LM/n! = ( M ) / n !
f o r t h e number o f labelled
N
LM = ( M )
.
Our aim i s t o
w e have
M
.
n
(1)
T h i s is c l e a r l y c o n s i d e r a b l y s t r o n g e r t h a n t h e a s s e r t i o n t h a t t h e automorphism group o f a . e .
GM
E
G(n,M)
is trivial. n is t r i v i a l then the
I f t h e automorphism group o f a g r a p h of o r d e r
g r a p h c o n t a i n s a t most one i s o l a t e d v e r t e x and a t most one v e r t e x o f degree
n-1
.
I t is e a s i l y s e e n (see e . g .
Erdds and RBnyi C61
C71) t h a t t h i s happens i f and o n l y i f
2 n
log n +
-
and
a
log n -+
m
or
.
I t i s s u r p r i s i n g t h a t t h i s s i m p l e n e c e s s a r y c o n d i t i o n on M i s s u f T h i s i m p o r t a n t r e s u l t i s due t o Wright ClOI; f i c i e n t t o imply (1).
E a r l i e r ~ 6 l y a(see C81) and O b e r s c h e l p C91 had proved (1) under cons i d e r a b l y stronger c o n d i t i o n s on
M.
Our aim i s t o g i v e a com-
b i n a t o r i a l proof of t h i s theorem. Consider t h e symmetric group Sn a c t i n g on V = I 1 1 2 , . . . , n ) . w E Sn l e t G’(w) be t h e set of g r a p h s i n GM i n v a r i a n t under and p u t
1
W E Sn
where
I ( w ) = IG(w) ~ ( w )=
a(G)
1
G E GM
I
.
a(G)
.
Since
w
Then
,
i s t h e o r d e r of t h e automorphism group o f
a ( G ) = lAut G I
For
GM
has exactly
n!/a(G)
G
E
G,
:
g r a p h s isomorphic
Distinguishing vertices of random graphs
to a given graph =
I:
G
E
35
GM ,
(n!/a(G))-'
"
=
GeGM For the identity permutation holds.iff
- Z n!
a(G) =
1
.
,I
~(w) n I(1) = LM , so (1)
W€S
G€GM 1 E Sn we have
The number I(w) depends only on the size of the orbits of w acting on V ( 2 ) , the set of N pairs of vertices, because a graph GM belongs to G ( w ) if and only if the edge set of GM is the union of some entire orbits of w acting on V ( 2 ) Let A be a fixed set with a elements. We shall consider set S Denote by systems A = {A1,A2,..,As} partitioning A : A = u Ai i=l the collection of those b-element subsets F(b;A) = F(b;A l,...,As) of A which are the unions of some of the sets Ai Thus B E F(b;A) iff IBI = b and for every i we have Ai :, B or B n Ai = @ Suppose there are ml sets Ai of size 1, m2 sets of size 2,..., mt sets of size t and no set with more than t elements. Note that these parameters satisfy t a = C jmj (3) j=1 The definition of F(b;A) implies that if f(b;A) = IF(b;A) I then
.
.
.
.
.
I"' ] .k
f(b;A) = f(b;ml,m 2,...,m ) =
(1.)
3
t where the summation is over all (ij)2 t t k = k((i.1 = C ji 5 b , b-k 5 ml J 2 j=2 j
3=2 [TI) 3
satisfying 0
.
(4)
, 5
ij
5
m j and
If A' is obtained from A by decomposing an Ai into some sets (that is if the partition A' is a fiedinernen-t of A) then clearly F(b;A) c F(b;A') , SO f(b;A) 5 f(b;A') (5) In particular, if m1+2m2 = a and 0 5 k 5 m2 then
.
f (b;ml,m2)
5
f (b;ml + 2k, m2-k)
.
Furthermore, if A is the union of two disjoint systems, say and A" , then b f (b;A) = C f (b-i;A')f (i;A") i=O
.
(6)
A'
36
E. Bollobh
Consequently if m
...,m
j
=
m! i b
+
m'! i
with m! 3
t 0
and
3
2
0 then
..
,...
1
ml
.
t 7) f (b-i;mi ,m' t f (i;my,. ,m") i=o we need is not The last property of the function f(b;ml,...,mt) quite immediate so we state it as a lemma. Lemma 1. Suppose ml s a-2 If a-ml is odd, f(b;ml,
) =
.
..
f (b;ml,. ,mt) s f (b;ml+l,#(a-ml-l)) and if a-ml is even then f (b;ml,m2,. ,mt) s f (b;m1+2,#(a-ml-2))
..
Proof. Since a partitioned into contains no sets Furthermore, if by (6) f (b;ml,m2
set Ai sets of of size m3 = 0
,...,mt)
.
with more than 3 elements can be size 2 and 3, by ( 5 ) we may suppose that A greater than 3 , that is m4 = 5 =.. mt = 0. we are home since then a-ml = 2m2 and
.=
= f (b;ml,m2) s f (b;ml+2,m2-l)
.
.
Now suppose that m3 > 0 We distinguish two cases according to the parity of m3 , which is also the parity of a-ml First suppose that m3 is odd, say m3 = 2k+l 2? 1 We know from (7) that b f (b;ml,m2,2k+l) = 1 f (b-i;ml,m2)f(i;0,0,2k+l) i=o and b f (b;ml+l,m2+3k+l) = 1 f (b-i;ml,m2)f (i;1,3k+l) i=o Hence it suffices to show that
.
.
.
.
f (i;O,O,Zk+l) s f (i;1,3k+l) (8) In order to show ( 8 ) we use (3) to write out the two sides explicitly. If the left-hand side of ( 8 ) is not zero then i is a multiple of 3. Furthermore, if i = 6j then ( 8 ) becomes (2k+l 3k+l) 2j ( 3j and if i = 6j+3 then ( 8 ) is 2k+l 3k+l (2j+l) (3j+l) Both inequalities are obvious. seandly, if m3 is even, say, m3 = 2k+2 then f (b;ml,m2*,2k+2)s f (b;ml+l,m2+1,2k+l 5 f (b;m1+2,m2+3k+2) ,
-
completing the proof. 0 Let us return to our central theme, relation (2). We shall prove it by estimating I ( w ) in terms of the number of vertices fixed by w Denote by the set of permutations moving m vertices,
.
Sp'
Distinguishing vertices of random graphs
37
.
that is fixing n-m vertices. Then S Furthermore, r ' = a} and SA1)= pI if w E t h e n t h e m v e r t i c e s moved by w can be s e l e c t e d i n n (m) ways and t h e r e a r e a t most m! p e r m u t a t i o n s moving t h e same set of m v e r t i c e s . Hence
Sp)
Lemma 2. Let 2 i m integer satisfying
("im) +9
f o r which Then f o r
N1
E'
n
, and
n
<
m+4 ("im) +2
2
3
d e n o t e by
N1
= Nl(m)
the
is a l s o an i n t e g e r .
N2 = (N-N1)/2
w
I(W)
5
,
i
w e have
:S
s f(M;NitN2)
a
Proof. Suppose w E Sn(m) a s a permutation a c t i n g on v ( 2 ) h a s o r b i t s of s i z e i, i = 1 , 2 , n S i n c e a graph GM b e l o n g s ' t o G ( w ) i f f i t s edge s e t is t h e union of some e n t i r e o r b i t s ,
..., .
Mi
.
I ( w ) = I G ( w ) I = f(M:M1,M2,...,Mn)
(N-q)/2
Hence i f M! is chosen t o be M l + l o r M1+2 so t h a t % = 3 is an i n t e g e r , t h e n by Lemma 1 w e have I ( w ) 5 f(M:Mi,Mi) I f a permutation f i x e s t w o p a i r s of A t most how l a r g e is M1 ? v e r t i c e s , s a y {x,y} and {y,z} , t h e n it a l s o f i x e s y , t h e common v e r t e x , and so x and z a s w e l l . Consequently a p a i r f i x e d by w E e i t h e r c o n s i s t s of two v e r t i c e s f i x e d by w o r it is d i s j o i n t from a l l o t h e r p a i r s f i x e d by w Hence
.
Sp'
and so Mi
5
N1
.
.
T h e r e f o r e by i n e q u a l i t y ( 6 ) we have I ( w ) 1; f(M:Mi,Mi) 5 f(M:N1,N2)
.
0
I t so happens t h a t (2) c a n be proved f a i r l y e a s i l y under a somewhat Here we o n l y s t a t e t h i s s t r o n g e r assumption on M t h a n n e c e s s a r y . r e s u l t : f o r a proof w e r e f e r t h e r e a d e r t o Wright C101. As a consequence of t h i s p r e l i m i n a r y v e r s i o n of t h e main r e s u l t i n t h e proof o f t h e main theorem w e may assume t h a t M = 0 ( n l o g n ) Theorem 3. If In log n < M i N In log n then uM LM/n! 0
.
-
-
.
B. Bollobds
38
Now we are r e a d y t o p r o v e t h e fundamental theorem o f Wright C101. Theorem 4 .
Suppose
+
#n(log n Then UM
$(n))
M
5
and
+ m
-
N
5
+
#n(log n
.
Jl(n))
.
LM/n!
N
$(n)
The map s e n d i n g a g r a p h i n t o i t s complement p r e s e r v e s t h e
Proof.
automorphism group and sets up a 1-1 c o r r e s p o n d e n c e between
GM and
.
Hence i n p r o v i n g t h i s r e s u l t w e may assume t h a t M s N / 2 GN-M Furthermore, because o f Theorem 3 w e may and s h a l l assume t h a t +n l o g n + $ ( n ) n s M 5 2n l o g n
.
.
W e s h a l l a l s o assume t h a t
$(n)
n
no
2
where
e a s y t o check t h a t t h e r e is s u c h an Let
no
depends on t h e f u n c t i o n
and i s chosen so t h a t a l l o u r i n e q u a l i t i e s h o l d .
N1
= Nl(m)
L(m) =
and
1 L(m,i)
N2 = N 2 ( m )
. be a s i n Lemma 2 and s e t
= #(N-N1)
I
where t h e summation is o v e r a l l
.
no
I t w i l l be
i
satisfying
0
5
i
5
and
N2
5 21 5 M I f w E S,I"' t h e n by Lemma 2 w e have I ( w ) s L(m) Hence by ( 2 ) and ( 9 ) t h e theorem f o l l o w s i f w e show t h a t
M-N1
.
n For t h e s a k e of convenience l e t u s r e c a l l t h e i n e q u a l i t y satisfies: m(n
-
-
m
2)
-
2 s N-N~
5
m(n
-$-
2)
-
#
N1
.
(11)
Using s t a n d a r d e s t i m a t e s o f b i n o m i a l c o e f f i c i e n t s w e f i n d t h a t I,
Set
!L(m,i)
=
L(m,i) L (m ,i-1)
-
(N1/NlM
N-N1-2 i+2
2i
E(m,j)
i s 7 !L(m,i)
3
i
.
m m+5 exp I-2M -n( 1 - -)2n
}
.
(12)
(M- 2 i +2 ) (M-2 i+l ) (N1-M-2i-1) (N1-M+2i)
m t h e r a t i o !L(m,i) I n f a c t , i f 1 5 i 5 j then
and n o t e t h a t f o r a f i x e d v a l u e of c r e a s i n g function of
5
i s a de-
39
Distinguishing vertices of random graphs
I n o r d e r t o prove (10) w e s h a l l decompose t h e r a n g e of
m
into
t h r e e i n t e r v a l s and show t h a t t h e sum o v e r e a c h of t h e s e i n t e r v a l s
is
o(1) (a)
.
2 i m
Suppose t h a t =
(1(m,l)
5
(log n )
Then
N-N1
mn (2n l o g n; ( N-INI-M)
2(m,l)
.
n
M(M-1) 2 (N1-M+l) (N1-M+2)
10
.
2 ~
-1 i a(m,l) ,
5
t h i s shows t h a t
L ( m ) 5 c L(m,O)
L(m)/LM
1 (n)
m
T h e r e f o r e by ( 1 2 )
m ;(1 - z)}
m
5
c exp{-2M
5
c exp{-m(log n
+
+ ( n ) )(1
5
c exp{-m log n
-
~ ( n ) m 2%
i
expI-m log n
-
-
$1
-
++(n)ml
I l o g n}
.
i n our range we f i n d t h a t
m
Summing o v e r t h e
m
.
c
f o r some a b s o l u t e c o n s t a n t
L(m)/LM s
1 exp{-#+(n)mI
m
= o(1)
.
(b) Now l e t u s t u r n t o t h e l a r g e s t i n t e r v a l w e s h a l l c o n s i d e r : 10
n
5
(log n )
m
i
n
-
n'
( l o g n)
2
.
I t i s e a s i l y checked t h a t i n t h i s r a n g e
Put j = L22(m,l)J
.
Then
so, e s t i m a t i n g r a t h e r c r u d e l y ,
L(m,l)
2
4
.
B. Bollobds
40
we have logIL(m,O)/LM)
5
2M 109 n n-m + M n-f
.
Furthermore I 2 N1 - M > T(n-m) 2 so
Hence by (14) we have 1 ("Im L(m)/LM = o(1) for the sum over the range at hand, if, say, log n
(m+2)log n + 14 (logn)
+ 2M
log n-m
+
Mn"
< 0
.
(15)
The derivative of the left-hand side with respect to m is log n - 2M/(n-m) < 0 so it suffices to check (15) for the minimal value of m , namely m = rlon/(logn)21 Since n-m 5 - m 2M 2 n(1og n + $(n)) and log n n , the left-hand side of (15) is at most
.
2 log n + 14n/(log n)' as required.
-
+ 2x1'
$(n)m
l o g n < 15n/(l0gn)~- m < O ,
(c) Finally, suppose that m >- n nt (log n)2
-
.
In this range the crudest estimates will do. Since N2 5 N/2 M-21 L(M,i)/LM = 5 -@& M' = 1. (M-Zi)! (N-M)M h(m,i)
k2)[5i)/(i) -
If
2(i+1)
5
M
, we
5
.
have
as N1 > n/2 holds for every m value of i , LM/2J , we have h(m,LM/ZJ)
,
n4 #/2/(N-M)M/2
.
Furthermore, at the maximal s
n-3n
Consequently in our range we have 2h(mILM/2J) s n-n I(n), L(m)/LM 5 This completes the proof of our theorem.
.
41
Distinguishing vertices of random graphs
52.
DISTANCE SEQUENCES OF GRAPHS I N
G(n,M)
R e c a l l t h a t t h e d i b t a n c e beqUenCe of a v e r t e x x i s ( d i ( x ) ) , Our aim i s t o show t h a t i n a where d i ( x ) = {z E V : d ( z , x ) = i ) c e r t a i n range of M almost e v e r y random graph GM is such t h a t i t s vertices are u n i q u e l y determined by t h e i r d i s t a n c e sequences. The d i s t a n c e sequence of an i s o l a t e d v e r t e x i s O,O,..., so a graph w i t h two i s o l a t e d v e r t i c e s does n o t have t h e p r o p e r t y above. The n e x t r e s u l t shows t h a t i f M i s n o t t o o l a r g e b u t it i s l a r g e enough t o e n s u r e t h a t almost no GM c o n t a i n s two i s o l a t e d v e r t i c e s t h e n a l m o s t e v e r y graph i s such t h a t i t s v e r t i c e s a r e u n i q u e l y determined by t h e i n i t i a l segments of t h e i r d i s t a n c e sequences.
.
Theorem 5.
Suppose
$(n)
and 13/12 n
+
m
#n(log n + $ ( n ) ) s M 5 Then a . e . graph i n G(n,M) i s such t h a t f o r e v e r y p a i r of v e r t i c e s ( x , y ) t h e r e i s an L s !Lo = 3 r (logn/log(M/n)) 4 1 w i t h d,(x) ;t da(y) Pmf . -
.
For t h e s a k e of convenience w e s h a l l prove t h e analogous 2 It w i l l r e s u l t f o r t h e model G(n,P(edge) = p) , where p = 2M/n be c l e a r t h a t t h i s d o e s n o t change t h e e s s e n c e of t h e proof o n l y t h e c a l c u l a t i o n s become a l i t t l e more t r a n s p a r e n t . Furthermore, t h e d i f f i c u l t i e s depend on t h e s i z e of M : f o r M r a t h e r c l o s e t o t h e lower bound w e need a s l i g h t l y d i f f e r e n t argument. (a) Assume f i r s t t h a t pn/(logn)3 + and p s 2n -12/13 P i c k t w o v e r t i c e s i n V , s a y x and y I t s u f f i c e s t o show t h a t i n G(n,P(edge) = p ) t h e p r o b a b i l i t y o f d i ( x ) = d i ( y ) f o r a l l Set i 5 to = 3 r (logn/log(M/n)) 4 1 i s o ( n m 2 )
.
-. .
ra(Xly) =
(2
: d(z,x) = d(z,y) =
Aa(X)
=
I2
: d(Z,X) =
a , d(Z,y)
7
8
Ag (Y)
= {z : d(z,y) = a , d(z,x)
7
,
Aa(x)
= IAa(x) I
and
8
6 a ( ~ )= ~ A , ( Y )
I
.
Easy c a l c u l a t i o n s show t h a t i n o u r range (2pn)'o
5
.
n
By Lemmas 3 and 5 of [ 4 1 we f i n d t h a t f o r some f u n c t i o n with probability 1 - o(n? we have f o r a l l R s 0 :
a
16a(x)-(pn)
I
s c ( p n I R and
S i m i l a r r e l a t i o n s hold f o r
d,(x)
~ Q ( Y ) and
-
dg(Y)
5 *
E(pn)
a
E
.
= dn)
-+
0
(16)
42
B. Bollobds and suppose w e are g i v e n t h e e d g e s of a
0 s L! s to-1
Now l e t
random g r a p h
G
P
G(n, P ( e d g e ) = p )
E
spanned by t h e u n i o n o f t h e
s e t of v e r t i c e s a t d i s t a n c e a t most v e r t i c e s a t d i s t a n c e a t most
R+1
from
R
y
from
.
and t h e s e t o f
x
S the set s = IS1
Denote by
o f v e r t i c e s n o t b e l o n g i n g t o t h e u n i o n above and set s 2 n/(pn).l
[ I f (16) holds then c l e a r l y What i s t h e p r o b a b i l i t y t h a t
, conditional
d!L+l(x) = dR+l(y)
t h e s e t of edges g i v e n so f a r ?
As t h e e v e n t
in
AR(y)
.
on
dR+l(x) = dR+l(y) c o n d i t i o n a l on t h e e d g e s so f a r d e t e k m i n e h t h e number o f v e r t i c e s j o i n e d t o some v e r t e x i n
S
most max b ( k ; s, 1 - ( 1 - p ) k where
b(k; s l y )
6Q(Y)
,
t h i s probability is a t
1 ,
d e n o t e s t h e k t h t e r m of t h e b i n o m i a l d i s t r i b u t i o n :
s-k
k
b ( k ; s l y ) = ( E l Y (1-y)
Consequently t h e p r o b a b i l i t y of
dR+l(x) = d!L+l(y)
c o n d i t i o n a l on
( 1 6 ) i s a t most ( pn)
- ( R+l) /2 di(x) = di(y)
T h e r e f o r e t h e p r o b a b i l i t y of
for
i = 1,2,...,R0
I
i s a t most o(n-3)
+
II
( p n )-‘/2
since
II
R=1
= o(n-2)
!L=1
( p n ) ‘I2 t p n ( p n )
,
2 g0/4
T h i s completes t h e proof o f t h e a s s e r t i o n i n o u r r a n g e . (b) log n
Now assume t h a t + $ ( n ) s pn 5 ( l o g n ) 4
f o r some f u n c t i o n
Jl(n)
+.
a
.
a s s u m p t i o n s a . e . random g r a p h
I t i s e a s i l y s e e n t h a t under t h e s e
G
P
following properties: ( i ) Gp
E
G(n; P ( e d g e ) = p )
h a s no i s o l a t e d v e r t e x ,
( i i ) e v e r y v e r t e x h a s d e g r e e a t most
3pn
has the
,
(iii) f o r any two a d j a c e n t v e r t i c e s t h e r e a r e a t l e a s t
# l o g n/log log n
vertices a d j a c e n t t o a t l e a s t one o f them,
( i v ) f o r e v e r y p a t h o f l e n g t h two t h e r e a r e a t l e a s t v e r t i c e s a d j a c e n t t o a t l e a s t one of them, (v)
f o r every integer
2 s 20
spans a subgraph of s i z e a t most
L!
,
.
e v e r y set of
I?
# log n
vertices
43
Distinguishing vertices of random graphs
Denote by A the event that a random graph G has the properties P above. If A holds then we can proceed more or less as in part (a). Conditional on A , with probability at least 1 - n-3 we have C (d,(x) 1
+ dR(y)) = o(n)
and mint6R(x), 6R(~)l
.
for all R 5 R0 the probability of
2
8-3 (pn)
As in (a)I these imply that, conditional on di(x) = di(y) for all i 5 Lo is at most
Lo-3 59.:/21 II (pnliI2 = o(pn) i=l
-2
= o(n
1
-
P(A)
+
n2 o(n-*) = o(1)
,
1 .
Hence the probability that for borne pair of vertices for all i 5 9.0 is at most
di(x) = di(y)
A
(x,y) we have
.
Remarks 1. Theorem 5 is essentially best possible. One can show that a.e. random graph GM E G(n,M) contains two vertices x and y such that di(x) = di (y) whenever i 5 (log n/log (M/n) ) Furthermore,
’.
it is easily seen that if M = Ln11/8J I sayl then a.e. GM has diameter 3 and a.e. GM contains vertices x and y with di(x) = di(y) for all i [In fact di(x) = di(y) = 0 for
.
Thus Theorem 5 does not hold for large values of i 2 4.1 though the bound n13/12 could easily be improved.
M
,
2. The proof shows that the vertices can be distinguished by other For example if M is in parts of the distance sequence as well. our range, R 1 and R2 are natural numbers, (4M/n) and
111L2 > n5 (M/n)
then almost no random graph GM contains two vertices with di(x) = di(y) for all i : L1 5 i 5 R 2 13.
x
and
y
DISTANCE SEQUENCES OF RANDOM REGULAR GRAPHS
The study of random regular graphs was started only recently. The main reason for this is that the asymptotic number of labelled r-regular graphs of order n was found only in 1978 by Bender and
B. Bollobhs
44
.
Canfield C11 Even more recently in C31 a simpler proof was given for the same formula; furthermore in C31 a model was given for the set of regular graphs with a fixed vertex set, which can be used to investigate random labelled regular graphs. In particular, Bollobds and de la Vega C51 studied the diameter of random regular graphs. Our approach here is fairly close to that of C31. We shall give a brief description of the model mentioned above, in a form suitable for the study of distance sequences. Let r 2 3 be fixed and denote by G(n,r-reg) the probability space of all r-regular graphs with a fixed set of n labelled vertices. We shall assume that rn is even and any two graphs have the same As customary, we shall say that a t m o b t e u e t y (a.e.) probability. r-regular graph has a certain property if the probability of having the property tends to 1 as n -+ m Let W1,W 2,...,Wn be disn A c o n d i g u h a t i o n i b a p a f i t i t i o n 06 joint r-element sets. W = u Wi i n t o unotdehed paihLd, called e d g e d . If (a,b) is an edge of 1 a configuration F then we say that a is j o i n e d to b Denote by R the set of configurations and turn it into a probability space by giving all configurations the same probability. Given a configuration F E R , define $(F) to be the multigraph having vertex V = {W1,W2,...,Wn) in which Wi is joined to Wj if some Clearly element of Wi is joined to some element of W in F j every r-regular graph with vertex set V is of the form $(F) for some F E R and I$-l(G) I = (r!In for every r-regular graph with vertex set V In fact, as shown in C31, for about
.
.
.
.
e-(r2-1)’4 of all configurations F is $(F) an r-regular graph. an immediate consequence of this we see that a.e. r-regular graph has a certain property if and only if a.e. configuration has the corresponding property. The property we are interested in is that of having two vertices with the same distance sequence. Let F be a configuration. In order to define the distance sequences of F we define the d i b t a n c e d(Wi,Wj) = d (W ,W.) between two classes WilWj as the F i l minimal k for which there are classes W = Wi , Wi1,...,W\ = Wj io such that for every II 8 0 s II < k , an edge of F joins an element Equivalently, dF(Wi,W.) is to an element of Wi Of wi, II +1 I As
.
.
the distance between Wi and Wj in the graph $(F) For Wi E V set d,(Wi) = {Wi : d(Wi,Wi) = E l and call (d,(Wi))y=l the d i b t a n c e A f Z q U e n C e 0 6 Wi Thus Wi has the same distance sequence
.-
~
45
Distinguishing vertices of random graphs
i n a configuration
F
Theorem 6 .
2
to = L order
r
Let
(#+E),&J
lo
n
as i n t h e graph 3
.
and
E > 0
.
be f i x e d .
Set
Then a.e. r - r e g u l a r l a b e l l e d graph of
i s such t h a t e v e r y v e r t e x
n
$(F)
x
i s u n i q u e l y determined by
Proof. Analogously t o t h e proof of Theorem 5, o u r aim i s t o show t h a t t h e p r o b a b i l i t y of two f i x e d c l a s s e s Wi and W j of a random configuration satisfying 1 i II 5 to , da(wi) = dg(wj) t
.
i s o(n-2) Let 1 5 i j 5 n be f i x e d i n t e g e r s . We shall s e l e c t t h e edges of a random c o n f i g u r a t i o n one by o n e , t a k i n g t h o s e n e a r e s t t o Wi and W j f i r s t . T h i s approach i s c l o s e l y modelled on t h a t i n Bollobds and d e l a Vega C51. Suppose w e have s e l e c t e d a l l edges a t l e a s t one e n d v e r t e x of which belongs belongs t o a c l a s s a t d i s t a n c e less t h a n k from t h e s e t {Wi,Wj) Take a l l c l a s s e s a t d i s t a n c e k from { W i , W j ) and one by one select p a i r s f o r t h o s e e l e m e n t s i n t h e s e c l a s s e s t h a t have Having done t h i s c o n s e c u t i v e l y f o r n o t been p a i r e d so f a r . k = Oflf...,n-2 and n-1 , w e have c o n s t r u c t e d t h e union of t h e and W i n a random c o n f i g u r a t i o n . I n f a c t it components o f Wi j i s r a t h e r t r i v i a l t h a t a . e . c o n f i g u r a t i o n i s connected so a f t e r t h i s sequence o f o p e r a t i o n s w e a l m o s t s u r e l y end up w i t h t h e e n t i r e r a n dom c o n f i g u r a t i o n . As i n C51, w e c a l l a n edge i n d i b p e n d a b l e i f it i s t h e f i r s t edge t h a t e n s u r e s t h a t a n o t h e r c l a s s WE i s a t a c e r t a i n d i s t a n c e from E q u i v a l e n t l y , an edge i s d i s p e n s a b l e i f each c l a s s coniWi,Wj) t a i n i n g a n endvertex of t h e edge i s e i t h e r one of Wi and W or j else it c o n t a i n s a v e r t e x i n c i d e n t w i t h a n edge w e have a l r e a d y chosen. Note t h a t a n edge j o i n i n g two v e r t i c e s o f t h e same c l a s s is dispensable. What i s t h e p r o b a b i l i t y t h a t t h e k t h edge w e select is dispensable? A s t h e k-1 edges s e l e c t e d so f a r a r e i n c i d e n t with v e r t i c e s i n a t most k + l c l a s s e s , t h i s p r o b a b i l i t y i s a t most ( k + l ) (r-1) (n-1) r Therefore t h e p r o b a b i l i t y t h a t more t h a n 2 of t h e f i r s t ko = Ln1’6J edges are dispensable is a t most
.
.
-2)
I
B. Bollobds
46
t h e p r o b a b i l i t y t h a t more t h a n
!Z1
= Ln1’81
of t h e f i r s t
e d g e s a r e d i s p e n s a b l e i s a t most
kl = Ln6/13J
L1+l = o(n-2)
(18)
k 2 = Ln2I3J
of t h e f i r s t
!Z2 = Ln5/13J
and t h e p r o b a b i l i t y t h a t more t h a n
edges are d i s p e n s a b l e i s a t most i2+1 = o(n-2)
.
Let
A
be t h e event t h a t a t m o s t
2
of t h e f i r s t
ko
edges a r e
and a t most k 2 of t h e d i s p e n s a b l e , a t most 111 o f t h e f i r s t kl f i r s t k2 By ( 1 7 ) - ( 1 9 ) w e f i n d t h a t t h e p r o b a b i l i t y o f A i s T h e r e f o r e i t s u f f i c e s t o show t h a t t h e p r o b a b i l i t y o f 1 o(n-2) f o r 1 s II s IIo c o n d i t i o n d o n A i s o ( n - 2 ) d,(Wi) = d,(Wj)
-
. .
.
In order t o estimate t h i s conditional probability w e describe a n o t h e r , s l i g h t l y d i f f e r e n t way o f s e l e c t i n g t h e e d g e s o f a configuration. As b e f o r e , suppose w e have s e l e c t e d a l l e d g e s i n c i d e n t w i t h v e r t i c e s i n classes a t d i s t a n c e less t h a n k from {Wi,Wj) P a r t i t i o n t h e elements t h a t a r e unpaired a t t h i s s t a g e i n t o edges and p u t a n edge i n t o o u r random c o n f i g u r a t i o n under c o n s t r u c t i o n i f
.
it s a t i s f i e s one o f t h e f o l l o w i n g t h r e e c o n d i t i o n s .
( i ) One o f
t h e v e r t i c e s i n c i d e n t w i t h t h e edge b e l o n g s t o a c l a s s a t d i s t a n c e k from Wi , ( i i ) b o t h v e r t i c e s i n c i d e n t w i t h t h e edge b e l o n g t o c l a s s e s a t d i s t a n c e k from W ( i i i ) if w e add t h e e d g e s 1 ’ s a t i s f y i n g ( i )t h e n one e n d v e r t e x o f o u r edge b e l o n g s t o a c l a s s a t d i s t a n c e k from W and t h e o t h e r b e l o n g s t o a c l a s s a t j W e c a l l t h i s the (2k)th opehation. d i s t a n c e k + l from Wi
.
Suppose t h a t a f t e r t h e c o m p l e t i o n o f t h e ( 2 k ) t h o p e r a t i o n t h e r e t h a t have a r e tk v e r t i c e s i n t h e c l a s s e s a t d i s t a n c e k from W j n o t been p a i r e d so f a r and t h e r e a r e sk classes c o n s i s t i n g ent i r e l y of u n p a i r e d e l e m e n t s . The ( 2 k + l ) h t o p e h a t i o n c o n s i s t s o f p a i r i n g t h e tk v e r t i c e s above w i t h tk e l e m e n t s o f t h e sk c l a s s e s and adding a l l t h e s e p a i r s t o o u r random c o n f i g u r a t i o n . Having completed t h e
( 2 k + l ) s t o p e r a t i o n , we have d e t e r m i n e d a l l
edges i n c i d e n t w i t h v e r t i c e s i n classes a t d i s t a n c e a t most k from so w e can proceed t o t h e (2k+2)nd o p e r a t i o n . Once a g a i n , iWi,Wj) i f w e perform c o n s e c u t i v e l y t h e 0 t h , l s t , Znd, (2n)th
...,
o p e r a t i o n s , t h e n w e c o n s t r u c t a l l e d g e s i n t h e components c o n t a i n i n g and Wj
Wi
.
47
Distinguishing vertices of random graphs W e may assume t h a t t h e
satisfies
0 <
E
Now l e t u s assume t h a t k
5
a p p e a r i n g i n t h e s t a t e m e n t o f t h e theorem
E
.
< l/8
Note t h a t t h e n
I t i s e a s i l y s e e n t h a t t h e n for
holds.
A
w e have
Lo
tk 2 (r-1)k-3
sk
and
.
n/2
2
(20)
( I n f a c t b o t h bounds are r a t h e r c r u d e . )
Note t h a t
is
dk+l(Wi)
determined b e f o r e w e b e g i n t h e ( 2 k + l ) s t o p e r a t i o n so i f dk+l(Wi) - dk+l(Wj) h a s t o h o l d t h e n t h e p a i r s o f t h e tk v e r t i c e s must belong t o a g i v e n number o f t h e
sk
classes.
Hence t h e p r o b a b i l i t y
t k and sk s a t i s f y i n g 3 ( 2 0 ) i s a t most t h e maximum of t h e p r o b a b i l i t y t h a t a tk element
that
dk+l(Wi)
= dk+l(W.)
sk
s u b s e t of t h e union o f exactly 11
R
tk and
c o n d i t i o n a l on
r
c l a s s e s of
e l e m e n t s e a c h meets
c l a s s e s , t h e maximum b e i n g t a k e n o v e r a l l v a l u e s of
rk
s a t i s f y i n g (20).
The f o l l o w i n g lemma g i v e s an
upper bound f o r t h i s maximum. S
L e t R = u Ri be a p a r t i t i o n i n g of an r s - e l e m e n t s e t Lemma 7. i n t o r-element s e t s , 1where r 2 3 i s f i x e d . Suppose t 5 c s 5 /a
.
c
f o r some c o n s t a n t
Denote by
s e c t e d by a random t - s u b s e t
T
of
XT R
.
t h e number o f
Rj's
inter-
Then
max P ( X T = R ) s co s 1'2/t 11
co
f o r some c o n s t a n t Proof.
.
W e may assume w i t h o u t loss o f g e n e r a l i t y t h a t
The p r o b a b i l i t y t h a t
T
has a t l e a s t
3
e l e m e n t s i n some
a t most
Since t h e lemma f o l l o w s i f we show t h a t PQ = P(XT = R
PQ = 0
.
c1
for some c o n s t a n t Clearly
and
if
R
maxlTnRil ,i t/2
5 2)
t
5
c1 s # /t
and o t h e r w i s e
2
s#
Ri
. is
B. Bollobas
48
For
1 s u
(t-1)/2
5
set
(t-2u)(t-2u-1) r-l (s-t+u+l)(u+l) 2r It is easily seen that if u is an integer then we have
0,
.
=
-
- Qu
P+u-l/Pt-u Define
,
uo
1
u0
c
t/2
, by
= l .
-
QUO
Then
and Pt-u increases with u for u 5 u0-1 and decreases with for u 2 uo-1 Furthermore, easy calculations show that for u o - u#o s u < u - l -
.
u
0
0, s 1 + c2uo -# and for Qu where
uo 2
c2
1
-
1
5
u s uo
+ u#
c2u-# 0
is some constant.
min{Pt-U
:
uo
- :u
Hence
s u s u0
f o r some positive constant we have
c3
so (21) implies that max PR s -. 1 s cu s # /t R c3u0 for some positive constant cu
.
+ u0# I
2
c3 max pR 11
(21)
Finally, by the definition of Pi
.
0
The proof of Theorem 6 is easily completed with the aid of Lemma 7 . The probability that d,(Wi) = d,(Wj) for R s Ro is at most 1 - P(A) + IIRO {co n# /(r-l) R-3 I , h where
h = l#’&jJ (r-1)&o
2
+
3
.
Since
n (1+€)/2
the sum above is ~ ( n - ~, )completing the proof of our theorem.
0
49
Distinguishing vertices of random graphs
Remark. A slight variant of the proof above gives the following extension of the theorem: Let r 2 3 and # < 6 < 1 be fixed and set ho = L 6 logn log(r-1) 1 6 hl = h0 ir-126-1 Then a.e. r-regular labelled graph of + 3 order n is such that every vertex x is uniquely determined Idi(x) : ho < i 5 hl)
’
.
.
REFERENCE E.A. Bender and E.R. Canfield, The asymptotic number of labelled graphs with given degree sequences, J. Combinatorial Theory (A) 24 (1978) 296-307. ’ c 2 1 B. Bollobds, Graph Theory - An Introductory Course, GTM vo1.63, Springer-Verlag, New York - Heidelberg - Berlin, 1979. B. Bollobds, A probabilistic proof of an asymptotic formula for C 31 the number of labelled regular graphs, Europ. J. Combinatorics 1 (1980) 311-316. B. Bollobds, The diameter of random graphs, Trans. Amer. Math. C41 SOC. to appear. C51 B. Bollobds and W.F. de la Vega, The diameter of random regular graphs, to appear. C61 P. Erdds and A. Rbnyi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960) 17-61. C71 P. Erdds and A. Rgnyi, On the strength of connectedness of a random graph, Acta Math. Acad. Sci. Hungar. 12 (1961) 261-267. C81 G.W. Ford and G.E. Uhlenbeck, Combinatorial problems in the theory of graphs, Proc. Nat. Acad. Sci. U.S.A. 43 (1957) 163-167. [91 W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann. 174 (1967) 53-78. c 101 E.M. Wright, Graphs on unlabelled vertices with a given number of edges, Acta Math. 126 (1971) 1-9. c11
-
This Page Intentionally Left Blank
Annals of Discrete Mathematics 13 (1982) 5 1-60 0 North-Holland Publishing Company
THE TRAIL NUMBER OF A GRAPH B6LA BOLLOBAS and FRANK HARARY University of Cambridge, England and University of Michigan, Ann Arbor, U.S.A. The t r a i l number of t r ( G ) of a graph G is t h e maximum l e n g t h of a t r a i l i n i t . Consider t h e s e t of a l l connected g r a p h s w i t h n v e r t i c e s and m edges. W e f i r s t e a s i l y d e t e r m i n e t h e e x a c t maximum v a l u e of t r ( G ) among t h e s e g r a p h s . Then w e g i v e t h e e x a c t minimum provided m 5 4 ( n - 1 ) / 3 . F i n a l l y w e prove an a s y m p t o t i c a l l y c o r r e c t e s t i m a t e of t h e minimum of t r ( G ) f o r a l l v a l u e s of n and m. 1.
INTRODUCTION
W e f o l l o w a m i x t u r e of t h e terminology and n o t a t i o n of C11 and C31. However, l e t G(n,m) be a c o n n e c t e d g h a p h w i t h n 2 3 u e h t i c e d a n d m e d g e d . A t h a i L o f G i s a walk i n which no edge o c c u r s more t h a n once. A c i h c u i t i s a c l o s e d walk. An e u t e h i a n t h a i l i s a c l o s e d t r a i l c o n t a i n i n g a l l t h e edges; a c o v e t i n g t h a i L i s such an open t r a i l . The thaiL numbeh t r ( G ) i s t h e maximum l e n g t h of a t r a i l i n G . Of c o u r s e t r ( G ) = m i f and o n l y i f G i s e u l e r i a n o r G h a s a c o v e r i n g t r a i l . Buckley C21 i n d e p e n d e n t l y proposed t h e d e f i n i t i o n of t r ( G ) and noted t h a t t h i s i s n o t a ramsey f u n c t i o n , a f a c t which does n o t concern us h e r e . Our o b j e c t i s t o i n v e s t i g a t e t h e maximum and minimum v a l u e s of t r ( G ) f o r G(n,m). W e f i r s t n o t e t h a t t h e maximum can be determined w i t h o u t any e f f o r t . 2.
THE MAXIMUM TRAIL NUMBER O F A GRAPH
T h i s q u e s t i o n i s s u f f i c i e n t l y t r i v i a l t h a t one can d e t e r m i n e t h e answer q u i c k l y and e x a c t l y . W e w i l l see t h a t t h i s is n o t so f o r t h e minimum t r a i l number. Theorem 1. The maximum t r a i l number of a g r a p h G(n,m) n i s odd o r m 2 + 1; o t h e r w i s e it i s );( -
(i)-
is
%+
m
if
1.
Proof. I f n is odd t h e n Kn i s e u l e r i a n . The f i r s t m edges of an e u l e r i a n t r a i l of Kn form a graph G(n,m) w i t h t r a i l number m. 51
B. Bollobcis and E Harary
52
If
m
5
-
n i s even, t h e n Kn (2 n n (2) 7 + 1 , the f i r s t m
n
-
form a g r a p h
G(n,m)
then
H H
G
Hence f o r
of order
m. On t h e o t h e r hand, t h e i s a t most - + 1.
n
is t h e subgraph formed by t h e edges o f a t r a i l i n h a s a t most two v e r t i c e s of d e g r e e n-1, so
e(H) 3.
is eulerian.
1)K2
edges o f an e u l e r i a n t r a i l a g a i n
w i t h t r a i l number
t r a i l number of a g r a p h For i f
-
5
71{ 2 ( n - l )
+
(n-2) 2 ) =
(y)
5
-
+
G
.
1
,
0
M I N I M U M TRAIL NUMBER FOR SPARSE GRAPHS.
Before c o n s i d e r i n g t h e minimum t r a i l number f o r a r b i t r a r y G(n,m) g r a p h s , w e f i r s t d e t e r m i n e it e x a c t l y f o r t h e b p a h d e g h a p h d d e f i n e d h e r e a s s a t i s f y i n g m = n+k where k i s a f i x e d i n t e g e r and n i s large.
Let
Theorem 2 .
p (k,n)
be t h e minimum v a l u e o f tr (G)
f o r G(n, n+k)
.
The minimum t r a i l number f o r t h e s p a r s e g r a p h s a r e g i v e n
e x a c t l y by t h e f o l l o w i n g s i x e q u a t i o n s : =
=
=
=
=
= Proof.
(-1). p(-1,n)
(0).
1;
2
n
t
3,
n = 3, n = 4,5,6, n a 7,
1: 1: 1’6
n = 4,5,6, n L 7,
n = 4 n a 5, n = 5,6, n a 7, k a 4
2
W e now j u s t i f y t h e a s s e r t i o n s (-1) A s a connected graph w i t h
i s a t t a i n e d when
G
m = n-1
is the s t a r
-
and
n
2
3k+4.
(4).
is a tree, t h e value of and is 2 .
Kl,n-l
m = n is u n i c y c l i c , so w e have n 2 3. When n = 3, the only unicyclic graph is the triangle K3 , so t r = 3. There a r e j u s t two u n i c y c l i c g r a p h s w i t h n = 4 , b o t h having t r = 4 . A connected g r a p h w i t h
Among t h e f i v e u n i c y c l i c g r a p h s
G
with
n = 5 , a l l b u t one have
t r ( G ) = 5 , t h e e x c e p t i o n b e i n g t h e graph o f F i g u r e l a which h a s
t r = 4 , so t h a t
p(0,5) = 4. There a r e e x a c t l y 10 u n i c y c l i c g r a p h s of o r d e r 6 , j u s t two o f which ( F i g u r e l b , c ) a t t a i n t h e s m a l l e s t
The trail number of a graph
53
possible trail number, 4. For n 2 7, one sees at once that p(O,n) = 5 as illustrated in Figure Id.
1
A JdA Figure 1.
The extremal graphs for k = 0.
(1).
If there are two edge-disjoint cycles in G, then there is nothing to prove as we must have tr(G) 2 6. If not, then by C1, p.1203, G > T(K4-e), that is G has a subcontraction to a topological copy (subdivision) of K4-e shown in Figure 2a. Since K4-e has a covering trail, G contains K4-e as an induced subgraph, for otherwise tr(G) 2 6.
(a) Figure 2.
(b)
(C)
(d)
Extremal graphs for k = 1.
For n = 4, tr(K4-e) = 5 as this graph has a covering trail. When n = 5 and 6, the graphs of Figure 2b,c are the unique extremal graphs which have trail number 5. For n 2 7 , the extremal graphs have the form of Figure 2d, with at least three endvertices; each such graph has tr= 6.
(2).
The complete graph K4 is the only graph with n = 4 and k = 2. Hence a 6 o h t t i o h i it is the unique extremal graph for those parameters and tr (K4) = 5.
B. Bollobiis and F. Harary
54
Now c o n s i d e r n 2 5 . F i r s t n o t e t h a t any l o n g e s t c i r c u i t o f G h a s T h e r e f o r e t h e r e a r e no two e d g e - d i s j o i n t c y c l e s . ' l e n g t h a t most 5 . Hence, by Theorem 3 . 1 of C 1 , p.1201, of t h e t h r e e m u l t i g r a p h s K4
( t h r e e edges j o i n i n g two v e r t i c e s ) ,
3K2
h a s t o o few e d g e s and
3K2
T(K4),
must b e homeomorphic t o one
K3,3*
Or
Now
G
K4.
a topological
a t most one edge of
K3,3
3
t h e hexagon, so w e have
C6,
But K 4 i t s e l f h a s tr = 5 so t h e r e i s which is s u b d i v i d e d and t h a t edge i s
T(K4) s u b d i v i d e d by a t m o s t one v e r t e x of d e g r e e 2 . L e t us r e c a l l t h a t t h e j o i n G2
i s t h e union of
similarly
G1
+ G
G1,
+ K2
+
+
K4
of two d i s j o i n t g r a p h s
G2
and t h e complete b i g r a p h u
G3 = (G1+G2)
Now i f j u s t one edge o f o b t a i n t h e graph
G2
GI
(G2+G3),
K(V1,
and
G1 V2);
and so f o r t h .
i s s u b d i v i d e d by e x a c t l y one v e r t e x , w e
z2 + Kl
n = 5
o f F i g u r e 3a which h a s
and
k = 2 . I f t h i s v e r t e x of d e g r e e 2 i s a t t a c h e d t o a new e n d v e r t e x , t h e n w e have K 2 + + K1 + K1 ( F i g u r e 3 b ) w i t h n = 6 and k = 2 , b o t h o f t h e s e e x t r e m a l g r a p h s having t r = 6 .
z2
Otherwise f o r a l l
n
2
5, w e have
K4
with a s t a r
t o e x a c t l y one of i t s v e r t i c e s , t h a t i s t h e g r a p h These are a l l t h e e x t r e m a l g r a p h s f o r
Figure 3c.
(a) F i g u r e 3.
(3).
K K3
l,r
+
attached
K1
+ Fr
of
k = 2.
(b)
The e x t r e m a l g r a p h s f o r
k = 2.
As u s u a l w e f i r s t c o n s i d e r s m a l l
n.
The smallest p o s s i b i l i t y
w i t h k = 3 is n = 5 f o r which t h e u n i q u e extremal g r a p h i s t h e wheel W5 = .K1 + C4 which 'has t r = 7 . When
n = 6,
t h e r e a r e j u s t t h r e e extremal graphs:
K1
+
K1
t h e g r a p h o b t a i n e d by adding a new e n d v e r t e x t o t h e c e n t e r of and t h e two c u b i c g r a p h s K and K3 X K 2 . 3,3
+
C4,
W5,
55
The trail number of a graph
For n 2 7 , j o i n n-5 independent v e r t i c e s t o a v e r t e x of C4 i n t h e wheel W5 = K1 + C 4 . The g r a p h o b t a i n e d has n v e r t i c e s and n + 3 edges and i t s t r a i l number i s 8. Thus ~ ( n , 3 )5 8. W e now prove t h a t
1.1 2 8 whenever n 2 I . any l o n g e s t c i r c u i t h a s l e n g t h a t most 6 .
Case 1.
Assuming t h a t
1-1
5
7,
There a r e no two v e r t e x - d i s j o i n t c y c l e s .
W e f i r s t n o t e t h a t t h e r e e x i s t i n t h i s c a s e two e d g e - d i s j o i n t c y c l e s . Otherwise, by C o r o l l a r y 3.3 of C1, p.1201, G contains T(K3,3). But a s a l o n g e s t c i r c u i t h a s l e n g t h 5 6 , t h e graph G must c o n t a i n
a s a subgraph. S i n c e p 313 adjacent t o ( a vertex o f ) K3,3.
K
t
t h e r e e x i s t s a new v e r t e x v Hence G h a s a t r a i l of l e n g t h 8 .
7,
Now w e have two e d g e - d i s j o i n t c y c l e s which a r e n o t v e r t e x - d i s j o i n t . Both t h e s e c y c l e s must be t r i a n g l e s because of t h e bound of 6 on t h e l o n g e s t c i r c u i t l e n g t h . Hence w e have a s a subgraph K3 = K2 + K1 + K2 and w e d e n o t e i t s v e r t e x s e t by W. K3 If P i s a p a t h of l e n g t h 2 having no edge i n K3 K3 but a t l e a s t one v e r t e x i n W t h e n K 3 * K 3 u P h a s two odd v e r t i c e s so i t s t r a i l number i s 8. Consequently i f we remove E ( K 3 * K 3 ) from G , a l l remaining edges a r e independent. T h e r e f o r e GCWI, t h e subgraph i n duced by W, i s t h e wheel W5 and t h e r e e x i s t a t l e a s t two more (independent) edges, each having e x a c t l y one v e r t e x i n W. A t l e a s t one of t h e s e edges i s i n c i d e n t w i t h a v e r t e x of C 4 so once a g a i n
o u r graph h a s t r a i l number a t l e a s t 8. Case 2 .
There a r e two v e r t e x - d i s j o i n t c y c l e s .
A s both t h e s e c y c l e s must be t r i a n g l e s , and as t h e y must be j o i n e d
by j u s t one edge ( f o r o t h e r w i s e G has an 8 - t r a i l , i . e . , a t r a i l of l e n g t h 8 ) , t h e subgraph implied by t h i s c a s e i s K2 + K 1 + K 1 + K 2 = F . L e t c be one of t h e two c u t v e r t i c e s of t h i s subgraph. I f there is a v-c edge t h e n w e have an 8 - t r a i l , so t h e r e i s none. F u r t h e r , t h e o t h e r v e r t i c e s , n o t i n W = V ( F ) must be independent f o r o t h e r w i s e t h e r e is a 2-step p a t h t o a non-cutvertex of F r e s u l t i n g i n an 8 - t r a i l . Also, no v e r t e x v { W c a n be a d j a c e n t t o two v e r t i c e s i n W a s t h i s would y i e l d e i t h e r an 8 - t r a i l or two d i s j o i n t c y c l e s one of which i s C 4 . T h e r e f o r e a l l added edges must form s t a r s a t t h e n o n - c u t v e r t i c e s of F. A s any l o n g e s t c i r c u i t h a s l e n g t h a t most 6 , t h e two a d d i t i o n a l edges t o be added t o t h e v e r t i c e s of F t o g e t k = 3 must g i v e t h e t r i a n g u l a r prism K 3 x K2 which has t r = 7 . F i n a l l y , t o have n = I , w e need one more v e r t e x j o i n e d
B. Bollobis and F. Harary
56
t o any v e r t e x of
t r = 8, com-
r e s u l t i n g i n a graph wi t h
K3x K2,
p l e t i n g Case 2 and t h e proof o f
(3).
(4).
k
We know from ( 3 ) t h a t i f
2 4
and
Thus a l l w e have t o show i s t h a t f o r
k
2
n 4
7
2
and
then n
2
p(k,n)
3k+4
2
8.
there is
L e t Go be a g r a p h of GI = ( k + l ) K 3 by a d d i n g a new v e r t e x xo t o G1 and j o i n i n g it by an edge t o e a c h t r i a n g l e i n G1. Then Go h a s 3 ( k + l ) + k + l = no + k e d g e s and a s xo i s a c u t v e r t e x , i t s t r a i l number i s 8. To o b t a i n G add n-no new
a graph
order
vertices t o 4.
w i t h t r a i l number 8 .
G = G ( n , n+k)
no = 3 ( k + l )
+
1 o b t a i n e d from
and j o i n e a c h t o
Go
il
xo.
M I N I M U M TRAIL NUMBER.
Our o b j e c t i s t o d e r i v e close u p p e r and lower bounds f o r t h e minimum
.
t r a i l number among t h e g r a p h s G(n,m) In t h e proof, we w i l l require a n e l e m e n t a r y lemma. I t i s c o n v e n i e n t t o w r i t e d ( G ) = d = 2m/n f o r t h e average of t h e degrees. Lemma 3 .
Every g r a p h
G(n,m)
h a s a component o f s i z e a t l e a s t
d ( d + l )/ 2 . Proof.
Let
G = G(n,m).
to
Gk
where
Define
ti
by
GI
Gi
= G(nifmi)
b e t h e components of
mi = t i ( t i + l ) / 2 , SO
ti
5
ni - 1
mi
5
n1 . t .1/ 2 .
and
Hence i f f o r e v e r y
i,
mi < n then f o r every
(%
+
1) = d ( d + 1 ) / 2 ,
if
0
a contradiction.
i s b e s t p o s s i b l e whenever b o t h
Note t h a t t h e bound
d(d+1)/2
d = 2m/n = t-1
n/(d+l) = k
and
are integers.
For then
kKt
has
57
The trail number of a graph
order n, edges.
t
and each component has exactly (2) = d(d+1)/2
size m,
Lemma 4. If G = G(n,m) satisfies 8d(d+l) 5 n(n-4) , then it has a subgraph of size at least d(d+l) which is the union of at most two components. Proof. Let the components of G be Gi = G(ni, mi), i = 1,. ..,k with ml 2 m2 z . . . ~ mk. Then by Lemma 3 , m1 Now if ml
2
d(d+l),
(11
d(d+1)/2.
5
there is nothing to prove,
Otherwise
and so we apply Lemma 3 to G - G1 to get m-ml 2(m-ml) f(nl, m,) = ml + n-nl + 1) 5 ml + m2.
( 7
f(nl, ml) is at most the size of G1 that it is at least d(d+l) , that is
As
g(n,m,nl,ml) = ml + 2 + -m-ml --
n-nl
u
G2,
(3)
it remains to show
m-ml ) 2 ( 7 "1
2m (-?i2m + 1)
5
0.
(4)
In proving ( 4 ) we shall not assume that the various parameters are integers, only that they satisfy the inequalities. Since f(nl, ml) is a decreasing function of nl, we may take ml = nl(nl - 1)/2. Then (1) gives n1
2
d
+
2m + 1, 1 = n
(5)
and by (21,
so
the assumption on
d
implies n > 4nl.
Now if m is as large as possible so that ( 5 ) is an equality, then ml = and ml + m2 2 d(d+l) as required. Hence it suffices to show that
B. Bollobh and F. Harary
58
i n t h e range allowed by (5). 2 n (n-n,)
Now
3g(n,m,nl,ml) = 4 (m-ml)n 2 +(n-nl)n2-8m(n-nl) 2-2n (n-n,)
am
5
4{ (rn-ml)n2-2m(n-nl) 2 I
5
4{-mn 2 +4mnn11 = 4mn(4nl-n)
so ( 6 ) h o l d s .
< 0,
Then so does (4) and t h e lemma i s proved.
0
I t i s convenient t o c a l l a v e r t e x o d d o r e v e n i n accordance w i t h t h e
p a r i t y of i t s degree. Then an e v e n ghaph h a s a l l v e r t i c e s even; i n p a r t i c u l a r a connected even graph i s e u l e r i a n . L e t e ( P ) be t h e l e n g t h of a p a t h P . W e have come t o t h e f i n a l lemma needed i n t h e proof of our main theorem. Lemma 5.
A graph
G
contains a forest
F
such t h a t
G-E(F)
is an
even graph. Proof. L e t E l c E ( G ) be a minimal s e t of edges whose d e l e t i o n l e a v e s an even graph. Then F = ( V ( G ) , E l ) is a f o r e s t . Indeed i f
is a cycle i n F t h e same i n G-E(F)
C
t h e minimality of Theorem 6 .
Let
E'
t h e n t h e p a r i t y of e v e r y v e r t e x v E V ( G ) is a s i n G-E (F-E(C) ) = G- ( E l - E ( C ) 1, contradicting
.
G = G(n,m)
be a connected graph s a t i s f y i n g
where Then
--Proof.
By Lemma 5 our graph a t l e a s t m ' = m-n+l edges. c o v e r i n g t r a i l so tr(G) 2 tr(G1)
2
G
If
c o n t a i n s an even subgraph G I with GI is connected t h e n it has a
m-n+l > d ' ( d l + l )
+
1.
Now suppose t h a t G ' i s disconnected. S i n c e d 1 = 2m1/n s a t i s f i e s (61, Lemma 4 i m p l i e s t h a t G ' has a subgraph of s i z e a t l e a s t d ' ( d ' + l ) which i s t h e union o f two components, s a y H1 and H 2 . As G i s connected it c o n t a i n s an H1-H2 p a t h P whose i n t e r n a l vert i c e s do n o t belong t o V(H1) u V ( H 2 ) . Then H1 u H2 u P i s
59
The trail number of a graph connected and h a s e x a c t l y two v e r t i c e s o f odd d e g r e e . tr(G)
Corollary 7. G = G(n,m)
tr(H1uH2uP) = e(H1uH2uP)
2
I f t h e average degree
d
2
d'(d'+l)
+
Consequently
1.
0
o f a connected g r a p h
satisfies n d s + 1, 2JT
then
tr ( G )
2
+
(d-1) (d-2)
1.
0
Remarks. (1) Theorem 6 is c l o s e t o b e i n g b e s t p o s s i b l e . F o r example i f n = k t + l , k 2 2 and m = k ( i ) + k t h e n t h e r e is a G = G(n,m) such t h a t t r ( G ) = t2
- t+2.
Such a g r a p h G i s o b t a i n e d from k Kt by adding one v e r t e x and k edges (see F i g u r e 4 ) . I n p a r t i c u l a r , i f 2k t t t h e n t h e
F i g u r e 4 . For t = 4 and k = 5 w e o b t a i n G = G ( 2 1 , 3 5 ) w i t h t r ( G ) = 1 2 .
average degree
is a t l e a s t
d
t r ( G ) s d(d+l)
Furthermore, i f
t
if
2k
2 t
+
2.
is even t h e n t r ( G ) = t2
SO
and so
t-1
-
2t
+
3.
then tr(G)
5
d2
+
4
60
B. Bollobh and F. Harary
Note t h a t t h e g r a p h d e f i n e d i n t h e proof o f ( 4 ) o f Theorem 2 i s e x a c t l y t h i s graph
G
for
t = 3.
( i i ) I t i s p o s s i b l e t o s t r e n g t h e n Theorem 6 i n v a r i o u s ways. For example one can p r o v e t h a t i f t 2 3 and i s odd, and k i s s u f f i -
c i e n t l y l a r g e , t h e n t h e minimum t r a i l number of any G ( k t + l , k(:)) i s e x a c t l y t 2 - t + 2. However o u r proof i s l e n g t h y and f a r from e l e g a n t , so it is o m i t t e d . REFERENCES 1. B. Bollobbs, Extremal Graph Theory. (1978).
Academic Press, London
2.
F. Buckley, A ramsey p r o p e r t y f o r g r a p h i n v a r i a n t s . Theory 4 (1980) 383-387.
3.
F. Harary, Graph Theory.
Addison-Wesley,
J Graph
Reading (1969).
Annals of Discrete Mathematics 13 (1982) 61-64
0 North-Holland Publishing Company
MESSAGE GRAPHS
J.H.CONWAY and M.J.T.GUY University of Cambridge England
In this paper, the term v-gtaph will mean finite directed graph with constant invalence v t 2 and the same constant outvalence. We shall write p q to mean that there is an edge directed from the vertex p to the vertex q. We shall think of such a graph as a system for transmitting messages, with p + q meaning that it is possible to send a message from p to q in a single step. Accordingly we shall call G a cumulative d-Atep (meAAage) ghaph if for every pair of vertices p,q there is a directed path of at most d edges from p to q. In a similar way, we shall call G a phecidely d-Atep (meAAage) gtaph if for every p,q there is a path of exactly d edges directed from p to q. Finally, we shall call G ttanAitiVe if its automorphism group is transitive on the vertices. -+
Our interest in message graphs was prompted by conversations with M. S. Paterson, who asked us how many vertices a transitive cumulative d-step 2-graph could have, and remarked that he knew of a transiWe found tive cumulative (3n-1)-step 2-graph with n. 3" vertices. ourselves just as interested in the non-transitive cases. We remark first that a precisely d-step v-graph can have at most vd vertices, and a cumulative one at most l+v+v2 +...+ vd A graph which meets this bound will be called t i g h t .
.
v = 2, d = 3
v = 2 , d = 2 Figure 1.
Two precisely d-step v-graphs 61
J. H.Conway and M.J. T. Guy
62
Then w e s h a l l p r o v e Theohem 1 . Tight precisely d-step graphs exist f o r a l l d. Theahem 2 . Theahem 3 .
gram
Tight. cunulative d-step exist only f o r d = 0 o r 1. I f there is a transitive precisely d-step v-graph w i t h N
vertices, then (for a l l n) there exists a t r a n s i t i v e curulative (dnin-l)-step v - g r e w i t h n.N" vertices. From o u r p o i n t of v i e w t h e g r a p h mentioned by ' P a t e r s o n is t h a t o b t a i n e d by t h e c o n s t r u c t i o n o f o u r Theorem 3 t o t h e b i - d i r e c t e d By a p p l y i n g it t o t h e g r a p h of F i g u r e l b t r i a n g l e (Figure l a ) . i n s t e a d , we o b t a i n a c u m u l a t i v e ( 4 n - l ) - s t e p g r a p h w i t h n.6" v e r t i c e s , which i s r a t h e r b e t t e r , s i n c e 61i4 > 31'3. T h i s is e a s y . W e t a k e t h e v e r t i c e s of G as Proof o f Theorem 1. a l l l e n g t h d s e q u e n c e s o f l e t t e r s from an a l p h a b e t of s i z e v , w i t h j o i n i n g r e l a t i o n aOal.,.adml + ala 2 . . . a d f o r e v e r y c h o i c e of d + l
letters
ao,al,
...,a .
(See F i g u r e 2 . )
10
Figure 2 .
A t i g h t p r e c i s e l y d - s t e p v-graph f o r d = v = 2
Proof of Theorem 2 .
For
d = 0
,
t h e graph i s j u s t a p o i n t j o i n e d
t o i t s e l f v t i m e s ( F i g u r e 3 a ) , and f o r d = 1 a complete g r a p h w i t h e a c h edge d o u b l y - d i r e c t e d ( F i g u r e 3 b ) . So t h e main problem (and indeed t h e main r e s u l t of t h i s p a p e r ) i s t o e s t a b l i s h t h e non-existencc f o r larger d.
(a) v = 3 , d = 0 F i g u r e 3.
(b) v = 3 , d = l
Tight cumulative d-step graphs
Message graphs
63
We consider the vector space of formal linear combinations of the vertices (and always refer it to the base consisting of the vertices). On this space we build the linear map a defined by a(p) = lq over all q for which p -+ q Then the tightness is expressed by the equation
.
l+a+a2
+...+
ad = J
,
where J is represented by the all 1s matrix, Multiplying by a - v , we deduce (a-v)(a'€) (a-E 2 1 . . (a-E d ) = 0 ,
so
that
aJ = VJ
.
.
where E is a primitive (d+l)st root of unity, and since the numbers v, E, E 2 , are distinct, that a is diagonalizable. It is easy to see that the multiplicity of the eigenvalue v is 1 (all coordinates of an eigenvector must equal the coordinate of maximal absolute value): let mk be the multiplicity of E~ (1 s k 5 d) Then by considering the trace of the maps 2 llala ,...,ad we deduce d , (0) 1 + ml + m2 + = l+v+...+v
...
.
... (1) v + mlE + m 2 ~ 2 + ... + m d ~ d = 0 , and so on up to d 2 (d) v + m l ~ d+ m 2 ~ 2 d+ ... + mdEd = 0 . (All the matrices a,a 2 ,...,ad have zero diagonal.) +
If we multiply the equation (i) by 1
+
E-lV
+
E -2kv2
+...+
-ki
,
and sum, we derive: d E -dkvd + (d+l)m, = l+v+. , .+v , E
which is equivalent to $+Ll E-kv-l
+
(d+l)mk =
$+Ll . v- 1
This, on the other hand, implies that E~ is real for k=l,...,dl and so d+l 5 2 , which proves the Theorem. 0 Proof of Theorem 3 . we let G and construct the new graph Gn are expressions of the form (iIPoP1.. .Pn-l) where i = O,l,...,n-1 relation is
be the graph given in the hypotheses, as follows. The vertices of Gn
I
and each
pk
is a vertex of
(i lPoP1. .Pn-l) * (i+1IPOP1. * .Pi-]9iPi+l.. .Pn,l
G.
The joining
J. H.Conway and M.J. T. Guy
64
.
It helps where the subscripts are read modulo n, and pi + qi to think of popl...pn-l as a "tape", and of i as a "reading head" which at each step "advances" the symbol pi currently scanned to one of the qi for which pi + qi , and then cycles one space. The graph is obviously transitive, since if TI is an automorphism of G and t any integer, the map ( i I pop1. .pn-l1 + (i+t 71 (Po)71 (pl) TI (pn-1)
.
is an automorphism of
.
I
..
Gn To see that it is a cumulative (dn+n-1)-step graph, let us suppose that we wish to find a path from (iIpopl...pn-l) to (jlqoql...qn-l). There is some t with 0 s t s n-1 and i+t 3 j (modulo n), and so any t-step path from (iIpopl...pn-l) will terminate at a vertex of the form (jlrorl...rn-l) Since we have used i n-1 steps we still have dn steps available. We can therefore perform d complete cycles of our "reading head", in which it will "advance" each of ro,...,rn-l exactly d times, and so can be used to transform them into qo,ql,...,qn-l respectively. This completes the proof. 0
.
We end with several problems: Is every tight precisely d-step v-graph isomorphic to Problem 1. the one we constructed in Theorem l? Problem 2.
IS there a transitive tight d-step v-graph?
(Affirmative answers to both problems are impossible.) We note that Figure lb is also a cumulative 2-step 2-graph Problem 3 . 2 Is there another cumulative d-step v-graph with v+v +. .+vd vertices?
.
Problem 4. Transitive precisely 4-step 2-graph with 12 or more vertices? (If so, application of Theorem 3 will produce a cumulative (5n-l)-step 2-graph with at least n.12" vertices, and we note that 121/5 > 61i4 .) Considerably more is known about similar problems concerning undirected graphs, see C13 and C2; Ch.1~1. REFERENCES C11 J.-C. Bermond and B. Bollobds, Diameters of graphs A survey, to appear, 123 B. Bollobbs, Extkemal Gaaph T h e o a y , London Math. SOC. Monographs, No.11, Academic Press, London, 1978.
-
Annals of Discrete Mathematics 13 (1982) 65-70 0 North-Holland Publishing Company
SETS OF GRAPHS COLOURKNGS DAVID E. DAYKIN and PETER FRANKL University of Reading, England and C.N.R.S., Paris, France
be a f i n i t e graph and b , c be p o s i t i v e i n t e g e r s . W e study t h e maximum number M of colourings of G so t h a t a d j a c e n t paikA of G g e t t h e same colour a t most b t i m e s . A colouring g i v e s each edge, v e r t e x of G one of c colours. P a i r means:- C a d & e . Two edges. CaAe v. Two v e r t i c e s . CaAe ev. Two edges o r two v e r t i c e s o r an edge and a v e r t e x . I n a l l c a s e s l i m ( b + m ) M / b is bounded.
Let
1.
G
INTRODUCTION
This paper has t h r e e c a s e s e, v, ev. W e have a f i x e d f i n i t e graph G and when w e say p a i h w e mean:- CaAe e . Two edges. ~ a s e V. TWO v e r t i c e s . Case ev. Two edges o r two v e r t i c e s o r an edge and a vertex. I n every case t h e members of a p a i r must be a d j a c e n t in G. By a c-cok?ouhing w e mean t h a t one of
c c o l o u r s has been given a s follows:- CaAe e. To each edge. CaAe v. To each v e r t e x . CaAe ev. To each edge and each v e r t e x . W e say a p a i r is bad i f both members of t h e p a i r g e t t h e same c o l o u r . W e w r i t e xe, xv, xev as t h e c a s e may be f o r t h e minimum number c f o r which t h e r e is a c-colburing with no bad p a i r . Usually xe, xv, xev a r e w r i t t e n x, x ' , x" and c a l l e d t h e chromatic numbers of G . A be t h e maximum v e r t e x degree i n G. Brooks C2, p.1281 proved t h a t xv 5 A + 1 and found when t h e i n e q u a l i t y is s t r i c t . Vizing C2, p.1331 proved t h a t A 5 xe i A + 1. W e say G is c l a s s 1 i f A = xe and class 2 otherwise. Erdds and Wilson C11 proved t h a t almost a l l G a r e c l a s s 1. The complete graph KP is c l a s s 1 i f f p is even. T r i v i a l l y A + 1 i xev and it has long been conjectured t h a t xev 5 A + 2. Let
65
D.E. Daykin and P. Frank1
66
w e c a l l a c-colouring e q u i t a b t e i f a t each vertex t h e W e will u s e a forthcoming theorem o f H i l t o n and d e Werra. I t s a y s t h a t if c does n o t d i v i d e t h e d e g r e e o f any v e r t e x of G t h e n G h a s an e q u i t a b l e edge c - c o l o u r i n g . e
I n case
numbers o f edges o f any two c o l o u r s d i f f e r by a t most one.
x
Consider any one of t h e t h r e e c a s e s and l e t
xev
a s t h e c a s e may be.
of c - c o l o u r i n g s s u c h t h a t each p a i r i s bad i n a t most Our o b j e c t i s t o s t u d y
c-colourings. assume t h a t 1
5
b and
2
5
xel xv
denote
or
M = M ( G , b , c ) b e t h e maximum number
Let
x.
c <
b
of t h e
To a v o i d t r i v i a l i t i e s w e
M.
Clearly
is superadditive,
M
namely M(bli
so
lim(b
+
C)
+ M(b2i
w)M/b e x i s t s .
f o r t h e complete g r a p h (1)
16
H
K
P
AubgKaph
i d u
Usually i n s t e a d of
06
then
G
we w r i t e
M
M(bl + b2i
5
C)
I n p a r t i c u l a r w e are i n t e r e s t e d i n M ( K 1 P i n view of t h e f o l l o w i n g o b s e r v a t i o n .
For example w e o b v i o u s l y have 2.
C)
Me,
b
MV,
MeV
5
M(G)
5
MeV Mel
5
M(H).
as t h e c a s e may b e . Mv < QD.
GENERAL RESULTS
W e f i r s t d e f i n e two f u n c t i o n s d = dl+ dc where t h e di
...+
Thus f o r example i f
d
f ( c , d ) and
F(c, d).
e d g e s of
have a common v e r t e x
G
f o r Case e i n any c - c o l o u r i n g t h e r e a r e a t l e a s t I t i s e a s y t o see t h a t
c < F
(3)
F(c, m c ) = ' F ( c , m c -1) = ( m c - l ) / ( m
Consider any c a s e .
Let
For 1 5 j 5 c l e t v j A s s u m e t h a t t h e numbers
h
be a mapping
x!/(m!)c.
-
1) f o r 2
h:{l,
5
then
a.
m.
2,...,x}
+
{1,2,...,c}.
be t h e number of i w i t h h ( i ) = j . v d i f f e r by a t most one. L e t p = p
be t h e number o f c h o i c e s f o r s u c h p =
ci
bad p a i r s a t
F(c,d) = c,
lim(d
0 )
f
and
(2)
+
Put
d i f f e r by a t most one, t h e n
h.
F o r example i f
x
= mc
(x,
c)
then
Sets of graph colourings
x
Now by t h e d e f i n i t i o n of
there is a
67
X-colouring o f
G
w i t h no
bad p a i r . Using t h i s f i x e d c o l o u r i n g , f o r e a c h of t h e p mappings h w e change t h e c o l o u r i n g t o a c - c o l o u r i n g i n t h e o b v i o u s way. By
f3
symmetry e a c h p a i r i s bad i n t h e same number
of t h e
11
derived
Thinking a b o u t Case e, i f w e had x e d g e s w i t h a a t h e n e a c h h would i n d u c e e x a c t l y f ( c , x ) bad
c-colourings. common v e r t e x
p a i r s (of e d g e s ) a t
and hence w e n o t i c e t h a t
a,
I t now f o l l o w s t h a t
x)
c < F(c,
(5)
s lim(b
+
m)M/b.
I f P i s t h e t o t a l number of p a i r s i n G and B i s t h e minimum number of bad p a i r s i n any c - c o l o u r i n g t h e n c l e a r l y
Sometimes t h i s bound i s s h a r p b u t c l e a r l y n o t always. 3. Now
CASE e . G
COLOURING EDGES
has a vertex
[A/cl
there are
r
rA/cl /c second f i r s t result.
7.
then
p a i r s (of e d g e s ) a t Me/b
Using ( 2 1 ,
(51,
lim(A
+
+
1 times w e o b t a i n o u r
ti
there a r e a t l e a s t
f (c, A )
sc
a
F(c, A)
5
b
and i n t h e
a,
Me = b .
c < A
A s remarked i n S e c t i o n 2 i f
(7)
I n t h e f i r s t c-colouring
A.
Repeating t h i s
cb+’ < A
16
THEOREM 1.
of d e g r e e
a
edges o f t h e same c o l o u r a t
for
c < A.
(7) w e have m ) { l i m ( b + m)M/b) = c
f o r Cases
e
and
ev.
W e can now s t a t e t h e main r e s u l t of t h i s s e c t i o n . THEOREM 2 .
14
id
(i) A =
xe
(iii)G = K
multiple Lim(b
+
06
Oh
P c,
-)Me/b
id ( i i )xe
with then
p
= F(c, A)
= A
o d d and
doh
+ 1
=
A
i b
c < A.
m c oh n o t an o d d .
bad
D.E. Daykin and P. Frank1
68
Proof. When G is class 1 we are in (i) and the result follows by (51, (7). In (ii) the result follows by (31, (51, ( 7 ) . In (iii) we have p = A + 1. If c does not divide A we get an equitable c-colouring of K as a special case of the Hilton and de Werra P theorem. If 2c divides b it is easy to construct one. The result follows by permuting the vertices of this equitable c-coloured K in all possible ways. 0 P Using the last technique and (6) we get THEOREM 3.
lim(b
+
0 )
0
Me(K )/b = P/B.
P
Let S denote a Class 2 graph with every vertex of degree 3, for instance the Petersen graph T. They call S a b n a h k . For c = 2 Theorem 2 (ii) applies, but an elementary argument about two adjacent vertices shows that snarks have equality in (7) for b = 1 and hence for all b. Similarly Me(K3, b, 2) = 3b and Me(K5, 1, 2) = 1 and Me(K5, 2b, 2) = 6b and Me(T, b, 3) = 15b. It would be nice to know Me(S, b, 3). We get strict inequality in (7) for K7 with c = 2 because P/B = 105/43 < 5/2 = F(2, 6). 4.
CASE v.
COLOURING VERTICES
Let Q be the largest integer q for which K is a subgraph of q G. Then corresponding to Theorem 1 we have the following observation. THEOREM 4. Lemma 5.
cb+’ < Q
16 16
Mv
5
then
H .LA a Aubghaph blE(H) I
0
Mv = b. 04
G
with
c < x(H)
then
Proof. Suppose the lemma is false, so that we have more than blE(H) I colourings. If ( a , B ) is in the set E(H) of edges of H there are at most b colourings in which a and B get the same colour. Hence there is one colouring in which a and B get different colours for every (a,B)rE(H). But this colouring contradicts c < x ( H L . 0 The lemma says in particular that if G has a triangle and then MV 5 3b. It also easily yields our next result.
c = 2
Sets of graph colourings
MV s blE(G) I
We have
THEOREM 6 .
+
= c
xv(G)
1 and
69
w i t h e q u a e i t y id6 d o h aLl x
xV(G\x) = c
E(G).
E
By s t r a i g h t f o r w a r d arguments on t h e d e f i n i t i o n s w e g e t LEMMA 7 .
The numbeh MV(b, c ) d o h which t h e h e e x i b t
(i)
xv( G\Ei)
s c
t h e maximum i n t e g e h m E l l . . .,Em c E ( G I buch t h a t
i b
1 s i s m,
doh
and
Next d e f i n e
x
that
06
each edge
(ii)
V
a s t h e minimum v a l u e of
A
(G\E) i
THEOREM 8 .
a t mob2
ib i n
G
MV/b
1 Eil
5
b
5
E(G)
c + l = x v then
16
Proof.
Take a
c
+
c o l o u r classes have a t most
w e choose t h e s e e d g e s f o r 16
G
i6
lim(b Let
+
E
I.
for
IE(G)I/(
A 5
A
5
G
C+l)
(E(G)
so
lEil
.
w i t h no bad p a i r .
c + 1 colour classes.
I/(:')
Some two
edges between them, and
o
A.
edge t h a n A i t i V e t h e n
=) MV/b = I E ( G ) I / A .
(El = A
have
c E(G)
L e t t h e r e l e v a n t automorphism g r o u p of = EOi
such
0
1 c o l o u r i n g of
Then p u t t h e v e r t i c e s i n t o t h e
Ei
E c E(G)
5
THEOREM 9 .
Proof.
0
C.
mA
THEOREM 10.
Ei.
the
06
over
IEl
I n t h e n o t a t i o n of Lemma 7 w e have
Proof.
Let
b
1s i
I;
m
G
and be
xV(G\E)
{el,...,
5
c.
Om}.
and a p p l y Lemma 7 .
0
As an example w e q u o t e t h e edge t r a n s i t i v e P e t e r s e n g r a p h T . Now I E ( T ) I = 1 5 and A = 3 when c = 2 . I t is n o t t o o h a r d t o cons t r u c t f i v e 2 - c o l o u r i n g s w i t h b = 1 so Mv = 5b. W e c o u l d n o t f i n d a r e s u l t f o r vertex t r a n s i t i v e graphs. t r i a n g u l a r pyramid
MV(y, b , 2 ) = 3b.
y
which h a s
6
This small graph
For instance consider t h e
v e r t i c e s and h a s y i s i n C2, p.1711.
D.E. Daykin and P. Frank1
I0
Now consider K We recall that the T u r h graph has the maximum P' number of edges among all graphs with p vertices and xv = c. Hence for K we have A = f (p, c) , so by Theorem 10 we have P lim(b
THEOREM 11.
+
m)
0
MV(K )/b = F(c, p). P
For c = 2 on K5 we found that MV is 1, 4 , 7, 10, according as b is 1, 2, 3, 4 . Using (l), ( 2 ) , (5) and Theorem 11 we see that lim(Q
(8)
16
Problem. 06
5.
deghee
3
CASE ev.
G
+ m)
i6
what i 6
(lim(b + m)Mv/b) = c.
edge t h a n d i t i v e w i t h
x
and evehy vehtex
= 3
A?
COLOURING EDGES AND VERTICES
Looking again at a vertex of degree Mev/b
5
F(c, A
+
1)
A
for
we see that c < A
+
1.
Using (5) as before we get THEOREM 1 2 .
I d (i) A + 1 lim(b
We have
xeV(K
P
) =
+
=
xev
oh
id (ii)
-)Mev/b = F(c, A + 1)
p = A
+
1
xev doh
= A
+
2 = m c then
c < A + 1.
so the theorem covers
K P'
REFERENCES
1.
P. Erd6s and R.J. Wilson, On the chromatic index of almost all graphs, J. Combinatorial Theory Series B, 2 ( 1 9 7 7 ) 2 5 5 - 2 5 7 .
2.
F. Harary, G h a p h T h e o h y , Addison-Wesley ( 1 9 6 9 ) .
Annals of Discrete Mathematics 13 (1982) 71-74 0 North-Holland Publishing Company
ON HADWIGER'S NUMBER AND THE STABILITY NUMBER P. DUCHET and H. MEYNIEL
E.R. 175, C.M.S. 5 4 Bd. Raspail, 75007 Paris, France
If a graph with n vertices and stability number a is not contractible to Kh+l then the following inequality holds: (2a
-
1) h
2
n.
Let G be a graph (multiple edges or loops are permitted); we use here the following parameters: n(G) a(G) y(G) h(G) B(G)
:
the number of vertices
: the stability number
the chromatic number the Hadwiger number ( = the greatest integer h such that G admits the complete graph Kh as subcontraction) : the minimal cardinality of a dominating set of G. :
:
For every graph we have (1)
a y
2
n.
As a special case of the four-colour problem, Erdds had conjectured
that for every planar graph the following inequality holds: (2) a 2 n
-. 4
This is actually implied by the four-color theorem C31. consequence of Hadwiger's conjecture C11: HADWIGER'S CONJECTURE : h
2
Thus, as a
y
a very natural question appears, namely n 2 g CONJECTURE :
.
In C21, Mader proved that a graph which is not contractible to has at most 8n rp log3pl edges. Therefore we have (3)
y
5
16
f(h+U(log3(h+l)1 71
.
K
P
P.Duchet and H,Meyniel
12
Hence
Our aim here is to improve inequality ( 4 ) as follows. THEOREM.
For every graph, (2a-l).h s n
(5)
.
LEMMA. For every connected graph, there exists a connected dominating subset with B ' vertices such that 8'
(6)
min(2a-1, 36-2)
5
.
PROOF OF THE LEMMA. We construct by induction two sequences of sets of vertices (Si) and (Di) with the following properties:
... ...
So c S1 c S 2
Do
c
D1
Si
c
Di
c
D2 and
are stable sets, are connected sets,
lDil s 21Sil
-1
for
i = 0,1,2,..
..
We begin with So = Do = {v} where v is any vertex of G . If Si is not a maximal independent set, we can choose a vertex x at distance two from Si since the graph is connected. Let y be a common neighbour of x and Si and put
si+l = si u
{XI,
Di+l = Di u {x,y}. The last Si is a maximal independent set and connected dominating set. In order to prove the inequality B ' i 3 8 dure is possible. We construct sequences properties: Co
c
C1
c
C2
Do
c
D1
c
D2
... ...
Di
c
Ci
and
lCil
-
Dk
is the required
2 an analogous proce(C,) and (Di) with the
are connected sets, are subsets of D, i
3 IDi!
-
2
for
i = 0,1,2,..
.
,
where D is a given dominating set of the graph which is minimal for inclusion. We start again with Co = Do = {vj, where v is any vertex of D. If Di # D, then, by the minimality of D, there exists a vertex y at distance two from Di; hence D contains a vertex x at
13
On Hadwiger's number and the stability number
distance at most 3 from Thus we put
is
y
or a neighbour of
is a common neighbour of
y
and
Di+l = Di u Ci+l = Ci u z
where
Dk = D, Ck
When
Di ( x
y).
{XI, Ex~Y~zI~ Di.
is the required connected dominating set.
REMARK. In (6), the inequalities are the best possible between 6 ' and 6 or a. Nevertheless, a slight refinement of 6 ' s 2a - 1 can be shown: a connected graph always contains a connected dominating subset with at most 2a - 1 vertices that contains a stable subset with a vertices. The proof is easy by induction on the number of vertices. PROOF OF THE THEOREM. Put G = (V, E). Clearly we can suppose that G is connected. If D is a connected dominating subset, then by considering the effect of the contraction by D we find that
where
GV-D
denotes the induced subgraph.
Since we can choose D with no more than 2a inequality (5) easily follows by induction on
-
1 vertices, the n(G)
.
REFERENCES
13 (1958).
C11
H. Hadwiger, Ungeldste probleme, Element. Math.
C21
W. Mader, Homomorphiesatze fur Graphen, Math. Annalen 178 (1968) 154-68.
C31
K. Appel and W. Haken, Every planar map is 4-colourabler 1 Illinois J. Math. 11 (1979), 429-490 and 491-567.
&
2
This Page Intentionally Left Blank
Annals of Discrete Mathematics 13 (1982) 75-80 0 North-Holland F'ublishing Company
A DEPTH-FIRST-SEARCH CHARACTERIZATION OF PLANARITY H. DE FRAYSSEIX and P. ROSENSTIEHL ColKge de France, Paris, France and Ecole des Hautes Etudes en Sciences Sociales, Paris, France
1.
INTRODUCTION
Hopcraft and T a r j a n [ l ] have produced t h e f i r s t a l g o r i t h m f o r t e s t i n g p l a n a r i t y of graphs i n l i n e a r t i m e . T h e i r proof o f l i n e a r i t y r e l i e s on a lemma a s s o c i a t e d w i t h t h e computing p r o c e s s . The main theorem o f t h i s p a p e r d i s p l a y s a p r o p e r t y o f t h e Trgmaux
trees o f a g r a p h , which c o n s t i t u t e s a new c h a r a c t e r i z a t i o n o f p l a n a r i t y , and j u s t i f i e s t h e s o - c a l l e d " L e f t - R i g h t Algorithm f o r P l a n a r i t y T e s t i n g and Embedding" 161. I n some way t h e theorem e n l i g h t e n s i n g e n e r a l t e r m s why a Tr6maux tree a l l o w s a l i n e a r computing t i m e . A theorem on Tr6maux trees is a l s o a theorem on D e p t h - F i r s t - S e a r c h C41, s i n c e t h e trees o f t h e maze s o l v e r - l e S i r d e Tremaux are e x a c t l y
-
t h e trees g e n e r a t e d by Depth-First-Search. L e t us s t a r t w i t h some d e f i n i t i o n s .
W e c o n s i d e r a s well known t h e
u s u a l d e f i n i t i o n s o f g r a p h , v e r t e x , e d g e , c y c l e , c o c y c l e and tree. By a tree we s h a l l mean a s p a n n i n g tree. Any r o o t e d t r e e ( T , r ) o f G d e f i n e s a p a r t i a l o r d e r on t h e v e r t i c e s o f G , f o r which r i s A Thdmaux t h e e ( o r Depth-dikht-heahch
t h e minimum e l e m e n t .
tkee)
o f G i s a r o o t e d t r e e ( T , r ) such t h a t e a c h c o t r e e edge a E T I is i n c i d e n t w i t h two comparable v e r t i c e s . I t follows t h a t every edge o f G , b e l o n g i n g t o T o r T I , i s i n c i d e n t w i t h a lower v e r t e x v - ( a ) and an upper v e r t e x v + ( a ) , and w e have
r
5
v-(a)
< v+(a)
The dundamentat c q c t e which meets
T1
in
.
T(a) a
.
, defined
for
a
E
T1
The dundamentat cocycee
,
.
i s the cycle , defined
T1(e)
A CkOhbabte f o r e E T , is t h e c o c y c l e which meets T i n e paih C a , $ l i s a p a i r o f e d g e s a,$ E T I i n c i d e n t w i t h f o u r d i f f e r e n t v e r t i c e s , and such t h a t T ( a ) n T ( $ ) i s n o t empty. A Thdmaux embedding Go of G i s a n embedding such t h a t t h e p a i r s o f edges which c r o s s have o n l y one c r o s s i n g and a r e c r o s s a b l e p a i r s . 75
16
The
H.de Fraysseix and P. Rosenstiehl Aet
Xo
is t h e s e t of c r o s s a b l e p a i r s t h a t c r o s s i n Ca,BI whose upper v e r t i c e s a r e n o t comparable, correspond a unique p a i r of tree edges, e and f , such t h a t v - ( e ) ' = v - ( f ) and a E T1(e) , f3 E T l ( f ) The s w i t c h i n g set l.i(e,f) a s s o c i a t e d w i t h such a c r o s s a b l e p a i r Ca,BI is t h e set of c r o s s a b l e p a i r s included i n t h e C a r t e s i a n product of T l ( e ) and T * ( f ) Geometrically, two Trgmaux embeddings Go and G' which d i f f e r by s w i t c h i n g o f e and f o n l y , have c r o s s i n g sets Xo and X' whose symmetrical d i f f e r e n c e i s e q u a l t o p ( e , f ) (Figure 1 ) . Go
ChOAbhg
.
To a c r o s s a b l e p a i r
.
.
a E T1 d e f i n e s an upper a n g l e e+(cr) and a lower a n g l e @'(a) with t h e tree c h a i n which j o i n s v + ( a ) t o v-(a) I n a n embedding Go , a n a n g l e is e i t h e r on t h e r i g h t o r on t h e l e f t o f t h e corresponding c h a i n . L e t u s n o t e t h a t a n edge
.
G'
Figure 1
-
G1
Switching
e
and
f
.
G s a t i s f i e s t h e c o n d i t i o n s o f t h e theorem. Go i s compatible w i t h t h e c o c y c l e D. XO = p ( e , f ) . X I = 0. G i s p l a n a r .
( S t r a i g h t l i n e s f o r tree and curved l i n e s f o r cotree edges.)
I1
A depth-first-search characterization of planarity L e t u s g i v e now a theorem on which r e l i e s t h e main theorem o f t h e
paper. Theohem 1 .
16 t h e chobbing
bet
Xo
06
a ThLmaux e m b e d d i n g
Go 0 6
p ( e , f ) , itthehe e a n d a h e t h e e e d g e b w i t h a common t o w e h u e k t e x , t h e n G L A p t a n a h . G
can be p a h t i t i o n e d i n t o b w i t c h i n g b e t b
f
The Wu-Liu c h a r a c t e r i z a t i o n theorem o f p l a n a r g r a p h s C31 d e f i n e s a v e c t o r s p a c e C on GF(2) g e n e r a t e d by t h r e e k i n d s o f s w i t c h i n g
sets, i n c l u d i n g t h o s e o f t h e p ( e , f ) t y p e , and a s s e r t s t h a t G Thus i t i n c l u d e s Theorem 1. p l a n a r i f and o n l y i f Xo E C
.
2.
is
THE MAIN THEOREM
L e t u s recall t h a t t w o a n g l e s are d e f i n e d f o r e a c h c o t r e e edge o f a
Trgmaux tree of G , and t h a t i n a g i v e n embedding some p a i r s o f a n g l e s a r e on t h e same s i d e , some o t h e r s a r e on d i f f e r e n t s i d e s .
G , a Trgmaux tree ( r , T ) o f G , and f i v e t y p e s o f p a i r s o f a n g l e s i n o r d e r t o d e f i n e two a b s t r a c t b i n a r y r e l a t i o n s on a n g l e s . The f i r s t c l a s s o f p a i r s w i l l be c a l l e d S (S f o r "same") and a second c l a s s D (D f o r " d i f f e r e n t " ) . S u D w i l l c o n s t i t u t e t h e e d g e s o f a g r a p h H whose v e r t i c e s are t h e set o f cotree a n g l e s . a, a' and $ d e n o t e c o t r e e e d g e s .
W e c o n s i d e r below a g r a p h
H. de Fraysseix and P.Rosenstiehl
(In the figures, circles are used to indicate that location and side are not considered. )
When Go is an embedding without crossing, it is easy to check that the set D (resp. S ) defined above in an abstract way is a subset of pairs of cotree angles disposed on different sides (resp. same side) in the embedding. The main theorem considers In some way the reciprocality Theonem 2 .
06
H
.
G
i b
pkknah i d and o n l y i d t h e b e t
D
i b
.
a cocycee
A depth-first-search characterization of planarity
I9
O u t l i n e of t h e proof be a p l a n a r embedding w i t h o u t c r o s s i n g . f o r each angle a s i d e : l e f t o r r i g h t . I.
Let
It defines
Go
I t is e a s y t o check f o r e a c h o f t h e f i v e t y p e s t h a t , whenever a
p a i r o f a n g l e s b e l o n g s t o D ( r e s p . S ) , t h e a n g l e s a r e a r r a n g e d on d i f f e r e n t s i d e s ( r e s p . same s i d e ) i n t h e embedding, f o r i f t h e y were n o t , c r o s s i n g s would a p p e a r . T h e r e f o r e D i s a c o c y c l e of H , which s e p a r a t e s t h e l e f t a n g l e s from t h e r i g h t o n e s . 11.
Let
G
be a graph such t h a t , f o r a g i v e n Tr6maux t r e e , i t s
associated set
D
is a cocycle of
H
.
F i r s t it i s e a s y t o
c o n s t r u c t a p l a n a r Tr6maux embedding Go ( w i t h p o s s i b l e c r o s s i n g s ) such t h a t , whenever a p a i r o f a n g l e s b e l o n g s t o D (resp. S) t h e a n g l e s a r e a r r a n g e d on d i f f e r e n t s i d e s ( r e s p . same s i d e ) . In o t h e r words Go s a t i s f i e s t h e f i v e t y p e s o f r e l a t i o n s . By Theorem 1 t h e proof w i l l be complete i f w e show t h a t t h e c r o s s i n g set
Xo
can b e p a r t i t i o n e d i n t o s w i t c h i n g sets
.
p
( e lf )
.
L e t Ca,BI E Xo From c o n d i t i o n s ( i )and ( i i ) ,v + ( a ) and v + ( B ) a r e n o t comparable. T h e r e f o r e t h e r e e x i s t s a unique p a i r e , f E T I
.
a s s o c i a t e d t o a and B From c o n d i t i o n s ( i i i ), ( i v ) and ( v ) it f o l l o w s t h a t p ( e , f ) c Xo : i n o t h e r words, f o r any c r o s s a b l e p a i r [cr',B'] w i t h a' E T ' ( e 1 , 8 ' E T ' ( f ) , w e have Ca',B'I E Xo (see
Figure 1).
3.
CONCLUSION
In t h e context of Depth-First-Search,
p l a n a r i t y t e s t i n g and
embedding r e q u e s t t o d e a l o n l y w i t h r e l a t i o n s on c o t r e e a n g l e s , d e f i n e d by o r d e r r e l a t i o n s a s d i s p l a y e d i n c a s e s ( i ), ( i i ) (iii) , ,( i v ) , (v). I n p a r t i c u l a r t h e embedding of t h e t r e e can be i g n o r e d , s i n c e tree a n g l e s do n o t a p p e a r i n t h e n e c e s s a r y c o n d i t i o n s o f Theorem 2 . Because o f t h e d r a s t i c r e d u c t i o n o f v a r i a b l e s t o d e a l w i t h , t h e l i n e a r computing t i m e c a n be a c h i e v e d .
H.de Fraysseix and P. Rosenstiehl
80
REFERENCES
,
Hopcroft, J . E .
c21
J . A C M , 2 1 ( 1 9 7 4 ) , 549-568. L i u , Y., A c t a M a t h . AppL. S i n i c a 1 ( 1 9 7 8 ) , 321-329.
C31
R o s e n s t i e h l , P. Wu-Liu,
C41
and T a r j a n , R . E .
, Efficient
c11
, Preuve
i n AnnaLA
T a r j a n , R.E.
06
planarity testing,
algebriqu e du critere d e p l a n a r i t e d e
Dibckete Mathematicb, 9,
, Depth-first-search
( 1 9 8 0 ) , 67-78.
and l i n e a r g r a p h a l g o r i t h m ,
S I A M J . Comput. ( 2 ) , ( 1 9 7 2 ) , 146-160. Toward a t h e o r y of c r o s s i n g numbers, t o t i a L T h e o h y , 8 , ( 1 9 7 0 ) t 45-53.
C51
T u t t e , W.T.,
[6 I
F r a y s s e i x , H.
de, a n d R o s e n s t i e h l , P., The l e f t - r i g h t a l g o r i t h m
f o r p l a n a r i t y t e s t i n g a n d embedding i n l i n e a r t i m e
(to a p p e a r )
J. Combina-
Annals of Discrete Mathematics 13 (1982) 81-88 0 North-Holland Publishing Company
GRAPH-THEORETICAL MODEL OF SOCIAL ORGANIZATION* JONATHAN L. GROSS Department of Computer Science Columbia University New York, N.Y. 10027, U.S.A. Mary Douglas h a s p r o p o s e d a t h e o r y o f c u l t u r a l a n a l y s i s t h a t f o c u s e s o n t w o mechanisms f o r t h e c o n t r o l o f s o c i a l b e h a v i o r . One i s t h e t o t a l i t y o f c a t e g o r i c a l d i s t i n c t i o n s , s u c h a s d a y and n i g h t . The o t h e r i s t h e e x t e n t t o which i n d i v i d u a l s e x p e r i e n c e t h e i r own i d e n t i t i e s t h r o u g h membership i n s o c i a l u n i t s . What d i s t i n g u i s h e s t h i s t h e o r y from most o t h e r s are t h e f u n d a m e n t a l h y p o t h e s e s t h a t t h e s e
t w o mechanisms are q u a n t i f i a b l e and t h a t t h e y are operationally orthogonal. Anticipating t h e i r precise f o r m u l a t i o n , s h e h a s i n t r o d u c e d t h e names
arid
and
g r o u p f o r measured v a l u e s of t h e t o t a l s t r e n g t h o f t h e s e t w o r e s p e c t i v e mechanisms. P o l y t e c h n i c models o f g r i d a n d g r o u p h a v e b e e n d e v e l o p e d and are d e s c r i b e d h e r e . The outcome o f a n e x p e r i m e n t p a r t i a l l y confirming t h e r e l i a b i l i t y of g r i d measurements’ i s a l s o p r e s e n t e d . 1.
INTRODUCTION C e r t a i n s o c i o l o g i c a l phenomena are a p p a r e n t l y b e t t e r u n d e r s t o o d
t h r o u g h a n a l y s i s of a c u l t u r a l e n v i r o n m e n t t h a n o f i n d i v i d u a l behavior.
The c l a s s i c case i s s u i c i d e , a s r e p o r t e d by E . Durkheim
[1897]. A n o t h e r s t r i k i n g example i s t h e s u c c e s s o f S . Minuchen i n t r e a t i n g a n o r e x i a by m o d i f y i n g t h e s t r u c t u r e o f t h e f a m i l y r e l a t i o n s h i p s , r a t h e r t h a n by i s o l a t i n g t h e p a t i e n t w i t h t h e p r e s e n t i n g symptom. Some o f t h e phenomena o f i n t e r e s t i n t h i s program o f r e s e a r c h
a r e c o m p e t i t i v e n e s s , b u r e a u c r a t i c s t a n d a r d i z a t i o n , and i d e n t i f i c a t i o n
*
D e d i c a t e d t o P r o f e s s o r Frank H a r a r y on t h e o c c a s i o n o f h i s s i x t i e t h birthday.
81
82
J. L. Gross
of individual i n t e r e s t with collective goals.
Mary Douglas h a s
i d e n t i f i e d t h e v a r i o u s k i n d s o f c u l t u r a l e n v i r o n m e n t s i n which e a c h o f these attributes is likely t o exist.
For t h e p a s t t h r e e y e a r s I have
b e e n d e v e l o p i n g m a t h e m a t i c a l models o f s o c i a l s t r u c t u r e a n d d e s i g n i n g experiments t o test h e r c u l t u r a l i n s i g h t s .
The p r e l i m i n a r y e m p i r i c a l
r e s u l t s are p o s i t i v e . I h a v e b e e n a b l e t o q u a n t i f y what Douglas [ 1 9 7 0 , 19781 c o n s i d e r s
t o b e t h e t w o f i r s t - o r d e r v a r i a b l e s of c u l t u r a l environment.
One,
c a l l e d grid, i s t h e amount o f c o n t r o l e x e r t e d by t h e p u b l i c l y acknowledged c a t e g o r i c a l d i s t i n c t i o n s s u c h as male and f e m a l e , workday a n d weekend, or e d i b l e and i n e d i b l e .
The measured v a l u e
d e p e n d s o n l y o n t h e e x t e n t t o which s u c h c l a s s i f i c a t i o n s c o n t r o l b e h a v i o r i n t h a t e n v i r o n m e n t , n o t on what p a r t i c u l a r c a t e g o r i e s e x i s t
o r how t h e y a r e o b s e r v e d .
The o t h e r , c a l l e d g r o u p , f i r s t - o r d e r
v a r i a b l e i s t h e amount o f c o n t r o l t h a t a r i s e s from a p e r s o n ' s e x p e r i e n c i n g h i s o r h e r i d e n t i t y t h r o u g h membership i n a s o c i a l u n i t . F o r i n s t a n c e , a w a g e - e a r n e r who v o l u n t a r i l y p u r c h a s e s l i f e i n s u r a n c e and names h i s w i f e a n d c h i l d r e n a s b e n e f i c i a r i e s m i g h t b e b e h a v i n g i n accordance w i t h h i s c u l t u r a l l y c o n s i s t e n t ex p er ien ce of t h e i r n e e d s a s h i s own. The q u a n t i f i c a t i o n o f g r i d a n d g r o u p i s b a s e d o n a g r a p h - t h e o r e t i c model, c a l l e d t h e EXACT model, which I d e v e l o p e d i n c o l l a b o r a t i o n w i t h S t e v e Rayner.
Once a s o c i a l u n i t is r e p r e s e n t e d i n terms o f a n
EXACT model, i t i s p o s s i b l e t o measure i t s g r i d score and i t s g r o u p
score.
Both o f t h e s e are d e r i v e d from e m p i r i c a l d a t a .
An e x p e r i m e n t
p r o v i d i n g evidence o f t h e r e l i a b i l i t y of one of t h e p r e d i c a t e s i s described. T h i s b r i e f s k e t c h i n d i c a t e s t h e basic f e a t u r e of a c o m b i n a t o r i a l model of s o c i a l o r g a n i z a t i o n .
D e t a i l s o f t h e e m p i r i c a l methodology
a n d j u s t i f i c a t i o n o f t h e c h o i c e o f p r e d i c a t e s are p r e s e n t e d i n a monograph by Gross a n d Rayner
19821.
F o r a g e n e r a l s o u r c e on g r a p h t h e o r y , t h e r e a d e r i s r e f e r r e d t o F. H a r a r y [ 19691.
Graph-theoretical model of social organization
2.
83
THE EXACT MODEL
The EXACT model f o r a s o c i a l u n i t h a s t h e f o l l o w i n g components: X = t h e s e t o f menbers o f t h e u n i t A = t h e set of d e f i n i n g a c t i v i t i e s of t h e u n i t
C = t h e s e t o f r o l e s p e r s o n s assume i n t h e s e a c t i v i t i e s T = a c u l t u r a l p a r t i t i o n of t h e annual t i m e c y c l e E = t h e s e t o f e l i g i b l e nonmembers
Thus, "EXACT" i s a n a p p r o x i m a t e acronym. Although X = { xl , . . . , x 1 , A = { a l , . . . , am} and P e n ) are simply sets, b o t h C and T have a d d i t i o n a l
E = {el,...,
structure.
ic
I n p a r t i c u l a r , t h e r e i s a subset o f r o l e s .
= {cl,
...
I
ic i. r1
The t i m e s p a n i s p a r t i t i o n e d i n t o associated w i t h a c t i v i t y ai. c u l t u r a l l y - d e f i n e d t i m e u n i t s s u c h a s months, weeks, a n d h o l i d a y s . I t a l s o c a r r i e s t h e s t r u c t u r e o f l i f e - c y c l e e v e n t s and t i m e - f a c t o r s i n social reciprocity. I n p r e c i s e terms, t h i s s t r u c t u r e i s r e p r e s e n t a b l e a s p r o b a b i l i s t i c d i s t r i b u t i o n s and a l g o r i t h m s . A m u l t i g r a p h a s s o c i a t e d w i t h t h i s model i s c a l l e d t h e EXACT g r a p h .
For every a c t i v i t y ai I t s v e r t i c e s are t h e members o f t h e u n i t . a n d e v e r y p a i r o f members x a n d xk who i n t e r a c t i n a c t i v i t y a i l j t h e r e i s a n e d g e l a b e l e d ai w i t h e n d p o i n t s x and xk. I n a m o r e
j
e l a b o r a t e v e r s i o n of t h e EXACT model, t h i s e d g e would a l s o c a r r y a weighting label t o i n d i c a t e t h e e x t e n t of t h e i n t e r a c t i o n . A s a s o c i a l u n i t , o n e m i g h t t h e o r e t i c a l l y s e l e c t any c o l l e c t i o n
o f p e r s o n s s u c h t h a t t h e EXACT g r a p h i s c o n n e c t e d .
I n p r a c t i c e , one aims f o r a c o l l e c t i o n o f p e r s o n s s u c h t h a t , s u b j e c t t o v a r i o u s c o n s t r a i n t s , t h e g r i d and g r o u p s c o r e s a r e l o c a l l y maximal. In a n t h r o p o l o g i c a l j a r g o n , o n e would s a y t h a t o u r s o c i a l u n i t s are d e f i n e d by t h e c u l t u r e . A l t e r n a t i v e b a s e s f o r d e f i n i n g s o c i a l u n i t s m i g h t b e g e o g r a p h i c ( e . g . a l l t h e p e r s o n s i n v i l l a g e Y) o r p o l i t i c a l ( e . g . e v e r y c a r d - c a r r y i n g member o f o r g a n i z a t i o n Z ) . Often these c r i t e r i a m i g h t y i e l d t h e same s e l e c t i o n o f a s o c i a l u n i t . However, i n cases o f j u x t a p o s e d c u l t u r e s , t h e y y i e l d d i f f e r e n t u n i t s .
J.L. Gross
84
3.
THE GROUP PREDICATES
A t p r e s e n t , t h e group score i s a c o m p o s i t e of f i v e " p r e d i c a t e " scores, e a c h a s s o c i a t e d w i t h t h e EXACT g r a p h G. Each p r e d i c a t e s c o r e c o r r e s p o n d s t o a p a r t i c u l a r s o c i a l a t t r i b u t e and i s a number between
0 and 1.
The f i r s t o f t h e group p r e d i c a t e s i s p r o x i m i t y , which i s t h e i n v e r s e o f d i s t a n c e . Of c o u r s e t h e d i s t a n c e d ( x i , x j ) between t h e i s t h e minimum number o f e d g e s one must t w o v e r t i c e s xi and x j
to x Thus, t h e a v e r a g e d i s t a n c e from j' t o t h e other v e r t i c e s is a positive integer. I t follows t h a t t h e xi r e c i p r o c a l o f t h e a v e r a g e d i s t a n c e i s a number between 0 and 1, which w e d e n o t e by p r o x ( x i ) . Then prox(G) i s d e f i n e d t o b e t h e a v e r a g e v e r t e x proximity. t r a v e r s e t o g e t from
xi
The second group p r e d i c a t e i s t r a n s i t i v i t y . From t h e p e r s p e c t i v e and xr b o t h o f xi , t h e i s s u e i s w h e t h e r when t w o p e r s o n s x j i n t e r a c t w i t h xi t h e y a l s o i n t e r a c t w i t h e a c h o t h e r . L e t Ni d e n o t e t h e induced ( s i m p l i c i a l ) graph on t h e n e i g h b o r s of xi
.
Then
trans(xi) = # E(Ni)/( #V (Nil)
Of c o u r s e , # E ( N i ) of
Ni
,
and
respectively.
#V(Ni)
mean t h e numbers o f e d g e s and v e r t i c e s
We now d e f i n e t r a n s ( G ) t o b e t h e a v e r a g e
vertex t r a n s i t i v i t y . A s graph i n v a r i a n t s ,
t h e r e i s no r e a s o n why h i g h p r o x i m i t y and
low t r a n s i t i v i t y c a n n o t c o e x i s t .
F o r i n s t a n c e a complete b i p a r t i t e
h a s p r o x i m i t y a t l e a s t 2 / 3 b u t t r a n s i t i v i t y 0 . On t h e graph K PIP o t h e r hand, imagine a g r a p h c o n s t r u c t e d b y r u n n i n g a l o n g e d g e p a t h Pu from one complete g r a p h K v t o a n o t h e r copy of Kv. Then t h e p r o x i m i t y is a p p r o x i m a t e l y 2/u. I f t h e number v i s much l a r g e r t h a n t h e number u , t h e n t h e t r a n s i t i v i t y score i s n e a r l y 1. The d i s t r i b u t i o n o f p a i r s o f v a l u e s ( p r o x ( G ), t r a n s ( G ) f o r a c t u a l s o c i a l u n i t s i s a matter f o r e x p e r i m e n t a l s t u d y . The t h i r d group p r e d i c a t e i s f r e q u e n c y .
W e define
f r e q ( x i ) t o be interacts with
t h e p r o p o r t i o n o f h i s o r h e r a l l o c a b l e t i m e t h a t xi t h e o t h e r members o f t h e u n i t . A l s o , f r e q ( G ) i s t h e a v e r a g e of t h e i n d i v i d u a l f r e q u e n c i e s . The o b v i o u s a l t e r n a t i v e s t o i n t e r a c t i n g w i t h
85
Graph-theoretical model of social organization
o t h e r members o f t h e u n i t a r e i n t e r a c t i n g w i t h o u t s i d e r s o r w i t h no one. I f a g i v e n s o c i a l u n i t , such a s f a m i l y , i s a s u b u n i t of a l a r g e r system, o r i f i t s membership o v e r l a p s w i t h o t h e r s o c i a l u n i t s , t h e n a p e r s o n ' s commitment might e x t e n t beyond t h e g i v e n u n i t . The f o u r t h p r e d i c a t e s c o r e , s c o p e , measures t h e d i v e r s i t y o f a p e r s o n ' s involvement i n a c t i v i t i e s of t h e u n i t r e l a t i v e t o t o t a l a c t i v i t y . More p r e c i s e l y , s c o p e ( x i ) i s t h e r a t i o of t h e number o f i n t r a - u n i t i n t e r a c t s with o t h e r members t o t h e t o t a l a c t i v i t i e s i n which xi internumber o f a c t i v i t y i n s i d e and o u t s i d e t h e u n i t i n which xi a c t s w i t h o t h e r p e r s o n s . Scope ( G ) i s t h e a v e r a g e i n d i v i d u a l s c o p e . The f i f t h group p r e d i c a t e score, i m p e r m e a b i l i t y , i s t h e o n l y one t h a t h a s no l o c a l v a l u e s .
During t h e a n n u a l t i m e c y c l e some s u b s e t
of e l i g i b l e non-members become members of t h e u n i t .
W e define
w e u s e t h e a v e r a g e of t h e f i v e group p r e d i c a t e s f o r t h e s c o r e group ( G ) . I t i s p o s s i b l e t h a t a d d i t i o n a l e m p i r i c a l A t present,
e v i d e n c e w i l l motive t h e a d o p t i o n of a weighted a v e r a g e . I f f o r some r e a s o n , n o t a l l f i v e p r e d i c a t e s a r e a p p l i c a b l e t o some s o c i a l u n i t , one a v e r a g e s t h e a p p l i c a b l e o n e s . The p r e d i c a t e s c o r e s a p p e a r t o b e c a l c u l a b l e o n l y i f one w a t c h e s everyone on t h e u n i t f o r a y e a r . o b t a i n approximate s c o r e s . 4.
Obviously one u s e s samples t o
THE G R I D PREDICATES
L i k e group, g r i d is a domposite of up t o f i v e p r e d i c a t e s , e a c h s c o r e d between 0 and 1. A p r e d i c a t e s c o r e f o r an e n t i r e s o c i a l u n i t i s c a l c u l a t e d a s a n a v e r a g e o v e r a l l members of t h e i r s c o r e s f o r t h a t predicate. The f i r s t g r i d p r e d i c a t e i s s p e c i a l i z a t i o n , t h e o p p o s i t e o f d i v e r s i f i c a t i o n . L e t C ( x i ) b e t h e set o f r o l e s assumed d u r i n g t h e t i m e c y c l e T . Then s p e c ( ) o = 1 - # C ( x i ) / #C. Thus, a p e r s o n who assumes v e r y few r o l e s w i l l s c o r e n e a r 1.
86
J. L. Gross
The second g r i d p r e d i c a t e i s asymmetry. x and x w e d e f i n e t h e i n t e r a c t i o n number i j t h e number o f a c t i v i t i e s i n which xi and x consequence o f t h e i r roles.
F o r e a c h p a i r of members inter(xi,xj) to be i n t e r a c t as a
j A l s o , w e d e f i n e t h e exchange number
exch(xi,xj) t o b e t h e number of a c t i v i t i e s i n which xi and x j exchange r o l e s and i n t e r a c t from b o t h s i d e s . For i n s t a n c e , i n a h o s p i t a l i t y a c t i v i t y i n many c u l t u r e s , it i s l i k e l y t h a t xi and exchange t h e roles of h o s t and g u e s t . I n a n employment a c t i v i t y i t i s u n l i k e l y t h a t t h e roles o f b o s s and worker are exchanged.
x
j
We
d e f i n e t h e asymmetry score exchanged. W e d e f i n e t h e asymmetry score
asym(xi) = 1
1 cx . E - #V(Ni) 1
V(Ni)
exch ( x i , x . ) inter(xi,x.) 3
The t h i r d g r i d p r e d i c a t e i s e n t i t l e m e n t .
W e calculate the score
f o r a member xi a s t h e p r o p o r t i o n of h i s o r h e r roles t h a t are a t t a i n e d by a s c r i p t i o n , r a t h e r t h a n by achievement. F o r i n s t a n c e , i f someone i s a u t o m a t i c a l l y a c a r p e n t e r b e c a u s e h i s f a t h e r w a s o n e , t h a t i s a s c r i p t i o n . On t h e o t h e r hand, i f h e g e t s a job b e c a u s e h e i s so s k i l l e d a t i t , t h a t i s achievement. I n g e n e r a l , w e d e f i n e 1 e n t i t l e (x-) C
=
#ascribed roles ( x i )
#roles (xi)
The f o u r t h g r i d p r e d i c a t e i s a c c o u n t a b i l i t y f o r o n e ' s immediate b e h a v i o r . The s c o r i n g i s b a s e d on s i t u a t i o n s i n which t h e o c c u p a n t s o f two roles i n t e r a c t .
I n an "accountability i n t e r a c t i o n " ,
one r o l e ( t h e "dominant" r o l e ) i n c l u d e s s a n c t i o n s t o b e u s e d i n e x a c t i n g performance f r o m t h e p e r s o n i n t h e o t h e r r o l e ( t h e " r e g u l a t e d " r o l e ) . F o r e a c h member xi w e d e f i n e t h e a c c o u n t a b i l i t y s c o r e a c c ( x i ) t o b e t h e p r o p o r t i o n of h i s i n t e r a c t i o n s t h a t a r e accountability interactions. The f i f t h g r i d p r e d i c a t e i s i n f o r m a t i o n t r a n s m i s s i o n . Phenomena s u c h as f o o d - t a k i n g , t h e w e a r i n g of c l o t h i n g , o r t h e a l l o c a t i o n o f s p a c e may b e r e g a r d e d as b e h a v i o r a l media f o r t h e t r a n s m i s s i o n of i n f o r m a t i o n a b o u t t i m e and s o c i a l roles. The g e n e r a l i d e a i s t h e r e e x i s t s o p t i m a l s t r a t e g i e s f o r g u e s s i n g what roles or e v e n t s a r e b e i n g r e p r e s e n t e d b o t h i n t h e a b s e n c e and i n t h e p r e s e n c e of c l u e s from s u c h media. From t h e p r o b a b i l i t i e s of c o r r e c t g u e s s e s w i t h and w i t h o u t t h e c l u e s , one c a n c a l c u l a t e t h e i n f o r m a t i o n
81
Graph-theoretical model of social organization
v a l u e o f t h e clues., i n t h e s e n s e of C. Shannon [1948]. t i o n score
b y t h e b e h a v i o r of 5.
The i n f o r m a -
i n f o ( x i ) i s b a s e d on t h e v a l u e of t h e c l u e s p r o v i d e d xi.
AN EXPERIMENT: INFORMATION I N FOOD BEHAVIOR
The w o r t h o f t h e i r q u a n t i f i c a t i o n o f g r i d and g r o u p depends on
t w o t h i n g s , t h e r e l i a b i l i t y o f t h e e m p i r i c a l measurements and t h e i r u s e i n t h e a n a l y s i s o f s u c h s o c i a l t r a i t s as c o m p e t i t i v e n e s s , b u r e a u c r a t i c s t a n d a r d i z a t i o n , a n d c o n s c i o u s n e s s of g r o u p w e l f a r e . An e x p e r i m e n t a l program s p o n s o r e d by t h e R u s s e l l Sage F o u n d a t i o n u n d e r t h e d i r e c t i o n o f Mary Douglas h a s p r o v i d e d e v i d e n c e t o s u p p o r t t h e r e l i a b i l i t y o f t h e s c o r i n g method f o r o n e o f t h e g r i d p r e d i c a t e s , information transmission. The method w a s a p p l i e d i n f o u r t o e i g h t h o u s e h o l d s i n e a c h o f t h r e e d i f f e r e n t American communities: a s o u t h e r n r u r a l community, a n e t h n i c s u b u r b o f a n o r t h e a s t e r n c i t y , and a m i d w e s t e r n I n d i a n reservation.
I n e a c h h o u s e h o l d o f e a c h community, t h e c l u e s t o
time-based e v e n t s were f i v e or more a s p e c t s o f f o o d - t a k i n g b e h a v i o r , i n c l u d i n g t h e d u r a t i o n o f t h e meals, t h e number o f p e r s o n s a t t e n d i n g a n d t h e number o f d i f f e r e n t f o o d v a r i e t i e s s e r v e d .
For each a s p e c t
a n d e a c h h o u s e h o l d a n i n f o r m a t i o n score w a s c a l c u l a t e d .
Due t o t h e
small s i z e of t h e s a m p l e , o n l y t h e r a n k s were c o n s i d e r e d s i g n i f i c a n t .
W e d i s c o v e r e d t h a t from o n e a s p e c t t o a n o t h e r , t h e i n f o r m a t i o n This tends t o confirm t h e r e l i a b i l i t y o f t h e method. What i t s u g g e s t s i s t h a t p e r s o n s i n t e r n a l i z e c o n s i s t e n t syndromes o f b e h a v i o r i n a c c o r d a n c e w i t h D o u g l a s ' s t h e o r y o f c u l t u r a l a n a l y s i s . F u l l d e t a i l s are p r e s e n t e d by Gross [ 1 9 8 1 1 . r a n k i n g s were c o n s i s t e n t .
J. L. Gross
88
REFERENCES M. Douglas [ 19701 I Natural Symbols, Penguin. M. Douglas [ 19781 I Cultural Bias, Royal Anthropological Institute, London.
E. Durkheim[l897], Le Suicide, translated into English as Suicide, Rout Ledge i3 Kegan Paul, London, 1952. J.L. Gross [1981], Measurement of calendrical information in food-taking behavior, manuscript. J.L. Gross and S.F. Rayner [1982], Grid/group analysis of social organization: an operational guide, manuscript.
F. Harary [1969], Graph Theory, Addison-Wesley. C.E. Shannon [1948] , A mathematical theory of communication, Bell System Technical Journal 27.
Annals of Discrete Mathematics 13 (1982) 89-100 0 North-Holland Publishing Company
ODD CYCLES OF SPECIFIED LENGTH IN NON-BIPARTITE GRAPHS ROLAND HAGGKVIST Institut Mittag-Leffler Djursholm, Sweden
I t is proved t h a t every s u f f i c i e n t l y l a r g e g r a p h 2n w i t h minimum d e g r e e more t h a n 2k+l e i t h e r c o n t a i n s no odd c y c l e of l e n g t h a t l e a s t k + l o r else it c o n t a i n s a ( 2 k - l ) - c y c l e . I t is a l s o shown t h a t e v e r y t r i a n g l e - f r e e g r a p h w i t h minimum d e g r e e more t h a n 3n is homomorphic w i t h a 5-cycle. A t t h e end some c o n j e c t u r e s a r e f o r m u l a t e d .
1.
INTRODUCTION
For unexplained terminology see C21. I n p a r t i c u l a r n o t e t h a t Cm d e n o t e s t h e c y c l e of l e n g t h m , t h a t 6 ( G ) d e n o t e s t h e minimum v e r t e x - d e g r e e i n t h e graph G , and t h a t G is H-free i f G h a s no subgraph isomorphic w i t h t h e g i v e n graph H. I n what f o l l o w s t h e number n w i l l always s t a n d f o r t h e number of v e r t i c e s i n t h e g r a p h G under c o n s i d e r a t i o n . One of t h e best-known r e s u l t s i n graph t h e o r y r e a d s : Every graph w i t h n2 edges c o n t a i n s a t r i a n g l e , u n l e s s it is b i p a r t i t e . a t least T h i s s p e c i a l c a s e of Turbn's theorem C81 was proved by Mantel i n 1907 C71, and h a s been t h e s t a r t i n g p o i n t of many i n v e s t i g a t i o n s , See Bollobds C3, Ch 111.5 and c h VII f o r a r e p r e s e n t a t i v e sample of t h e r e s u l t s which have been o b t a i n e d . Here I o n l y n o t e t h a t Erdds ( t o g e t h e r w i t h G a l l a i ) , and i n d e p e n d e n t l y Andrdsfai, proved t h a t a t r i a n g l e - f r e e n o n - b i p a r t i t e g r a p h h a s a t 2
most C *I( + 1 edges C5, Lemma 11; t h e same proof g i v e s under 2n , and i n t h e same assumptions t h a t t h e minimum d e g r e e is a t most 3 f a c t , a s noted i n A n d r i b f a i , ErdBs and S6s C 1 , remark 1 . 6 1 , t h a t 2n e v e r y n o n - b i p a r t i t e g r a p h w i t h minimum d e g r e e g r e a t e r t h a n 2k+l c o n t a i n s a n odd c y c l e of l e n g t h a t most 2k-1. 89
90
R. Haggkvist C2k-l-free
I n t h i s n o t e I s h a l l p r e s e n t some r e s u l t s on
graphs.
It
w i l l be shown t h a t i f such a g r a p h c o n t a i n s an odd c y c l e o f l e n g t h 2n k+l i f n is a t l e a s t 2 t h e n i t s minimum d e g r e e i s a t most 2k+l s u f f i c i e n t l y l a r g e w i t h r e s p e c t t o k. Hence, any 2-connected nonb i p a r t i t e s u f f i c i e n t l y l a r g e g r a p h w i t h minimum d e g r e e g r e a t e r t h a n
-
2 2k+l
c o n t a i n s a c y c l e of l e n g t h 2k-1. T h i s is s h a r p i n view of t h e graphs C2k+l(m), where H ( m ) d e n o t e s t h e g r a p h o b t a i n e d from H
by r e p l a c i n g e a c h x
X
t o e v e r y vertex i n
The assumption t h a t
V ( H ) by an m-tuple,
iff ( x , y ) is an edge i n H.
Y G
joining every vertex i n
i s 2-connected c a n b e dropped i f
k = 2,3,4
o r 5, b u t i s i n g e n e r a l n e c e s s a r y . I s h a l l a l s o demonstrate t h a t every t r i a n g l e - f r e e
degree g r e a t e r than
3n
i s homomorphic w i t h
C5
g r a p h w i t h minimum
(i.e. G
c
C5(n)).
T h i s r e s u l t should b e compared t o t h a t o f A n d r d s f a i , Erdds and S6s C11 which ( r e f o r m u l a t e d ) s t a t e s : Every Kr-free g r a p h w i t h minimum 3r-7 d e g r e e g r e a t e r t h a n -n i s homomorphic w i t h Kr-l (i.e. has c h r o m a t i c number less t h a n
r).
Restricted t o t h e case
r = 3
this
a g a i n g i v e s t h a t any t r i a n g l e - f r e e g r a p h w i t h minimum d e g r e e g r e a t e r 2n than 5 is b i p a r t i t e . F i n a l l y , a t t h e end some c o n j e c t u r e s a r e s t a t e d . 2.
MAIN RESULTS
L e t m e f i r s t g i v e a proof of t h e r e s u l t o f Erdds, G a l l a i , A n d r d s f a i
and S6s mentioned above. Theorem 1.
Every n o n - b i p a r t i t e g r a p h w i t h
odd c y c l e o f l e n g t h a t most Proof. v(S)
2
Let
5
S
6 > [=I2n
c o n t a i n s an
2k-1.
be a s h o r t e s t odd c y c l e i n
G.
W e may assume t h a t
s i n c e else t h e c o n c l u s i o n i n t h e theorem h o l d s t r i v i a l l y .
i s c l e a r l y c h o r d - f r e e , and no v e r t e x o u t s i d e S i s j o i n e d t o S by more t h a n two e d g e s , s i n c e S i s s e l e c t e d as a s h o r t e s t odd c y c l e S
in A
G.
Let
E(A,B)
and t h e o t h e r i n
d e n o t e t h e set of e d g e s i n B.
Then w e have
G
w i t h one end i n
91
Odd cycles of specified length in non-bipartite graphs Hence 6(G) 4
C d ( x ) 4 2 v ( G ) which i n p a r t i c u l a r i m p l i e s t h a t xrV(S) 2n C2k+11 i f S h a s l e n g t h a t least 2k+l. This proves
theorem 1. Remark.
The above p r o o f g i v e s immediately t h a t i f
odd c y c l e i n a t r i a n g l e - f r e e g r a p h
S
is a s h o r t e s t
C d ( x ) 5 2n. I n p a r xeV(S) t i c u l a r a n o n - b i p a r t i t e g r a p h G which s a t i s f i e s any o f i/-iii/ below c o n t a i n s an odd c y c l e of l e n g t h a t most 2k-1. 4n i/ d ( x ) + d ( y ) > 2k+l f o r e v e r y p a i r of n o n - a d j a c e n t v e r t i c e s x,y in
4n 2k+l
f o r e v e r y p a i r of a d j a c e n t v e r t i c e s
x,y
G.
C d(x) > 2 $ ~ ~ ~f o’r ne v e r y i n d u c e d p a t h of f i x e d xcv (P) l e n g t h t , t 5 2k.
iii/
(kzl)
Theorem 2 . G
G.
d(x)+d(y) >
ii/
n >
in
then
G
Let
G
be a graph w i t h
(2k+l) (3k-1).
Then, e i t h e r
2n 6 > CI-2k+l G
and
contains a k
C2k-l
o r else
7.
c o n t a i n s no odd c y c l e of l e n g t h g r e a t e r t h a n
Theorem 2 h a s some immediate consequences.
By a theorem of Voss and
Zuluaga C91 e v e r y 2-connected n o n - b i p a r t i t e
g r a p h c o n t a i n s an odd
c y c l e of l e n g t h
min { 2 6 - l I n ) .
C o r o l l a r y 1. L e t G 2n 6 > and n >
Consequently w e have
b e a 2-connected n o n - b i p a r t i t e
(kil)( 2 k + l ) ( 3 k - 1 ) .
Also n o t e t h a t s i n c e e v e r y n o n - b i p a r t i t e
Then
G
graph with
contains a
C2k-l.
g r a p h c o n t a i n s an odd c y c l e
( o f l e n g t h a t l e a s t t h r e e ) w e have C o r o l l a r y 2. n >
Let
be a n o n - b i p a r t i t e
G
(2k+l) (3k-1).
Then, i f
2n 6 > CI-2k+l
graph with
k = 2,3,4 o r 5,
G
and
contains a
‘2k-1’ C o r o l l a r y 1 i s b e s t p o s s i b l e b e c a u s e of t h e f a m i l y of g r a p h s and C o r o l l a r y 2 i s a l s o s h a r p (Take e . g . t h r e e c o p i e s of C2k+l ( m ) ,
Km,m’
s e l e c t a v e r t e x i n e a c h copy and j o i n t h e chosen v e r t i c e s by
a l t o g e t h e r t h r e e edges.
minimum d e g r e e m, Proof of theorem 2.
The r e s u l t i n g g r a p h h a s
is clearly non-bipartite, F i r s t some lemmas.
6m
vertices,
and h a s n o 1 1 - c y c l e . )
92
Lemma 1.
R. Higgkvist
...m,
Let Ail i = 1,2, m m I u Ail 2 C lAil i=1 i=l
-
be sets. C
lfi< jsm
Then
[Ai n Ajl.
Proof. This is nothing else but the first two terms in the inclusion, exclusion formula. Lemma 2. Let S be a set of k+l vertices in G (the graph in Theorem 2). Then S contains a pair of vertices uIv for which IN(u) n N(v) I > 3k-2.
Hence
(kil)
2 n -2k +2k > (3k-1) - 2k-1. This implies C IN u)nN(v) > 2k+l 2k+l u+v U,VES Since the sum has summands, at least one of these must be greater than
-
> 3k-2,
since k > 1.
This proves the lemma. Definition. An (m,k)-starter in G is a set of k+l vertices, each pair of which are joined by a path of length 2j-1, for some j E {m,m+l,. k-1).
..
Lemma 3 . If G contains an G contains a C2k-l.
(i,k)-starter for some
i < k, then
Proof. Let S be an (ilk)-starter in G I with i maximum. By lemma 2 S contains a pair of vertices x,y such that IN(x) n N(y)l > 3k-2. By definition of an (ilk)-starter, x is joined to y by a path p:x-xl x2j-2 y of length 2j-1 for some j E {i,i+l; k-11. We consider two cases.
...
-...
-
Case 1. j = k-1. In this case let z be a vertex in N(x) n N(y)-V(P). We know that z exists since 3k-2 > 2j. The cycle C:z-x-xlx~~-*-Y-zis a cycle of length 2k-1, which proves the lemma in the first case.
...
93
Odd cycles of specified length in non-bipartite graphs
j < k-1.
Case 2.
( N ( x ) n N(y)
Then
-
-
> k+2.
V ( P ) ( > 3k-2-(2k-4)
V(P) c o n t a i n s a s e t Q of k + l Consequently N(x) n N(y) v e r t i c e s , any p a i r u , v of which are j o i n e d by a p a t h u-x-xlx 2 j - 2 -y-v of l e n g t h 2 j + l . I t f o l l o ws t h a t Q i s a ( j + l , k ) - s t a r t e r , c o n t r a r y t o t h e c h o i c e of S ( s i n c e i < j + l 5 k - 1 ) .
...-
T h i s c o n t r a d i c t i o n p r o v e s lemma 3. Lemma 4 . G
If
c o n t a i n s an odd c y c l e of l e n g t h a t l e a s t
G
contains a
(1,k)-starter.
C : C ~ - C ~ - . . . - C ~ ~ - ~ - C ~t o b e a s h o r t e s t admiss-
m i s s i b l e and choose i b l e cycle in
G.
W e examine f i v e cases.
2m-1 < k + l .
Case 1.
S i n c e by assumption
t h e r e e x i s t d i s t i n c t v e r t i c e s ui, such t h a t (Ci,ui) i s an edge i n V(C)
k+ 1 is ad7
W e s a y t h a t an odd c y c l e o f l e n g t h a t l e a s t
Proof.
u
... u
{U1'U2'
6(G)
> [&I
2m-1,
i = 1,2,...
f o r every
G
i.
~ + ~ -i s~ a~ ( 1 }-k)-starter i n
k + l < 2m-1 < 2k-1.
Case 2 .
k+l t h e n 2,
I n t h i s case
{c1,c2,
> 2(k+l) i n G-V(C) The s e t
G.
... c
~ + ~is} a
(l., k ) -starter.
2k-1 < 2m-1
Case 3.
and
t h e segment of even l e n g t h between l e n g t h of
Pc
i'Cj
(c , c . ) .
h a s a chord
C
i s n o more t h a n
c ki
i
l
Let
P
Ci'Cj
c a l o n g C . Then t h e j 1 a t most, s i n c e else C and
-
f a i l s to be a shortest admissible cycle.
W e may t h e r e f o r e assume
k
f o r some P u t s1 = { c l , c 2". c ~ ~ - ~NOW, ) . i f m-r then we p u t
r s u c h t h a t 2r-1 < I. i s odd, s a y m - r = 2q+l,
S 2 = {ci:
1
that
and i f
ci = c1
and
cj = '2r-1
i = 2r-1+4t-1
m-r
for
S2 = {ci: i = 2r-1+4p-3
t = 1 , 2 , ...q
m-r = 2q,
i s even, s a y
for
be
then w e put
p = 1,2,...
9).
I n b o t h cases p u t S3 = {ci: i = 2r-l+4p
for
p = 1,2,...
q}.
i s a s e t of a t l e a s t k + l v e r t i c e s ( e q u a l i t y h o l d s when ( r , m ) = ( 2 , k + l ) ) any p a i r o f which are j o i n e d by an odd p a t h of l e n g t h a t most 2m-5, u s i n g v e r t i c e s from C o n l y . I n d e e d , t h i s i s o b v i o u s l y t r u e f o r any p a i r of v e r t i c e s i n S2 u S3 s i n c e n o such p a i r of v e r t i c e s a r e o f d i s t a n c e 2 a l o n g C. Any p a i r of Then
S = S1 u S2 u S3
R. Haggkvist
94
vertices in
+ (C1rC2r-1) '~11~2r-1 of length at most 2r-3 and clearly if x E S1 and y E S2 u S3 then the distance along C is 2 only if x = czr-2 and Y = c 2r in which case ~ ~ ~ - ~ - c ~ ~ - ~ is- an odd path of length 2r-1, or if x = c2 and y = c2m-1 in which case c2-c3-. 2' r- 1-'lmC 2m-1 is the desired path. By lemma 2 S contains a pair of vertices x,y with at least 3k-2 common neighbours. By the preceeding paragraph x and y are joined by a path P ' : X - X ~ - X ~ - . . . X ~ ~ -of ~ - length ~ 2j-1 for some j < m-1. If j < k-1, then there exists a set of k+l vertices which are common neighbours to x and y, but which do not belong to P'; such a set is obviously a (1,k)-starter. If k-1 < j < m-1 then every neighbour z common to x and y belongs to P', since else G contains an odd cycle of length at least 2k-1, and at most 2m-3, contradicting the choice of C as a shortest admissible cycle. (The cycle in question would be z-x-xlx 2j-3-y-z.) This means that N ( x ) n N(y) c V(P') and hence that IN(x) n N(y) n V(C) 1 > 3k-2. This is easily seen to give a contradiction, since
S1
are joined by an odd path in
..
...
(1)
Indeed, if k 2p-1 < 2.
IN(x) n V(C) I < k+2
for every vertex
x
in
C.
x = cl, then x can only be joined to the vertices * c2p-1' and C2m-~lc2m-2'c2m-41" - 1 C2m-2p for This contradiction proves case 3.
Case 4 . 2k-1 < 2m-1 and G contains an odd cycle C' which intersects C in a path of length v(C')-2, and C' is shorter than C. In this case C' must have length at most shorter admissible cycle than C.
...-
$
, since
else
C'
is a
Let c1-c2c2r-2 be the path in common for C and C', and let v be the vertex in V(C')-V(C). Put S = (ci:i = lI2,...2r} u u {ci: i = 2r+4p, {ci: i = 2r+4p-1, p = 1,2,... [2m-1-2r1-1} 4 u {v). Then S is a set of at least k+l p = 1,2,... [2m-:-2rl-l} vertices, any pair of which are joined by an odd path of length at most 2m-5, using only vertices in {v} u V(C). By lemma 2 S contains a pair of vertices x,y with at least 3k-2 common neighbours. As in the last part of case 3, N(x) n N(y) c V(C) u (v}. Consequently IN(x) n v(C) I > 3k-3 and IN(y) n V(C) I > 3k-3.
95
Odd cycles of specified length in non-bipartite graphs
S i n c e one of
x,y
belongs t o
w e again reach a contradiction t o
C
(1). T h i s p r o v e s c a s e 4 . 2k-1 < 2m-1,
Case 5.
is joined t o
is chord-free,
C
and no v e r t e x i n
-
G
V(C)
by more t h a n two edges.
C
This i s c l e a r l y t h e l a s t case t o consider.
The arguments i n t h e
proof of Theorem 1 g i v e immediately t h a t
c
d(x)
5
2n,
and hence t h a t
6(G)
5
[fm-ll 2n s
2n 2k+l I
[-
X€V (C)
c o n t r a r y t o assumption.
I
T h i s f i n i s h e s t h e proof of lemma 4 .
Theorem 2 i s an immediate consequence of lemmas 3 and 4 . Theorem 3.
3n 6 > [TI
Every t r i a n g l e - f r e e g r a p h w i t h
i s homomorphic
w i t h a 5-cycle. Proof of Theorem 3 .
F i r s t a lemma.
t a i n e d from an 8 - c y c l e and
Every t r i a n g l e - f r e e
subgraph f u l f i l s Proof.
...-x8-x1
xl-x2-
H
d e n o t e t h e g r a p h ob-
by adding t h e edges (x,,x,)
Then w e have
(x2,x6).
L&ma 5.
Let
Z
xrV ( H I
Notice t h a t
H
d(x)
5
graph 3n.
G
which c o n t a i n s
In particular
6(G)
H 5
x1-x2-
...-x8-x1
are
~x2,x4,x6,x8~
~ x 1 , x 3 , x 5 , x 7 ~ , n e i t h e r of which i s i n d e p e n d e n t i n
for a l l
x
c
xcV(H)
E
V(G),
dG(x) =
IEG(x,H) I
c
as a
c o n t a i n s no i n d e p e n d e n t s e t of f o u r v e r t i c e s
s i n c e t h e o n l y such s e t s i n and
3n [TI.
X C V( H )
5
3.
IE~(x,v(G))
H.
Thus,
I t follows t h a t
I
=
c
Y E V( G I
IE~(H,Y)I
5
3n.
T h i s p r o v e s t h e lemma. L e t G be t h e g i v e n t r i a n g l e - f r e e graph i n theorem 3. Since 2n , Theorem 1 g u a r a n t e e s t h a t h a s minimum d e g r e e l a r g e r t h a n 7
G G
G is b i p a r t i t e . In t h e l a t t e r case G i s homomorphic w i t h K2 and t h u s w i t h C5 , and t h e r e i s n o t h i n g t o prove. Hence w e may assume t h a t G c o n t a i n s t h e 5 - c y c l e C:vl-v2v5-v1. L e t D be t h e s e t of v e r t i c e s i n G j o i n e d t o
contains a 5-cycle, u n l e s s
...-
vertices i n
by e x a c t l y two e d g e s .
C
S i n c e no v e r t e x can b e j o i n e d
to
C
by t h r e e or more edges w i t h o u t f o r c i n g a n o n - e x i s t i n g t r i a n g l e
in
G
w e have
(3)
5 (Dl
2
C d(vi)
i=1
-n > C 77n 1
since
6(G)
2
3n
[TI
+ 1.
R.Haggkvist
96
The sets Di
= N(vi,l)
n N ( V ~ + ~ ) i, = 1 , 2 , . . .
5
with i n d i c e s counted modulo 5 p a r t i t i o n D. Moreover GCDI i s homomorphic with C5, s i n c e every p a i r of v e r t i c e s x,y e i t h e r belong t o Di u Di+2 f o r some i, o r else one belongs t o Di and t h e o t h e r t o Di+lr and only i n t h e second c a s e can x and y be Consequently w e joined by an edge, because Di u Di+2 c N(vi+l). may assume t h a t T = V ( G ) - D i s non-empty. L e t x be a v e r t e x i n T. By (3), IT1 = n ID1 S [;I < d ( x ) . I t follows t h a t x is joined t o some Di, say t o D1. W e claim t h a t
-
(4)
x
i s joined t o d i f f e r e n t
Dils.
This can be seen as follows. W e a l r e a d y know t h a t x i s joined t o D1. Consider t h e v e r t i c e s v3 and v 4 . Since n e i t h e r v3 nor v4 can be joined t o D1 it follows t h a t N(V3)
We a l s o know t h a t and hence
.
N(v4)
c
N(v3) n N(v4) =
-
+ IT1
IDl[ I t follows t h a t proves (4)
U
x
1s n
-
V(G)
- D1.
0, because of t h e edge
3n 2C71
n 3n + 2 + Cgl1 s [TI
m u s t have a neighbour o u t s i d e
T u D1,
( v 3 , v 4 ),
< d(x).
which
W e now consider two c a s e s .
Case 1.
x
i s joined t o
D1
and
D2.
.
Then, let x1 E D1 n N(x) and x2 E D2 n N(x) The graph w i t h vert i c e s Ix,xl,x2,v1,v2,v3,v4,v5~ and edges I(vi,vi+l) , i = 1 , 2 , . ..,5} u { (x,xl), ( x , x 2 ) , (x1,v2) , (x2,v1), (x2,v3) 1 i s a subgsaph of G i s o morphic t o t h e graph H i n lemma 5 . W e deduce t h e r e f o r e t h a t 3n 6 ( G ) 2 [TI c o n t r a r y t o assumption. Hence w e have reached Case 2. case
No v e r t e x i n
T
i s joined t o consecutive
Dils.
In t h i s
91
Odd cycles of specified length in non-bipartite graphs
Ti = {x
E
T: N(x) n Di-l $. 0 and
N(x) n Di+l
is a partition of TI since every vertex in Di's by ( 4 ) .
T
9
@)
is joined to two
We claim that (5) Ti u Ti+2 is an independent set of vertices in G for i = 1,2,...5. To see this, fix i and assume that (x,y) is an edge between vertices in Ti u Ti+2. Let xi+l E N(x) n Di+l and Yi+l E N(y) n Di+l. Then the graph with vertices {(vi,vi+l)l (V(C) - vi+l) u ~xlY,xi+l'Yi+l1 and edges (E(C) (vi+11vi+2)1 ) u { (vilYi+l)I (Yi+l'Vi+2)I (vilxi+l)lXi+1'Vi+2) I (x,xi+l)l(xly)l and (yIyi+l)) is a subgraph isomorphic to H in G, which again leads to a contradiction. This proves ( 5 ) .
-
It is obvious that Di u Ti and Di u Ti+2 are independent sets of vertices for all i, since otherwise some vertex in T is joined to consecutive Di's and this is forbidden in case 2. We deduce that G has edges between the independent sets Di u Ti and Di+l u Ti+l only, for i = 1,2,...,5. Hence G is homomorphic to c5' 3.
REMARKS AND CONJECTURES
1.
The truth of Theorem 1 prompts the following conjecture.
Conjecture 1. Every hamiltonian non-bipartite graph with minimum 2n is pancyclic (i.e. contains cycles of all degree more than 3 lenghts k, 3 s k < n). It has been shown by Bondy C l O l that every hamiltonian non-bipartite n2 edges is pancyclic. This was sharpened graph with at least CTI in C61 to the essentially best possible statement of its kind, namely: every hamiltonian non-bipartite graph with more than
IC*(
2
+ 1 edges is pancyclic.
Note however, that while conjecture 1 would be best possible for k = 3 , there seems to be no such tight fit for k = n-1. In fact I know of no example of a hamiltonian non-bipartite graph with minimum degree larger than which fails to contain a Cn-l. 2. Erdds and Simonovits considered the function $(n,Kr,t) defined as the maximal value of &(GI, where G is a graph on n vertices which is Kr-free and has chromatic number less than t. They
98
R. Hdggkvist
conjectured in particular that
$(n,K3,t)
3
for
t
2
4.
10n This conjecture must be modified slightly, since $(n,K3,4) 2 29 because of the graph in figure 1. It is natural to conjecture 10n therefore that $(n,K3,4) = 1 9 , but the following is more likely to be true. Conjecture 2 . Every triangle-free graph with 6 > 10n is homomorphic with a MClbius ladder on 8 vertices, i.e. an 8-cycle together with the chords joining vertices of distance 4 on the cycle.
Figure 1:
A 4-chromatic triangle-free graph on on 29m vertices each of degree 1Om.
Odd cycles of specified length in non-bipartite graphs
99
REFERENCES 1.
B. Andrbsfai, P. ErdBs and V.T. S ~ S ,On the connection between chromatic number, maximal clique and minimal degree of a graph, Discrete Mathematics 8 (1974) 205-218.
2.
J.A. Bondy and U.S.R. Murty, Graph theory with applications, MacMillan, London, 1976.
3.
B. Bollobbs, Extremal graph theory, Academic Press, London 1978.
4.
P. Erdds and M. Simonovits, On a valence problem in extremal graph theory, Discrete Mathematics 5 (1973) 323-334.
5.
P. Erdds, On a theorem of Rademacher-Turbn, Illinois Journal of Mathematics 6 (1962) 122-126.
6.
R.J. Faudree, R. Hdggkvist and R.H. Shelp, Pancyclic graphsconnected Ramsey number, submitted.
7.
W. Mantel, Problem 28, solution by H. Grouwentak, W. Mantel, j.Teixeira de Mattes, F. Schuh and W.A. Wythoff, Wiskundige Opgaven 10 (1907) 60-61.
8.
P. Turbn, On an extremal problem in graph theory, Math. Fiz. Lapok 48 (1941) 436-452. (in Hungarian with German abstract.)
9.
H.J. Voss and C. Zuluaga, Maximale gerade und ungerade Kreise in Graphen I, Wissenschaftliche Zeitschrift der Technisce Hochschule Ilmenau, Heft 4 (1977) 57-70.
10.
J.A. Bondy, Pancyclic graphs I, journal of Combinatorial Theory B 11 (1971) 80-84.
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Annals of Discrete Mathematics 13 (1982) 101-1 10 0 North-HollandPublishing Company
SIMPLICIAL DECOMPOSITIONS: SOME NEW ASPECTS AND APPLICATIONS R. HALIN Mathematisches Seminar, University of Hamburg West Germany
1.
INTRODUCTION
Decompositions and unique representation theorems arising from them A play a great role in algebra. It is natural to ask for similar results in graph theory. There is for instance the decomposition induced by the Cartesian product. Another kind of decompositions having some formal similarities with algebraic factorizations is the simplicia1 decomposition (abbreviated: s.d.) which is the topic of this paper. S.d.'s were the first time extensely used by K. Wagner in 1931 in his studies related to the Four-Colour-Problem. In the first place I am going to present some new results, aspects and applications which are not yet in my paper C6l; however a few basic results will be repeated here because I cannot assume familiarity with the previous paper. 2.
NOTATION
Graphs are allowed to be infinite;,IGI denotes the cardinality of V(G). H 5 G means that H is a subgraph of G; H c G indicates that H is a proper subgraph of G, and H 2 G that G contains H as an induced subgraph. G 2.u H expresses that G contains a subdivision of G. xlly indicates that vertices x and y are not adjacent. A complete graph will be called a simplex; we reserve the letter S for simplices S ( a ) denoting the simplex of order a. a.T.b (G) means that vertices a,b are separated in G by T 5 G. pG(a,b) is the Menger number of vertices a,b in G, i.e. the maximum number of internally disjoint a,b-paths in G. For a cardinal V b its immediate successor is denoted a+. 3 . THE NOTION OF SIMPLICIAL DECOMPOSITION
Let G be a graph and let GX ( A < a ) be a family of induced subgraphs of G, where u is an ordinal > 0. We say that this family 101
R. Halin
102
of GX forms a A i m p e i c i a t decompobition (s.d.)of ing conditions are fulfilled: 1.
u
G =
For all
3.
Each
if the follow-
GA;
X
2.
GI
1,
0<
is
ST
c
< uI
'I
(
u
X
GI) n GT = ST c
is a simplex;
GT.
X
Thus G is built up from the Gh's by an infinite process (if U is infinite) in each step of which GX is pasted to the part constructed before along a simplex, like an infinite cactus is composed of its branches. It is clear from the construction that every S A separate8 G, and that for instance the chromatic number of G equals the least upper bound of the x(GX), if u is finite or all the SX are finite. (This may be wrong in the case of infinite u and GX's.) The SX are also called the simplices of detachment of the s.d. in question. 4.
SIMPLICIAL SUMMANDS
The subgraphs which may occur in some s.d. of a graph may be considered as something like the direct summands of a group. We call them the AimpeiCia$ Aummandd of the graph in question. They can be characterized in several ways. Let us call
H 5 G
Aepakation-invakiant in
a.T.b
(H) =>
GI
if
a.T.b (GI.
Furthermore, H 5 G will be called convex in G, if for any a,b E V(H) and anychordless a,b-path P 5 G (i.e. P Q G) P 5 H holds. Then we can state: (1) Let H 5 GI H are equivalent: a) b) c)
H H H
not a simplex.
also
The following statements
is a simplicia1 summand of G ; is separation-invariant in GI is convex in G.
Further it is easy to show The intersection of any family of convex subgraphs of G is again a convex subgraph of G. (2)
This motivates the introduction of the convex cLoAuRe
CG(H)
of some
103
Simplicia1decompositions
subgraph H contain H.
of G as the intersection of all convex subgraphs which (Mind that G itself of course is convex in G.)
We can give an estimate of the order of CG(H) by the Menger numbers of non-adjacent vertices in G: N o and assume (3) Let be a regular cardinal > pG(x,y) < I8 for all xlly, x,y E V(G). Then for any H 5 G with IHI < I8 we have
From this we can prove ( 4 ) First decomposition theorem.
Let
be a regular cardinal > N 0' Let G be a graph with IGI 2 *c , pG(x,y) < Ub for all xlly, and S ( * ) 4 G. Then G has a s.d. GA(h < u , where u' is the initial ordinal of IG() in which all IGh I < UL
.
5.
PRIME-GRAPH DECOMPOSITIONS
G is called p h i m e (with respect to s.d.) if there is no s.d. of G with at least 2 members (or, equivalently, if G has no separating simplex). A s.d. GA ( A < u ) of G is called a p R i m e - g h a p h d e c o m p o b i t i o n (pgd) if every G is prime. h
When does there exist a pgd? It is easy to show that every finite graph has a pgd. On the other hand I have an example of a countable graph which contains an infinite simplex and which does not possess a pgd. I proved: (5) Second decomposition theorem. If G does not contain an infinite simplex then G has a pgd. This pgd may be chosen reduced, i.e.: no member is contained in another member. Each reduced pgd. contains exactly all the maximally prime subgraphs of G. Thus, if no S( N o ) in G exists, the members of any reduced pgd are uniquely determined, though their order may vary. If G S ( No), even if a reduced pgd exists, its members need not be uniquely determined. Let us consider an example. Let T3 be the tree which is regular of degree 3. If a root r in T3 is chosen, T3 determines an order theoretical tree. Let G be the comparability graph of the partial order determined by r, i.e. vertices of T3 are connected by an edge in G iff they are comparable (i.e. iff they lie on a path of T3 starting in r). Then all infinite paths of T3 starting fn r determine maximal prime
R. Halin
104
subgraphs of G (namely infinite simplices), and every maximal prime subgraph of G is obtained in this way. Thus G is countable and has N1 maximal prime subgraphs. G has a pgd with maximally prime members, which can be shown by exhausting step by step its vertices by maximally prime subgraphs; it does not matter in which order this is done. By reasons of cardinality no pgd of G can have more than N o members. Therefore G has uncountably many pgd's which are essentially different. 6.
A PARTIAL ORDER ON THE SET OF MAXIMAL PRIME SUBGRAPHS OF
G.
One could ask whether the non-existence of pgd's for certain graphs is due to the fact that we stipulated, in pgd, a certain well-ordering of its members (or, more precisely, a tree order, i.e. a partial order in which for each x the elements preceeding x form a wellordered set), and that perhaps, if the conditions on the order of the members in a s.d. are weakened, all graphs have a pgd. From this point of view the following relation in the set % of maximal prime subgraphs of G may be of interest. let S ( P , Q ) denote the simplex 5 Q For any distinct P , Q E t which is spanned by the vertices of Q which are adjacent to that component of G-Q which contains V ( P - Q ) . (This component is uniquely determined, by the primity of P. ) Now fix an R E P (as a root), and put, for different P , Q E p {R), P < Q iff neither S ( Q , P ) contains S ( R , P ) nor S ( R , P ) contains S ( Q , P ) . Then P separates R and Q . Furthermore, put R < P for each P E '8 - {R].
-
(6) <
is a partial order in
.
For Q E , the set of all P 5 Q need not be a chain, but in this set "uncomparable" is an equivalence relation. I t would be interesting to obtain further properties of this partial order. 7.
TRIANGULATED GRAPHS
The two decomposition theorems were used in (6) to solve some colouring and connectivity problems, which are not repeated here. I now want to talk on triangulated graphs (or rigid circuit graphs). A graph G is called t h i a n g u t a t e d if each circuit of length 2 4 has a chord (i.e. if no circuits other than triangles are induced subgraphs of G I . It is not difficult to prove
Simplicia1 decompositions
105
(7) A finite graph is triangulated iff it has a pgd consisting only of simplices. This can be generalized by the 2nd decomposition theorem: (7') A graph without an infinite simplex is triangulated iff it has a pgd consisting only of simplices. Perfect graphs and s.d.'s are related by the following fact: (8) Let G1 perfect iff each p.349.)
,...,Gr Gi
(r finite) be a s.d. of G. Then G is is perfect (i = 1,. ,r). (See Berge C21,
..
This implies an early result of Hajnal and Suranyi: (8') Every finite triangulated graph is perfect. There are infinite, non-perfect triangulated graphs (see section 8 below). S. Wagon put the question C111, whether every triangulated graph without an infinite simplex is perfect. By using the 2nd decomposition theorem I could give a positive answer C71: (9) Every triangulated graph without an infinite simplex is perfect
.
8. COMPARABILITY GRAPHS OF TREES. partial order X € V
A
s
in a set V
is called theetike if for every
is a chain (with respect to s ) . (V, 5 ) is called an ohdeh t h e o a e t i c a t t h e e if every A(x) is well ordered. The following statements for a treelike order (V, 5 ) are readily verified: i) Each chain K of (V, 5 ) is contained in a maximal chain of (V, 5 ) . ii) If K is a maximal chain and x E K, then A(x) 5 K. iii) If Ki (i E I) is a family of maximal chains and U Ki are comparable, then there is an i E I with x,y € K Y x,y € i€I As a consequence we have (10) If G is the comparability graph of a treelike order 5 , then there exists a pgd of G in which all the members are simplices and correspond to certain maximal chains with respect to s: the
R. Halin
106
order in which these simplices occur is arbitrary. Proof. Well order V(G), by step, maximal chains.
and then exhaust V(G)
by choosing, step
We have seen that in the infinite case not all the maximal chains (i.e. the maximal simplices of G) may appear in the pgd. However, for finite graphs the converse of (10) also holds, and we get the following characterization of finite comparability graphs (besides the criterion of E . S . Wolk): (11) Let G be a finite graph. The following statements are equivalent a ) G is the comparability graph of an order theoretical tree: b) G is connected and has a pgd each member of which is a simplex such the order of these members may be chosen arbitrarily; c) G is connected and has a pgd G1, Gr, each Gi being a simplex, such that the set of simplices of detachment contained in each Gi is a chain with respect to inclusion.
...,
The members of the pgd just correspond to the maximal chains of the order theoretical tree in question. By using the lSt decomposition theorem I could prove the following (x denoting the chromatic number) (12) x(G)
2
No =>
G u)
S(*)
for every
c
x (GI.
The question whether always G ZuS(x(G)) for x(G) infinite, can be considered as an infinite extension of Hadwiger's conjecture. Of course the conjecture is trivially wrong in the case that x(G) is a limit cardinal. However the conjecture fails in the other cases as well, hence (12) is best possible. I have been led to this by an example of R. Laver (see Cll). Let E be the set of all injective functions I of some countable ordinal 6 into N such that N 2 ( 6 ) is infinite (6 varies!). ( E , 5 ) is an order theoretical tree (here 5 is the inclusion for functions which are considered as sets). Let G denote the comparability graph of (E, 5 ) . Then x (G) = N 1, G 3uS ( N 1) as is shown by some tricks.
-
Therefore we may say: Hadwiger's conjecture fails for infinite chromatic numbers.
Simplicia1 decompositions
9.
107
INTERVAL GRAPHS
G = (V, E) is called an i n t e h v a C g h a p h if for each v be found an interval I(v) on the real line such that
..., GQ
Let us call a finite s.d. Go, ment SX a c o n b e c u t i v e s.d., if holds. Then we have
E
V
there can
with the simplices of detachX = 1, II
...,
Sx 5 GXml for all
(13) A finite graph G is an interval graph iff secutive pgd consisting of simplices.
G
has a con-
This result is closely related to the criterion of Fulkerson-Gross C31, which is however primarily formulated in the language of matrices. It is not difficult to deduce the criterion of Gilmore-Hoffman (for the finite case) from (13). (The latter says C41 that G is an interval graph iff G is triangulated and the complement E is a comparability graph (of a suitable partial order).) Much more difficulties arise if one tries to derive the criterion of LekkerkerkerBoland C91 from (13) and thus to obtain a simpler proof for the latter. It would be of great interest to get such a proof, in order to simplify and unify the theory of interval graphs. The proofs of the results so far (with the exception of section 6 ) can be found in C81, Chap. X.
10.
7
-DECOMPOSITIONS
In the final sections I will discuss some generalizations of the s.d. The separating simplices of a graph G have the property that none of them is separated by another one. Let us call T G a bepatatoh of G if there are vertices x,y of G such that T is minimally separating x,y in G, i.e.: x.T.y (G), but i x.T-t.y(G) for every t E V(T). Let 7 be a system of separators of G. 7 is called g h a d e d if 1. No T E 7 is separated by a TI E 7 ; 2. If T E 3 and TI 2 T is a separator of G, then TI E
7.
Obviously the separating simplices of graded system of separators of G.
a
G
form an example of a
Let be a graded system of separators of G; let G be finite. We now define a ? -decomposition of G quite analogously to a s.d.
R . Halin
108
of G; only instead of the simplices of attachment we allow now merely elements of
2.
Let Go, ...,Ga be induced subgraphs of G. We say that Go, (in this order) form a 7 -dccompobition of G, if 1.
ua
A =O
G A = G;
u -1
2. 3.
...,G1
U G A and G,, contain T,, properly; A =O
,
with 1 s A 11 S there is a 4. For every p , 0 such that T c G holds. (If T,, is a finite simplex, 4. follows P - A from the other conditions!). Furthermore, we call an induced subgraph H of G 7 -phime if no pair of vertices of H is separated in G be a memberof ? . A 7 -decomposition of G is a 7 -phime ghaph decompobition if each c of its members is d -prime. (Remark: It is not even required that each member of this decomposition be connected.) Then we can prove the following theorem:
7
(14) Let G be a finite graph, a graded system of separators of G. Then there exists a reduced 7 -prime graph decomposition, whose members must be exactly all maximal 7 -prime subgraphs of G. This result can be extended to infinite graphs to obtain a result similar to the 2nd decomposition theorem if each T E is required to be finite and that there is no properly ascending infinite chain T~ = T* = T~ of elements of 7 .
=,...
11. ARTICULATORS separator T of G is called an a h t i c u t a t o h if for any nonadjacent vertices x,y of T there holds P~(x8y) > ITI. It is not difficult to prove C51:
A
(15) The set of separators of G.
of articulators of
G
forms a graded system
Obviously 61 contains all separating simplices of G; the latter form a graded subsystem of a. It is also clear that, if we only take the elements of n or less elements in a graded system of separators, we again get a graded system. Hence an, the set of articulators with at most n vertices, is again graded.
Simplicia1decompositions
109
It is not difficult to prove that a 2-connected graph which is not 3-connected is either a circuit or has an articulator with 2 elements (see Tutte ClOl, theorem 11.34). Furthermore, one can prove that if G is n-connected and e is an edge such that G-e is (n-1)-separated, then G-e has at least one (n-1)-articulator. From this in C51 another successive construction of the finite 3-connected graphs is derived. 12.
SEPARATING EDGE-SETS
The main notions of sections 10, 11 can be carried over also to separating edge-sets. Let be a system of edge-sets of G. We call 7 a ghaded Aybtern of separating edge-sets of G, if is a cut of G; 1. each T E is separated by a T' E ; 2. no T E 3. if T E 7 and TI 5 T is a cut, then also T' E
a .
Then we can prove a decomposition theorem analogous to (14); here an induced subgraph of G is called -prime if no pair of its edges can be separated in G by a member of We can reduce the proof to (14) by considering the graph 6 which arises from G by inserting a vertex ve on each edge e and then show that the set of all ? := {vele E T) for T E 7 forms a graded system of separating vertex sets in 6 .
7.
T 5 E(G) is called an e d g e - a h t i c u t a t o h of G, if, for any TI 5 E(G) which separates two edges of T (in G) , necessarily IT'I > IT1 holds. Then again we find that the set of edge-articulators forms a graded system.
A cut
I do not know whether these decompositions can be carried over to matroids or hypergraphs. REFERENCES C11
J.E. Baumgartner, Results and independence proofs in combinatorial set theory, Thesis, University of California 1970.
C21
C. Berge, Graphes et hypergraphes, Paris, Dunod 1973.
C31
D.R. Fulkerson and O . A . Gross, Incidence matrices and interval graphs, Pacific J. Math. 15 (19651, 835-855.
C41
P.C. Gilmore and A.J. Hoffman, A characterization of comparabi,lity graphs and interval graphs, Canad. J. Math. 16 (19641, 539-548.
[5]
R. Halin,, Uber eine Klasse von Zerlegungen endlicher Graphen, Elektron. Informationsverarbeit. Kybernetik 6 (1970), 15-28.
It
R . Halin
110
C61
C71
R. Halin, Simplicia1 decompositions of infinite graphs. Advances in graph theory, Annals of Discrete Mathematics 2 (19781, B. Bollobds ed., p.93-109. R. Halin, A note on infinite triangulated graphs, Discrete Math.
31 (1980), 325-326.
C81
R. Halin, Graphentheorie 11. Wissenschaftl. Buchgesellschaft Darmstadt, to appear.
C91
C. Lekkerkerker and J. Boland, Remesentation of a finite sraph
by a set of intervals on the realLLine, Fundam. Math. 45-64.
51
(i962),
ClOI W.T. Tutte, Connectivity in graphs, Univ. of Toronto Press,
Toronto-London 1966.
C111 S. Wagon, Infinite triangulated graphs, Discrete Math. 183-189.
22
(19781,
Annals of Discrete Mathematics 13 (1982) 1 1 1-120
0 North-Holland Publishing Company
ACHIEVEMENT AND AVOIDANCE GAMES FOR GRAPHS FRANK H A R A R Y ~ University of Michigan Ann Arbor, U.S.A.
Consider t h e f o l l o w i n g games p l a y e d on t h e complete graph X by two p l a y e r s c a l l e d Alpha and B e t a . P green, F i r s t , Alpha c o l o r s one o f t h e edges o f X P t h e n B e t a c o l o r s a d i f f e r e n t edge r e d , and so f o r t h . A graph F w i t h no i s o l a t e d p o i n t s i s g i v e n . The f i r s t p l a y e r who can complete t h e f o r m a t i o n o f
F
i n h i s c o l o r i s t h e winner. The minimum p f o r which Alpha can win r e g a r d l e s s of t h e moves made by B e t a is t h e achievement number of
F.
In t h e corres-
ponding misere game, t h e p l a y e r who f i r s t forms F i n h i s c o l o r i s t h e l o s e r and t h e minimum p f o r which B e t a can f o r c e Alpha t o lose i s t h e a v o i d a n c e number o f F. S e v e r a l v a r i a t i o n s on t h e s e games a r e p r o p o s e d , and p a r t i a l r e s u l t s p r e s e n t e d . 0.
PRELIMINARIES.
is t h e s m a l l e s t p o s i t i v e i n The cLabbicaL hambey numbeh r ( X m , X n ) t e g e r p s u c h t h a t e v e r y 2 - c o l o r i n g o f E ( X ) , i n g r e e n and r e d P w i t h o u t l o s s of g e n e r a l i t y , must c o n t a i n a g r e e n I$,,o r a r e d Xn. Its e x i s t e n c e is a s s u r e d by t h e c e l e b r a t e d theorem of Ramsey C271. T h i s number was p r e v i o u s l y d e n o t e d by r ( m , n ) , a s f o r example i n C2,8,103. I n November 1 9 6 2 , w h i l e l i s t e n i n g t o a l e c t u r e C61 by P a u l Erdds on t h e numbers r ( m , n ) , I f o r m u l a t e d t h e " g e n e r a l i z e d ramsey number" r ( F , H ) f o r any two g r a p h s F and H , n o t n e c e s s a r i l y comp l e t e , w i t h no i s o l a t e d p o i n t s . When H = F, w e o b t a i n t h e ( d i a g o n a l ) hambey numbeh r ( F ) = r ( F , F ) . C h v d t a l and I C53 w h i m s i c a l l y d e f i n e d a " s m a l l g r a p h " a s one w i t h a t most f o u r p o i n t s and n o i s o l a t e s and w e n o t e d t h e i r ramsey numbers.
Overseas Fellow 1980-81, C h u r c h i l l C o l l e g e , U n i v e r s i t y of Cambridge. 111
F. Harary
112
Achievement and avoidance games and numbers W e o n l y c o n s i d e r g r a p h s F w i t h no isolates. The game 06 achieving F on K is written (F,Kp,+) and i s p l a y e d a s f o l l o w s . Beginning P w i t h p i s o l a t e d v e r t i c e s , t h e f i r s t p l a y e r (Alpha o r A ) draws a green l i n e segment j o i n i n g t w o of them, t h e second (Beta o r B) a r e d l i n e j o i n i n g a n o t h e r p a i r of vertices, etc. The p l a y e r ( i f any) who f i r s t c o n s t r u c t s a copy of F i n h i s c o l o r i s t h e winner.
I n t h e avoidance game i n h i s color loses.
F
-1, t h e f i r s t p l a y e r ( i f any) who forms P' These games were f i r s t mentioned i n C111.
(F,K
The achievement numbek a ( F ) is t h e minimum p f o r which Alpha can win t h e game ( F , K p , + ) r e g a r d l e s s of what moves a r e made by B e t a . The avoidance numbclr a ( F ) is t h e s m a l l e s t p f o r which Beta can p l a y i n a way t h a t f o r c e s t h e formation o f a green F r e g a r d l e s s of t h e moves made by Alpha.
.
a = a ( F ) , a = a ( F ) and r = r ( F ) I t i s obvious t h a t a s r and a 5 r , a s n e i t h e r of t h e games ( F , K r ( F ) , + ) o r (F,Kr(F),-) can end i n a draw.
Let
Problem 0. Determine a ( F ) and a ( F ) f o r v a r i o u s f a m i l i e s of graphs. This a p p e a r s h a r d even f o r trees. 1.
AVOIDING
F
ON
r(F)
POINTS.
The f i r s t graph avoidance game was c a l l e d S I M by SIMMONS i n a n o t e C29J which asked f o r t h e outcome and b e s t s t r a t e g y f o r t h e game L a t e r b o t h Simmons and o t h e r s proved by b o t h p e r s o n a l (K3,K6,-). e x h a u s t i o n and w i t h computer a s s i s t a n c e t h e f o l l o w i n g d e f i n i t i v e result. Theorem 1. I n t h e t r i a n g l e avoidance game ( K 3 , K 6 , - ) , i f both p l a y ers p l a y r a t i o n a l l y i n a way t h a t d e l a y s a l o s s as l o n g as p o s s i b l e , t h e n t h e f i r s t p l a y e r Alpha w i l l lose on t h e l a s t p o s s i b l e ( f i f t e e n t h ) edge of K6, a t which t i m e h e w i l l form t w o green t r i a n g l e s . C o n j e c t u r e 1. Theorem 1 remains t r u e when " t r i a n g l e " and r e p l a c e d by " q u a d r i l a t e r a l " and C 4 .
K3
are
'
C l e a r l y t h i s c o n j e c t u r e can b e r e a d i l y s e t t l e d w i t h computer a s s i s tance.
113
Achievement and avoidance games for graphs
2.
ACHIEVEMENT AND AVOIDANCE NUMBERS FOR SMALL GRAPHS.
-
We now give the values of a and a for those small graphs for which they have been determined by exhaustion, as mentioned in C131. 3'
2K2
a
K2 2
5
a
2
r
2
K4-e
K4
K1,3 5
K3
4'
K1,3+e
5
4 ' 5
5
6
5
?
?
3
5
5
5
5
6
5
?
?
3
5
5
6
6
6
7
10
18
Shader and Cook C273 gave a strategy for Beta to force Alpha to lose the avoidance game (KlI3+e, K6,-). They thus showed that a(F) < r(F) when F = K1,3+e. UP2 (Unsolved Problem 2 ) for both K4-e and K4. 3.
Complete Table 2 by determining
a
and
-
a
ECONOMICAL G R A P H S .
When p = r(F) , the smallest possible number of green moves required to form F in the achievement game (F,Kp,+) is called the move numbelr, m(F) The d i z e of F is the number of lines, q(F) We call graph F economicaC if m(F) = q(F). The economical small graphs are K2, P3, 2K2, P4, K1,3 and K1,3+e. It appears that all paths Pn are economical.
.
Call green hence Clark
.
F u ~ Z i m a Z e ~economical y if for some p, Alpha can make a F in only q(F) moves. It is easy to see that all trees and all forests are ultimately economical, as noted by Curtis (unpublished)
.
UP3
Which graphs F are economical? (As this depends on the previous determination of a(F), it is more reasonable to ask which trees and which forests are economical.) Which graphs are ultimately economica1? 4.
INEQUALITIES.
Conjecture 4 . will have a
5
-It seems intituitively obvious that for all F a. But there is no proof or disproof as yet.
we
If this is true as we believe, then we would be able to partition all graphs F into the following four classes.
114
I. 11.
F. Harary
-= -a < a =
a = a
r,
as for
C4
a =
r,
as for
K
r,
which I think is an empty class.
111. a <
IV. a < a < r, also by K4-e. 5.
1,3
with with
r = 6. a = 5, r = 6.
which is probably satisfied by
K4
and perhaps
FIRST PLAYER ACHIEVEMENT.
B6la Bollob6s suggested a simpler achievement game in which only Alpha aims to form a green F, and the goal of Beta is to try to stop him. Question 5. For which graphs F does the achievement number under these new rules equals the previous number a(F)? Bollobds and Papaioannou C3J have determined these achievement numbers for the paths Pn (the answer being n unless n = 4 ) , the "bars" nK2, and other families of graphs. Perhaps surprisingly, they proved that the hamiltonian cycle Cn can be achieved on Kn when n t 8. (See also Papaioannou C26J for a weaker result in this direction.) The corresponding avoidance games, where only Alpha tries to avoid forming a green F, have not yet been studied. 6.
GAMES ON NONCOMPLETE GRAPHS.
has been defined both by Erdbs, The d i z e hambCy numbeh :(F) Faudree, Rousseau and Schelp C71 and independently by Miller and me C21J as the minimum size q(G) of a graph G such that G -+ F, meaning that every 2-coloring of E(G) must have a monochromatic F. This suggests achievement and avoidance games played on graphs G which are not complete, which are denoted similarly by (FIG,+) and (FiGr-). In view of the fact that F does not necessarily require r(F) points to be achieved on a complete graph, it is also meaningful to consider playing (F,H,+) for graphs H f . F . Defining the d i z e achievement and a v o i d a n c e numbehb as expected, several questions suggest themselves. Open Question 6. Determine the size achievement and avoidance numbers of small graphs, trees, and several families of graphs.
115
Achievement and avoidance games f o r graphs
7.
INDUCED SUBGRAPHS.
We write
F c G
when
i s a subgraph of
F
i s an induced subgraph. spanning s u b g r a p h s
u E2
E = El
of
factorization
W e now w r i t e u G2,
G = G1
F < G
and
G
A s u s u a l , a 6 a c t o t r i z a t i o n of
when
i n t o two
G
i s determined by a p a r t i t i o n
G1 u G2
.
In these t e r m s , G F i f f o r every G = G1 u G 2 , w e have F c G1 o r F c G 2 . E = E(G)
+
G
F
w e have
-f
t o mean t h a t f o r e v e r y f a c t o r i z a t i o n F < G
F < G2.
or
A s p e c i a l c a s e of a
b a s i c r e s u l t of Nesetril and Rddl C25J p r o v e s t h a t f o r e a c h there e x i s t s
F
such t h a t
G
G
+
F ;
F,
see a l s o Graham, R o t h s c h i l d
and Spencer C81. T h i s f a c t s u g g e s t s s e v e r a l q u e s t i o n s i n v o l v i n g achievement and avoidance games.
C a l l GI
t h e induced ramsey numbetr some
G
with
p
points,
a c h i e v e m e n t numbetr
ai(F)
smaller than ri(F)
G2
p1 < p2.
if
a s t h e minimum
.
p
Define
such t h a t f o r
+ F Similarly t h e induced is t h e minimum o r d e r of a g r a p h H
G
t h a t Alpha can c o n s t r u c t a g r e e n induced subgraph a l t e r n a t e l y c o l o r i n g edges of
H
less of t h e moves made by B e t a .
F
g r e e n and r e d a s f o r
of
such (by
H KP)
The c o r r e s p o n d i n g number
regard-
is
&(F)
defined a s expected. Open Q u e s t i o n 7. ri(F), ai(F)
For each graph
and a i ( F ) .
F,
d e t e r m i n e t h e induced numbers
( W e have a l r e a d y found most o f t h e s e
numbers f o r s m a l l g r a p h s and s t a r s , and a r e s t u d y i n g t r e e s and various families of graphs.)
8.
TREES.
w e l l known adage i n graph t h e o r y s a y s t h a t when a problem is new and d o e s not reveal i t s secret r e a d i l y , it s h o u l d f i r s t be s t u d i e d f o r trees where it w i l l g e n e r a l l y b e easier t o h a n d l e . A companion
A
motto u r g e s t h a t e a c h q u e s t i o n f o r g r a p h s a l s o b e s p e c i a l i z e d t o b i p a r t i t e g r a p h s and g e n e r a l i z e d t o d i r e c t e d g r a p h s .
Thus t h i s sec-
t i o n d i s c u s s e s trees and t h e n e x t two d e a l w i t h b i g r a p h s and d i g r a p h s .
I
C o n j e c t u r e 8.
The minimum v a l u e of
a(T)
among a l l trees
n i s r e a l i z e d when T = P the path. n' is a t t a i n e d f o r T t h e s t a r K ( 1 , n-1).
order
T
The maximum of
of a(T)
R e l a t e d q u e s t i o n s a s k f o r a l i s t of a l l t h e n - p o i n t trees f o r which
i s m i n i m u m o r maximum, f o r t h e s e t o f a l l v a l u e s of a ( T ) , and The c o n j e c t u r e when c o r r e s p o n d i n g r e s u l t s f o r t h e a v o i d a n c e number. a(T)
116
F. Harary
a(T)
is replaced by
r(T)
seems t o be t r u e .
We remark t h a t both trees of o r d e r 4 have a ( T ) = 5 t h r e e trees of o r d e r 5 have a ( T ) = 5, 5 and 7. 9.
and t h a t t h e
BIGRAPHS.
When both F and G are s p e c i a l i z e d t o b i p a r t i t e graphs i n Section 6 on noncomplete graphs, w e g e t achievement and avoidance games on bigraphs. One such game was analyzed i n C14 3 , namely ( K 1 , 4 t K5,5 I+), where it w a s shown t h a t t h e move number is 8 . I t i s a l s o easy t o v e r i f y t h a t t h e move number of (C4,K4,4,+) is 7 . The a s s o c i a t e d ramsey theory f o r bigraphs, proposed i n C123, was developed by Harborth, Mengersen and m e C193 where w e defined t h e b i p a h f i t e hambey b e t of a bigraph F a s t h e c o l l e c t i o n of a l l ordered p a i r s (m,n) with m 5 n such t h a t K -+ F but m,n Achievement games f o r bigraphs are being Km-l,n' Km,n-l 4 F i n v e s t i g a t e d by Exoo and myself. B i p a r t i t e achievement (and analogously avoidance) numbers can e i t h e r be defined a s t h e minimum n such t h a t K n I n y F (where t h e n o t a t i o n as expected m e a n s t h a t Alpha can make a green F playing on K n I n ) o r a s t h e minimum p = m + n such t h a t KmIn 7 F
.
.
Open Question 9 . Determine b i p a r t i t e achievement and avoidance numbers f o r trees ( e s p e c i a l l y stars and p a t h s ) , f o r even c y c l e s , and f o r o t h e r f a m i l i e s of b i p a r t i t e graphs. 10.
DIGRAPHS.
The digraph PG of a graph G is d e f i n e d t o have t h e same p o i n t set V(G) with each edge of G replaced by a symmetric p a i r of a r c s . For a given digraph D , again with no i s o l a t e s , t h e hambey numbeh r ( D ) , i f any, i s t h e minimum p such t h a t gKP + D , where t h e arrow now r e f e r s t o t h e c o l o r i n g of a r c s . I t was independently and simultaneously discovered by Bermond C 1 3 and by H e l l and m e C203 t h a t r ( D ) e x i s t s if and only i f D i s a c y c l i c . I t is easy t o see t h a t f o r t h e c y c l i c t r i p l e
achievement game (C3, P K 5 , + ) t h a t t h e move number is 5.
even though
C3
C3, Alpha can win t h e is n o t a c y c l i c and
Obviously no digraph containing a symmetric p a i r of arcs can be achieved. However, Andrew Thomason has proved t h a t any asymmetric digraph ( o r i e n t e d graph) D can be achieved on PKP f o r some p.
Achievement and avoidance games for graphs
117
The corresponding statement for avoidance is still open. Question 10. What are the achievement and move numbers for directed paths and directed cycles?
11. MULTIPLICITY GAMES. The lrambey m u L t i p t i c i t y of F, written R(F), is the minimum possible number of monochromatic copies of F in any 2-coloring of Kn where n = r(F). This concept was introduced in C233 where R(F) was determined exactly for all small F except K4. A survey of results subsequently derived about R(F) is given by Burr and Rosta C43.
A m u e t i p e i c i t y game for F is played like an achievement (or an avoidance game) except that the winner (or loser) is not the first player to complete a monochromatic F, but the player who completes in his color the most (least) copies of F. Open Question 11. Let F be a graph with n = r(F). What are the outcomes of the F-multiplicity achievement and avoidance games played on Kn? Of course similar questions can be asked for digraphs, bipartite graphs, induced subgraphs, and so forth. 12. CONCLUDING REMARKS. Many other achievement and avoidance games suggest themselves naturally. These include games designed from theorems C151 not only about graphs as in the present communication, but also concerning number theory, group theory, Latin squares, combinatorial designs and finite geometries. All the games about graphs presented above involve coloring the edges using two colors. This procedure can be modified in two ways. First, the vertices can be colored instead, as pointed out by J. Ngsetzil and V. R6dl (verbal communication). Second, the number of colors may be different from two. When only one color is used, we obtain games on n points, with objective graph F having order at most n, in which both players A and B use white chalk and the first player to complete F either wins or loses. When the identification of subgraph F is an NP-hard problem, e.g., if F is a hamiltonian cycle Cn, then these have been called "hard games'' by M. Grdtschel (verbal communication).
F. Harary
118
It is well known C9J that the 3-color ramsey number of a triangle, written r(K3,3), is 17. Nevertheless it has been found empirically that when three players A , B and r play triangle achievement (using blue, green and red chalk without loss of generality) on only seven points, the result does not terminate in a drawn game. Other graphical concepts also provide interesting material for games, such as convexity C22,171, connectivity, colorability, and so forth. Several combinatorial settings which are not explicitly graph theoretic are eminently appropriate for the design and play of achievement and avoidance games. Thus there are games involving chess pieces C18,241, games with square-celled animals C163, and games on triangular animals and polyhexes being explored with Heiko Harboth. REFERENCES. 1.
J.-C. Bermond, Some ramsey numbers for directed graphs.
M a t h . 9 (1974) 313-321.
DiAchete
2.
B. Bollobds, ExthemaL Ghaph Theohy.
3.
B. Bollobds and I. Papaionnou, Achievement numbers for graphs, to appear.
4.
S.A.
5.
V. Chv6tal and F. Harary, Generalized ramsey theory for graphs 11: Small diagonal numbers. Phoc. Ameh. M a t h . S a c . 32 (1972) 389-394.
6.
Academic Press, London (1978).
Burr and V. Rosta, On the ramsey multiplicities of graphs Problems and recent results. J.Ghaph Theohy 4 (1981) 347-361.
--
P. ErdBs, Applications of probabilistic methods to graph theory.
A Seminah o n Ghaph Theohy (F. Harary, ed.), Holt, New York (1967)
60-64.
7.
P. ErdBs, R.J. Faudree, C.C. Rousseau, R.H. Schelp, The size ramsey number. P e h i o d i c a M a t h . Hungah. 9 (1978) 145-161.
8.
R.L. Graham, B. Rothschild, J. Spencer, R a m ~ e y T h e o h y , Wiley, New York (1981).
9.
R.E. Greenwood and A.M. Gleason, Combinatorial relations and chromatic graphs. Canad. J. M a t h . 7 (1955) 1-7.
10. F. Harary, Ghaph Theohy.
Addison-Wesley, Reading (1969).
11. F. Harary, Recent results on generalized samsey theory for graphs. Ghaph Theohy and A p p L i c a t i o n A . Springer Lecture Notes 303 (1972) 125-138. 12.
F. Harary, The foremost open problems in generalized ramsey theory. Phoc. Fi6,th B h i t i A h C o m b i n a t o h i a l Conbehence. Utilitas, Winnipeg (1976) 269-282.
119
Achievement and avoidance games for graphs
13.
F. Harary, G e n e r a l i z e d ramsey t h e o r y f o r g r a p h s I t o X I I I . T h e T h e o h y and A p p L i c a t i o n 06 Ghaphb (G. C h a r t r a n d e t a l . e d s . ) Wiley, New York ( 1 9 8 1 ) , t o a p p e a r .
14.
F. Harary, A n achievement game on a t o r o i d a l b o a r d . PhoceedingF i h A t PoLibh S y m p O A i U m o n Ghaph t h e o h y (M. SysZo, e d . ) S p r i n g e r
L e c t u r e Notes, t o a p p e a r . 15.
F. H a r a r y , Achievement and avoidance games d e s i g n e d from theorems. R e n d i c o n t i Sem. Mat. F i b . MiLano (19821, t o a p p e a r .
16.
F. H a r a r y , Achieving t h e Skinny animal.
17.
F . H a r a r y , Convexity i n g r a p h s : Achievement and a v o i d a n c e games. AnnaLd D i b c h e t e Math. (1982) , t o a p p e a r .
18.
F. H a r a r y , Achievement and a v o i d a n c e games i n v o l v i n g c h e s s p i e c e s . J . RecheationaL M a t h . , t o a p p e a r .
19.
F. H a r a r y , H . H a r b o r t h , I . Mengersen, G e n e r a l i z e d ramsey t h e o r y f o r g r a p h s X I I : B i p a r t i t e ramsey sets. GeaAgow Math. J . 2 2 (1981) 31-41.
20.
F. Harary and P. H e l l , G e n e r a l i z e d ramsey t h e o r y f o r g r a p h s V: The ramsey number of a d i g r a p h . BULL. London Math. S O C . 6 ( 1 9 7 4 ) 175-182; 7 (1975) 87-88.
21.
F. Harary and 2 . M i l l e r , G e n e r a l i z e d ramsey t h e o r y V I I I : The s i z e ramsey number of s m a l l g r a p h s . S t u d i e d P n Puhe Mathematicb i n Memohy 06 Paue Tuhhn. T o a p p e a r .
22.
23. 24.
Euheha (19821, t o a p p e a r .
F. Harary and J . Nieminen, Convexity i n g r a p h s .
Geomethy (1981) , t o a p p e a r .
J . Ui&dehentiaL
F. Harary and G. P r i n s , G e n e r a l i z e d ramsey t h e o r y f o r g r a p h s I V : t h e ramsey m u l t i p l i c i t y of a graph. N e t w o h h b 4 (1974) 163-173. F. Harary and A.
Thomason, Covering w i t h i n d e p e n d e n t k n i g h t s .
J . Recheati0na.t M a t h . , t o a p p e a r .
25.
J. N e g e t z i l and V.
26.
A. Papaioannou, A Hamiltonian game.
27.
F.P. Ramsey, On a problem of f o r m a l l o g i c . S O C . 3 0 (1930) 264-286.
28.
L.E. Shader and M. Cook, A winning s t r a t e g y f o r t h e second p l a y e r i n t h e game T r i - t i p . Proc. Tenth S . E . condehence on Computing, Combinatohicb and Ghaph Theohy. U t i l i t a s , Winnipeg (1980).
29.
G.J.
Radl, The ramsey p r o p e r t y f o r g r a p h s w i t h f o r b i d d e n complete s u b g r a p h s . J . Combin. Theohy B20 ( 1 9 7 6 ) 243-249.
ed.)
66.
, Annals
of Discrete Math.
Simmons, The game of SIM.
Gkaph Theohy (B. Bollobbs,
(1982) , t o a p p e a r .
Phoc. L o n d o n Math.
J . R e c h u t i o n a L Math. 2 ( 1 9 6 9 )
This Page Intentionally Left Blank
Annalsof Discrete Mathematics 13 (1982) 121-138 0 North-Holland Publishing Company
EMBEDDING INCOMPLETE LATIN RECTANGLES A.J.W. HILTON University of Reading England
Cruse gives necessary and sufficient conditions for embedding saturated r x s latin rectangles on n symbols inside latin squares on t symbols, generalizing a theorem of Ryser. We give another proof of this result as well as a number of analogues and generalizations. 1.
INTRODUCTION
is an An r x s pahtiaL L a t i n hectangee R o n AymboLA ulI...,un r x s matrix in which some of the cells contain a symbol chosen from ~ull...,un~, whilst other cells may be empty, and which satisfies the conditions: each symbol occurs at most once in each row and at most once in each column. An r x s partial latin rectangle on ul,...,un is b a t u h a t e d if none of ul,...,un can be placed in any of the empty cells without violating the above condition. If there are no empty cells then we call R an incompLete r x s Latin hectangte. Let NR(ui) denote the number of times that the symbol di occurs in R. Ryser El23 proved the following theorem about completing latin rectangles. Theorem 1.1
An r x s incomplete rectangle R on symbols be completed to a t x t latin square on the same symbols if and only if U ~ , . . . ~ U ~can
NR( ui ) r r + s - t
(1 5 i
S
t).
Let DR denote the number of empty cells in a partial r x s latin rectangle R and let D p I Dc be the number of empty cells in r o w p and column c of R for 1 s p 2 r and 1 5 c S 6. Cruse C71 extended Ryser's theorem to the following. 121
A .J. W.Hilton
122
Theorem 1.2 A saturated r x s partial ,un can be completed to a symbols u1, if and only if the on symbols ul,...,ut hold : (i) DR t (r + s - t) (t - n), (ii) NR(ai) t r + s - t (iii) D 5 t - n and Dc 5 t - n
. ..
P
latin rectangle R on t x t latin square T following three conditions
(1 s i (1 S p
5
5
n), r, 1 5 c
5
s).
If t = n then Theorems 1.1 and 1.2 coincide. We give a new simpler proof of Theorem 1.2. If R is not saturated then the conditions of Theorem 1.2 are sufficient but not necessary for R to be completable to a t latin square T.
x t
If n = r = s then condition (i) becomes DR 2 (2n-t)(t-n), an easily sketched quadratic function of t, with leading term -t2 Clearly, for some R, if t is just greater than n the inequality may be satisfied and R can be completed to T, then, for somewhat larger values of t, the inequality may not be satisfied, but then for yet larger values of t the inequality may be satisfied again.
.
Later in this paper we discuss symmetrical analogues of this theorem and generalizations to (p, q, x) - latin rectangles. Whilst each theorem and proof has its own individual features, there is usually a lot of similarity between its proof and our proof of Theorem 1.2, and so we generally shorten the account, where possible, by saying, for example, that our proof of (x) in Theorem X is very similar to the proof of (y) in Theorem Y. In the analogues and generalizations it usually remains true that the conditions listed are sufficient but not necessary for the embedding if R is not saturated, and also one of the conditions is that DR has to be at least as large as an expression which is quadratic in t with leading term -t2
.
In Section 2 we collect all the results on edge-colourings of graphs we need to use. In Section 3 we prove Theorem 1.2. In Section 4 we consider symmetric analogues of Theorem 1.2. In Section 5 we consider generalizations of Theorem 1.2 to (p, q, x) - latin rectangles and in Section 6 we consider symmetric (p, p, x)- latin squares. Finally in Section 7 we consider an analogue to the case where the diagonal of T is almost completely specified.
Embedding incomplete latin rectangles
2.
123
EDGE-COLOURING THEOREMS
edge-colouhing of a multigraph G is a map f : E(G) + {Cl, C2,.. 3 where (C1, C2,...) is a set of colours. It is a p h o p e h edgecolouring if f (el) f f (e2) whenever el and e2 have a common vertex. The least number of colours for which G has a proper edge-colouring is denoted by x’(G). Let A = A(G) be the maximum degree of G and let m = m(G) be the maximum multiplicity of an i.e. the greatest number of edges joining any pair of edge vertices. For an edge-colouring of G, for each v E V(G), let Ci(v) be the set of edges incident with v of colour Ci and, for u, v E V(G), u $. v, let Ci(u, v) be the set of edges joining u, v coloured Ci.
An
-
An edge colouring is e q u i t a b l e if IICiCV)l
-
ICj(V)l
I
1
and it is balanced if also (V u, v
E
V(G), V i, j).
de Werra (C141, C171, C181) showed that, for any k, a bipartite graph has a balanced edge-colouring with k colours. If k is even then it is known (see C201) that any multigraph G A(G)1T co ours in which each vertex has at has an edge-colouring with \ most k edges of each colour on it. (This is an easy consequence of a theorem of Petersen C111). The well known theorem of Vizing C131 states that A (G) 5 x*(G) S h(G) + m(G). By identifying colours it follows that if k is odd and G is a simple graph then G can be edge-coloured with k colours so that at most edges of each colour are incident with each vertex. A graph is said to be of Class 1 if x’(G) = A(G), of Class 2 otherwise.
1-
An edge-colouring is equalized if the number of edges of colour Ci by at most 1 differs from the number of edges of colour C j (V i, j). Any of the types of colouring considered here can be equalized by a very simple argument (McDiarmid ClOl, de Werra C161). 3.
OUR PROOF OF THE PROTOTYPE THEOREM
In this section we prove Theorem 1.2, the prototype for the remaining theorems of this paper.
A.J.W.Hilton
124
Proof of Theorem 1.2 1. Necessity. (i) Each of the symbols U ~ + ~ , . . . , U must ~ be placed sufficiently often in R so that the Ryser condition
-
NR (u.) t (n + 1 s i I; t i 2 r + s is satisfied. The number of such symbols is (t-n). Consequently, the number of empty cells must be at least (r+s-t)(t-n). (ii) Since R is saturated, ul,...,un must each satisfy the Ryser condition. (iii) The vacant cells in any row or column have to be filled with the symbols LY~+~,...,U~, so there must be at most t-n of them.
2.
Sufficiency.
Suppose (i), (ii) and (iii) hold.
Consider the
bipartite graph G on vertices pl, . . . , p n, cl,...,cn formed by joining p i to c by an edge if and only if cell (i, j) is j empty. The graph has at least (r+s-t)(t-n) edges (by (i)) and has maximum degree I; t n (by (iii)). By Kdnig's theorem, it can be properly edge-coloured with t-n colours U~+~,...,U~, and by the theorem of McDiarmid and de Werra this proper edge-colouring can be equalized, and then there will be at least r + s - n edges of each colour. We can then put uk into cell (i, j) whenever picj is coloured u i . Then the Ryser condition will be obeyed by all the symbols, so, by Ryser's theorem, the embedding can be carried out.
-
4.
SYMMETRIC ANALOGUES OF THEOREM 1.2
In this section we make use of the following symmetric analogue of Theorem 1.2, due to Cruse C63. Theorem 4.1 A symmetric incomplete r x r latin square R on t symbols ul,...,ut can be completed to a symmetric t x t latin square on the same symbols if and only if NR(ui)
2
2s
-
t
(1 s i
and NR(ui)
t*(mod 2)
for at most
t
I;
t)
-r
symbols
ai.
Given a symmetric r x r partial latin square R on n symbols let G be the graph formed as follows: Let the vertices be vl, vr and let vi and v be joined by an edge if cells j (i, j) and (j, i) are empty. Let G* be the graph formed from G by adjoining one further vertex v* and the edge v*vi whenever
...,
125
Embedding incomplete latin rectangles
cell (i, j) is empty (1 s i s r). Let G’ be the graph formed from G by adjoining a vertex wi and an edge viwi for each i, 1 s i s r, such that cell (i, i) is empty. Recall that a graph Class 2 otherwise.
is of Class 1 if
G
A(G) = x’(G)
and is of
The results we obtain for the symmetric case are not so satisfactory as those of Theorem 1.2 because the necessary and sufficient conditions are in terms of G* or G’. However fairly good numerical conditions which are sufficient for the embedding to be possible are given in the corollaries. First we consider the case when
t
is odd,
Theorem 4.2 Let t be odd. A saturated symmetric r x r partial latin square on n symbols ul,...,un can be embedded in a symmetric t x t latin square T on t symbols ul,...,ut if and only if (i) DR 2 (2r-t)(t-n) (ii) NR(ui) 2 2r t (1 s i s n), (iii) Each symbol from ~ul,...,un~ occurs at most once on the diagonal of R, (iv) either A ( G * ) < t - n or A ( G * ) = t - n and G* is of Class 1.
-
Proof. 1. Necessity.
.
(i), (ii) The proof of these is as in the proof of Theorem 1.2. (iii). If T is symmetric and t is odd each symbol must occur exactly once on the diagonal of T. Therefore, in R, each symbol ull...,un occurs at most once on the diagonal. (iv). If R is embedded in T then we can obtain a proper edge-colouring of G* with colours U ~ + ~ , . . . , U as ~ follows: edge vivj (i f j) is coloured with al; if symbol uk is placed in cell (i, j) and edge v*vi is coloured with al; if uk occurs in cell (i, i). The edge-colouring is proper because (i) no two cells in the same row contain the same symbol and (ii) the symbols occurring in the diagonal cells of R are all different. The number of colours used is t - n. Therefore, either A ( G * ) = t - n and G* is of Class 1 or A ( G * ) < t n.
-
A .J. W.Hilton
126
2.
Sufficiency.
Suppose (i), (ii), (iii) and (iv) hold. G* can be properly edgecoloured with t - n colours U ~ + ~ , . . . , U ~ . This colouring can be 1 equalized, and then there will be at least 2(2r - t + 1) edges of each colour. Then, if i j, uk is placed in cells (i, j ) and (j, i) if edge v.v is coloured with u i and in cell (i, i) if = I Then NR(ai) 2 2r - t for each i, edge v*vi is coloured u<. 1 s i 5 t. Furthermore, no symbol will occur more than once on the diagonal of R. By Theorem 4.1, therefore, R can be completed to a symmetric latin square. Corollary 4.3.
Let
t
square on n symbols t x t latin square on
be odd. Ul I
t
. .. u
A symmetric
r
x
r
partial latin
can be embedded in a symmetric symbols if I
(i) DR 2 (2r-t)(t-n), (ii) NR(ui) z 2r - t (1 s i s r), (iii) Each symbol from {ul,...,an~ occurs at most once on the diagonal of R, (iv)
D
< t
P
-
n
(1
5
p s r).
If t is even, the result we obtain is even less pleasant. Let y be the number of symbols from {ulr..., un} which occur an odd number of times on the diagonal of R. Let the edges of G’\G be called p e n d a n t e d g e b of G’. In a proper edge-colouring of G‘ in which is one of the colours, let e(ai) be the number of edges of G coloured and let p(uf) be the number of pendant edges coloured
ui
ui
U’
i‘
Theorem 4.4. latin square symmetric t only if (i) (ii) (iii) (iv)
Let t be even. A saturated symmetric r x r partial R on n symbols ul,...,un can be embedded in a x t latin square T on t symbols ul,...,ut if and
DR 2 (2r-t)(t-n), NR(ui) 2 2r - t, y s t - n, the edges of G’ can be properly edge-coloured with
t
-
n
colours, say
U ~ + ~ , . . . , U ~ such ,
that
(a) 2e(ai) + p(ui) 2 2r - t (n+l 5 i 5 t), (b) at most t - n - y colours occur an odd number of times on the pendant edges.
127
Embedding incomplete latin rectangles
Proof.
1. Necessity. (Iand ) (ii). The proofs of these are as in the proof of Theorem 1.2. (iii). In T each symbol occurs an even number of times on the diagonal. Therefore, since R is saturated, each symbol from al,~..,un which occurs an odd number of times on the diagonal of R must occur at least once on the diagonal cells of T outside R, and so y 5 t-n. (iv). If R is embedded in T then we can obtain a proper edgecolouring of G’ with colours U ~ + ~ , . . . , U ~ as follows: edge if symbol ak occurs in cell vivj(i f j ) is coloured with a; (i, j) and edge viwi is coloured with a< if uk occurs in cell (i, i). Each edge is coloured since R is saturated, and the edgecolouring is clearly proper. Since NR ( 0i. ) > 2r t (by Theorem 4.1) it follows that
-
2e(oi) + p(a;)
t
2r
-
t
(n+l
5
i
2
t).
Since the number of symbols which occur on the diagonal of R an odd number of times before and after R is filled is y and at most t - n respectively (by Theorem 4.11, the number of colours occurring on an odd number of pendant edges is at most t n y.
- -
2. Sufficiency. Suppose (i), (ii), (iii) and (iv) hold. Let the empty cells of R be filled as follows: if edge vivj (i j) is coloured al;, then place symbol uk in cells (i, j) and (j, i), and if edge viwi is coloured a i then place symbol ak in cell (i, i). Then R is filled symmetrically. Condition (iv(a)) implies that NR (ai ) 2 2r - t (n+l s i 5 t) and condition (iv(b)) implies that at n colours occur an odd number of times on the diagonal of most t R. By Theorem 4.1, therefore, R can be embedded in a symmetric latin square of side t.
-
Corollary 4.5. Let t be even. A symmetric r x r partial latin square on n symbols ulr...,u n can be embedded in a symmetric if t x t latin square on t symbols ul,...,ut (i) (ii) (iii) (iv)
DR 2 (2r-t) (t-n), NR(ui) 2 2r - t y 5 t-n D s t - n - 2 P
(1 s i s r),
(1 s p
5
r).
A.J. W.Hilton
128
Proof. From G’ form a graph G” by removing as many pairs viwi and v.w of pendant edges as possible until there is at most one 1 1 pendant edge left, and replacing these pairs by edges vivj. Then may contain some double edges but no triple edges. By Vizing’s G” theorem, this graph can be edge-coloured with t - n colours and then this edge-colouring may be equalized in such a way that, if there is a solitary pendant edge e, then no colour class contains more edges than the colour class in which e lies. This edgecolouring induces a corresponding edge colouring of G’ which satisfies the conditions of Theorem 4.4, so R can be embedded in a symmetric latin square of side t. 5.
EMBEDDING PARTIAL (p, q, X)
-
-
LATIN RECTANGLES.
...,
An r x 8 p a h t i a l (p, q, x ) l a t i n a e c t a n g l e on symbols al, ‘n is an r x s matrix in which each cell contains at most x symbols (counting repetitions), each symbol occurring at most p times in each row and at most q times in each column. An r x s i n c o m p l e t e (p, q, x) l a t i n h e c t a n g l e on symbols al, ,an is an r x s partial (p, q, x) latin rectangle in which each cell contains precisely x symbols (counting repetitions). If each symbol occurs exactly p times in each row and exactly q times in each column and each cell contains exactly x symbols (counting repetitions) then the (p, q, x) latin rectangle is called e x a c t and in that case it is shown in C21 that, for some positive rational number t, r = pt, s = qt and n = xt. We shall assume throughout that t has the property that xt, pt and qt are all integers. We may require that no symbol in any cell is repeated, in which case the latin rectangle is said to be w i t h o u t h e p e t i t i o n ; other(p, q, x) wise we call it a (p, q, x)- latin rectangle, h e p e t i t i o n p e h m i t t e d . Please note that the terminology here is a slight variation of that in C21.
-
...
-
-
-
r X s partial (p, q, x)- latin rectangle, repetition permitted, on symbols al,...,an is datuhated if, for each cell c containing fewer than x symbols (counting repetitions), each of the symbols al,. ..,an either occurs p times in the row in which c lies or occurs q times in the column in which c lies.
An
r x s partial (p, q, X I - latin rectangle without repetition on symbols al, an is datuhated if, for each cell c containing fewer than x symbols, each of the symbols al, an either occurs in the cell c, or occurs p times in the row in which c
An
...,
...,
129
Embedding incomplete latin rectangles
lies or occurs q
times in the column in which
c
lies.
First we consider the problem of embedding r x s partial (p, q, x)- latin rectangles, repetition permitted. The following theorem C31 is needed. Theorem 5.1. A n r x s incomplete (p, q, x)- latin rectangle R, repetition permitted, on symbols ul, uxt can be embedded in an exact (p, q, XI- latin rectangle, repetition permitted, on the same symbols if and only if
...,
NR(ui)
2
qs + pr
-
pqt
(1 5 i c t).
Let the deficiency of the cell in row p and column c be x minus the number of symbols (counting repetitions) in the cell; let S r it be denoted by D Let D = 1 Dpc, Dc = 1 Dpc and PC r c=l p=l DR = 1 Dp. We now give the following generalization of Theorem 1.2.
p=l
Theorem 5.2. A saturated partial (p, q, XI- latin rectangle R un (repetition permitted) of size r x s on n symbols ul, can be embedded in an exact (p, q, x)- latin rectangle (repetition permitted) on xt symbols ul,. ..,uXt if and only if
...,
(i)
DR
(ii)
NR(ai)
(iii) Dp
(qs+pr-pqt)(xt-n),
2
5
2
qs + pr
p(xt-n)
-
and
pqt D,
(1 s i S
q(xt-n)
5
n),
(1
5
p
5
r, 1
5
c
S
s).
We omit the proof as it is very similar to that of Theorems 1.2. and 5.4. We next consider the problem of embedding r x s partial (p, q, x)latin rectangles without repetition. We use the following theorem C31. Theorem 5.3. A n incomplete r x s (p, q, XI- latin rectangle R without repetition on symbols ul, uXt can be embedded in an exact (p, q, x)- latin rectangle without repetition on the same symbols if and only if the following four conditions are obeyed:
...,
A.J. W.Hilton
130
(i)
NR(ui)
qs + pr
-
pqt
(ii)
NR(ui) s qs + pr
-
pqt + (pt-s)(qt-r)
2
(1 s i s Xt),
(1 S i
(iii) N (a,)
2
p + s
- pt
for all rows
(iv)
2
q + r
-
for all columns c
P
Nc(ui)
qt
p
S
xt),
R (1 s i
of
of
5
xt),
R 0 Si Sxt).
Here N (ai) and Nc(ui) are the number of times ai occurs in P row p and column c respectively. The result we deduce is the following, Theorem 5.4. A saturated partial (p, q, XI- latin rectangle R without repetition of size r x s on n symbols al, un can be embedded in an exact (p, q, XI- latin rectangle T without repetition on xt symbols 618 uXt if and only if the following eight conditions are obeyed:
...,
...,
(i)
DR
2
(qs+pr-pqt)(xt-n),
(ii)
DR
5
(qs+pr-pqt+(pt-s)(qt-r)) (xt-n),
(iii) NR(ui)
2
qs + pr
(iv)
NR(ui)
s
qs + pr
(v)
Dp s p (xt-n)
and
Dc (vi)
and
and
D
P
(1 s i s n),
pqt
pqt + (pt-s)(qt-r) (1 s i s n),
2
(p+s-pt)(xt-n)
2
(q+r-qt)(xt-n)
(vii) Np(ai)
2
Nc(Ui)
2
PC
s xt
- pt q + r - qt
p + s
-
n
p
of
R),
(V columns c
of
R),
(V rows
p
of
R),
(V columns
c
of
R),
(
q(xt-n)
Dc
(viii) D
-
(V
rows
p
Y rows
of R, 1
5
i s n)
(V columns c of R, 1 s i -s n),
(V rows
p
and columns c
of
R).
1. Necessity. (i), (iii) and (v) are just slight variations of the conditions in Theorem 1.2. (iii), (iv) and (vii). Since R is saturated, symbols ul, must satisfy the Ryser-type conditions of Theorem 5.3.
...,un
(ii) and (iv). Each of U ~ + ~ , . . . , Umust ~ ~ satisfy the Ryser-type conditions of Theorem 5 . 3 , so the deficiences DR, Dp and Dc must be sufficiently large for this to be possible.
Embedding incomplete latin rectangles
131
(viii). The deficiency of each cell has to be filled with the symbols U ~ + ~ ~ . . . , Unone ~ ~ ,being used more than once in any cell. 2. Sufficiency. Suppose conditions (i)- (viii) all hold. Consider the bipartite graph G on vertices PlI...,~,, cl,...,cs formed by joining pi to c by k edges if the deficiency of the cell j (i, j) is k. Then, by (i) and (ii I 2
IE(G)I
and, by (v) and (vi), (p+S-pt) (xt-n)
5
D
P
5
p(xt-n)
(V rows
p
of
R)
and (q+r-qt)(xt-n) 5 Dc s q(xt-n) By de Werra's theorem, G has a balanced edge-colouring with xt-n and this balanced colouring can be equalized. colours u ; + ~ , pqt Then the number of edges of each colour will at least qs + pr and at most qs + pr - pqt + (pt-s)(qt-r); furthermore, at each vertex pi the number of edges of each colour will be at least P + S - pt and at most p, and at each vertex cj the number of edges of each colour will be at least q + r - qt and at most q.
...,
-
We also note that the number k of edges joining pi to c is j the deficiency of the cell (i, j) and so, by (viii), is at most xt - n, so each parallel edge of a multiedge receives a different colour
.
We then put u k into cell (i, j) whenever there is an edge joining pi and c coloured u i Then the Ryser-type conditions of j Theorem 5.3 will be obeyed by all the symbols and there will be x symbols in each cell, at most p occurrences of each symbol in each row and at most q occurrences of each symbol in each column. Then, by Theorem 5.3 R can be embedded in an exact (p, q, X I - latin rectangle without repetition on symbols ul,...,uXt.
.
6.
SYMMETRIC (p, p, X)- LATIN SQUARES.
If p is odd then the theorems one could obtain concerning embedding partial symmetric ( p , p, x)-squares would be generalizations of Theorems 4.2 and 4.4 and their corollaries, Corollaries 4.3 and 4.5. Thus the theorems would be in terms of graphs: numerical conditions
A .J. W.Hilton
132
by themselves would be sufficient but not necessary for the embedding. As the results to be expected are therefore rather weak, we do not give them here. However, if p is even, then we have quite a different story, and we obtain numerical conditions which are necessary and sufficient for the embedding. First we consider symmetric ( p , p, XI- latin squares, repetition permitted. We shall need the following theorem C31. Theorem 6.1. Let p be even. An incomplete symmetric (p, p, x)latin square R of size r x r on symbols ul,...,a can be emXt bedded in an exact symmetric (p, p, XI- latin square, repetition permitted, if and only if (i)
NR(ui)
2pr
1
-
p2t
(1 s i
(ii) there are at most x(pt-r) which NR(ai) is odd.
5
xt),
values of
i, 1
5
i
5
xt, for
From this we deduce the following result about partial symmetric (p, p, XI- latin squares. .Theorem 6.2. Let p be even. A saturated symmetric partial (p, p, x)- latin square R of size r x r on n symbols U ~ , . . . , U ~ can be embedded in an exact symmetric (p, p, XI- latin square T on the symbols al,...,uXt of size t x t with repetition permitted if and only if the following four conditions hold: (i) (ii)
DR
2
(2pr-p t) (xt-n), NR(ui) b 2pr - p 2t
(iii) D p
2
i
p(xt-n)
Y' Here yR
(1 (V
rOWS
5
i
i
p of
n), R) ,
is odd.
is the number of values of
i, 1 s i
2
n,
occurs an odd number of times on.the diagonal of
R
such that
ui
and
Proof.
1. Necessity.
.
(i), (ii) and (iii) Theorem 1.2.
The proofs of these are as in the proofs of
Embedding incomplete latin rectangles
133
(iv). If z is the number of symbols ai with n+l 5 i 5 xt which occur on the diagonal of R an odd number of times then, by Theorem 6.1 and since R is saturated, y+z S x(pt+r). If Ddiag is even then z could be 0, but if Ddiag is odd then z must be at least one. Sufficiency. Suppose (i), (ii), (iii) and (iv) hold. Let G be the multigraph on vertices vl, vn’ as well as some further vertices, formed as follows. Vertices vi and v are joined by k edges if j cells (i, j) and (j, i) each have deficiency k; to vertex vi adjoin Dii pendant edges. Now form a multigraph G’ by pairing the pendant edges off as much as possible, so that there is at most one pendant edge left over, and replacing each pair viwa, v.w of J B pendant edges by an edge vivj. There will be one pendant edge left over if and only if Ddiag is odd. Then the maximum degree of G is s p(xt-n) (by (iii)) (each loop counting two towards the degree) 1 2t) (xt-n) (by and the number of edges of G is at least ~(2pr-p (i)). Since p is even, the edges of G’ can be coloured with xt n colours U ~ + ~ , . . . , Uso ~ ~that not more than p edges of each colour appear at any vertex (C101, C181). Then this colouring can be equalized, whilst retaining this latter property. Then we place uk in cells (i, j) and (j, i) (j i) R times each if 9. edges joining vi and v are coloured a;, and ak is placed j in cell (i, i) R times if there are R pendant edges of G (in the edge colouring of G corresponding to that of G ’ ) coloured a .; Then no symbol appears more than p times in any row and each symbol appears at least 2pr p 2 t times altogether in R. Furthermore the number of symbols from U ~ + ~ , . . . C which J ~ ~ appear on the diagonal of R an odd number of times is
...,
-
-
I
0
if
Ddiag is even,
if Ddiag
is odd;
so the number of symbols appearing an odd number of times on the diagonal of R is at most x(pt-r) By Theorem 6.1, R can now be embedded in an exact symmetric (p, p, x)- latin square on the symbols ul, oxt of size t x t, repetition permitted.
.
...,
If the symmetric (p, p, x ) - latin square T is to be without repetition, then there does not seem to be a completely satisfactory analogue of Theorem 6.2. The trouble seems to lie in the fact that there are multigraphs of maximum degree p (p even) which have
A.J. W.Hilton
134
equitable edge-colourings with p/2 colours, but do not have balanced edge-colourings with p/2 colours. An example is given in Figure 1.
Figure 1.
A multigraph with an equitable, but without a balanced,
edge-colouring with two colours. There is good reason to suppose that there is a good analogue of Theorem 6.2 in the case when the symmetric (p, p, x)- latin squares are permitted to have symbols occurring up to two times in a cell, but we have not tried to work out the details of this. The same remark applies to the symmetric blocked (p, p, XI- latin squares we are about to discuss.
~ y m m e t h i cbeocked (p, p, x)- l a t i n Aquahe on the symbols is a partial symmetric (p, p, X I - latin square on those symbols in which the diagonal cells are all empty. It is Aatuhated if to each nondiagonal cell with positive deficiency each symbol either occurs p times in the row in which c lies or the column. An i n c o m p l e t e AqmmetRic b l o c k e d (p, p, x) - l a t i n Aquahe on symbols (p, p, XI- latin square U~,...,U~ is a partial symmetric blocked in which each non-diagonal cell contains x symbols exactly (counting repetitions). An exact Aymmethic b l o c k e d (p, p, x)- l a t i n AQUahe on symbols al, is an incomplete symmetric blocked an (p, p, XI- latin square on those symbols in which each symbol occurs exactly p times in each row. The existence of an exact symmetric blocked (p, p, x)- latin square implies the existence of a positive rational number t such that n = xt and the number of rows is pt + 1; conversely if t is a positive rational such that xt and pt are positive integers then there is a (pt+l) x (pt+l) symmetric blocked (p, p, XI- latin square on xt symbols (see C21). As before we shall always assume that xt and pt are positive integers. Exact symmetric blocked (p, p, x)- latin squares have a natural interpretation as edge-colourings of multigraphs in which each pair A pahtiak?
ul,...,un
...,
135
Embedding incomplete latin rectangles
of vertices are joined by precisely x edges; in particular exact symmetric blocked (p, p, 1)- latin squares correspond to types of edge-colourings of complete graphs. The following theorem from C31 is required. An incomplete symmetric blocked (p, p, X I - latin Theorem 6.3. square R of size r x r on symbols ul,...,uxt can be embedded in an exact symmetric blocked (p, p, XI- latin square, repetition permitted, on the same symbols if and only if (i)
NR(Ui)
2
(ii) p(pt+l)
2pr
-
(1 s i
p(pt+l)
5
xt),
is even.
From this the following theorem may be deduced; we omit the proof as it is so similar to that of Theorem 6.2. Theorem 6 . 4 . Let p be even. A saturated blocked symmetric partial (p, p, x)- latin square on n symbols ul,...,un can be embedded in a symmetric blocked (t+l) x (t+l) (p, p, 1)- latin square T on xt symbols ul,...,uXt if and only if (2pr-p(pt+l)) (xt-n),
(i)
DR
(ii)
NR(ui)
(iii) D
P
2
5
2
2pr
-
(1 s i
p(pt+l)
p(xt-n)
(V
rows
p
5
of
n), R).
For a partial symmetric blocked (p, p, 1) latin square R of side vr formed by r x r, let GR be the graph with vertices vl, joining vi and v if cells (i, j) and (j, i) are empty. We j have the following theorem about partial symmetric blocked (p, p, 1)latin squares.
...,
Theorem 6.5. Let p be odd. A saturated blocked symmetric (p, p, 1)- partial latin square on n symbols al,...,on can be embedded in an exact symmetric blocked (t+l) x (t+l) (p, p, 1 latin square T, repetition permitted, on t symbols ul,... at if and only if
-
(i)
t
is odd,
(ii)
DR
2
(2pr-p(pt+l) (t-n),
(iii) NR(ui) (iv)
G
2
2pr
-
(1 s i
p(pt+l)
can be edge-coloured with
t
-
n
i
n),
colours so that
A.J. W.Hilton
136
(a) no colour appears on more than p edges at any point 1 (b) each colour appears on at least ~(2pr-p(pt+l)) edges. We omit the proof as it is similar to others in this section. The fact that t is odd comes from the fact that pt + 1 must be even in Theorem 6.3. Corollary 6.6. Let p be odd. A blocked symmetric (p, p, 1)partial latin square on n symbols ulI...,un can be embedded in a symmetric blocked (t+l) x (t+l) (p, p, 1)- latin square T on t symbols al, at if (i), (ii) and (iii) of Theorem 6.5 hold and the number of vacant cells (excluding the diagonal cell) in each row is < p(t-n).
...,
Proof. By Vizing’s theorem for a simple graph G, if A(G)
-
7.
EMBEDDING WITH SPECIFIED DIAGONALS.
In this section we give a theorem about embedding a partial r x s latin rectangle R in a latin square T in which some of the cells of T outside R are also specified. We use two functions f : {ul,
...,utl *
(0, 11
and
g
:
{ul,
...,utl
+
(0, 1,
If r > s we call the set E(r+l, s+l), (r+2, s+2),...,(n, of cells, the (r-s)-th lower diagonal outside R. Theorem 7.l.
r > s L 1 ,
Let
E
i=1
...,tl. n-r+s))
if if
s(ui)
and let R be an incomplete r x s latin square on the symbols ul, ut. Then R can be embedded in a latin square T on ul, ut with the symbol ui occurring at least g(oi) times on the (r-s)-th lower diagonal outside R (1 s i s t) if and only if
...,
...,
NR(ui) If
r = s,
2
r
+
s
-
t + g(ai)
in the inequality
(1 s i s t).
t
1
i=l
g(ui)
I
t
-
r
-
1, one cannot
Embedding incomplete latin rectangles
replace the bound by
t
-r
-
137
for a discussion of this see C11.
From Theorem 7.1 follows
f
{
t - r - l i f r=sr f(ui) 5 t - r if r + s i=l and let R be a saturated partial r x r latin square on the symbols ulr...,an' Then R can be embedded in a latin square T on at in which symbol ai occurs at least f(ai) the symbols ul, times on the (r-s)-th lower diagonal outside R if and only if the following three conditions hold:
Theorem 7.2.
Let
r
2 s 2
1, let
...,
(r+s-t)(t-n) + f
(i)
DR
(ii)
NR ( ai. ) 2 r D < t - n P
(iii
2
+
s
-
and
t + f(ui) D c < t - n
.
+. .+ f (ot), (1 < i s n), (V rows p of R and columns c of R).
The proof of this is completely straightforward following the proof of Theorem 1.2. An interesting question is: to which functions g can Theorem 7.2 be generalized? The generalization to any function g such that
does not seem to be valid: one would need to properly edge-colour a bipartite graph G with t - r colours so that the i-th colour class has at least r + s - t + g(ui) colours. For a discussion of this colouring problem see C71, C151 and C191. REFERENCES L.D. Andersen, Latin squares and their generalizations, c13 Ph.D. Thesis, Reading, 1979. C21
L.D. Andersen and A.J.W. Hilton, Generalized latin rectanqles I: construction and decomposition, Discrete Math. 31 (1980) 125-152.
C31
L.D. Andersen and A.J.W. Hilton, Generalized Latin rectangles 11: embedding, Discrete Math. 31 (1980) 235-260.
C4 1
L.D. Andersen and A.J.W. Hilton, Thank Evans!, to appear.
C53
L.D. Andersen, R. Haggkvist, A.J.W. Hilton and W. Poucher, Embedding incomplete latin squares in latin squares whose diagonal is almost completely prescribed, Europ. J. Combinatorics 1 (1980) 5-7.
C61
A.
Cruse, On embedding incomplete symmetric latin squares, J. Combinatorial Theory (A), 16 (1974) 18-22.
138
A .J. W.Hilton
C71
A. Cruse, On extending incomplete latin rectangles, Proc. 5th Southeastern Conf. on Combinatorics, Graph Theory and Computing, Florida Atlantic University, Boca Raton, Florida (1974) 333-348.
C81
J. Folkman and D.R. Fulkerson, Edge-colourings in bipartite graphs, Combinatorial Mathematics and its Applications (Univ. North Carolina Press, Chapel Hill, 1969).
C91
D. Kdnig, Theorie der Endlichen und Unendlichen Graphen (Chelsea, New York, 1950).
C101 C.J.H. McDiarmid, The solution of a time-tabling problem, J. Inst. Maths. Apalics. 9 (1972) 23-34. c111
J. Petersen, Die Theorie der regularen Graphen, Acta Math. 15 (1981) 193-220.
C121
H.J. Ryser, A combinatorial theorem with an application to latin squares, Proc. Amer. Math. SOC. 2 (1951) 550-552.
C131
V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz. 3 (1960) 25-30.
C141
D. de Werra, Balanced schedules, Information J. 9 (1971) 230-237.
C151
D. de Werra, Investigations on an edge-colouring problem, Discrete Math. 1 (1971) 167-179.
L161
D. de Werra, Equitable colorations of graphs, Rev. Franc. Rech. Oper. 5 (1971) 3-8.
El71
D. de Werra, A few remarks on chromatic scheduling, in: B. Roy, ed., Combinatorial Programming: Methods and Applications (D. Reidel Publ. Co., Dordrecht-Holland, 1975) 337-342.
C181
D. de Werra, On a particular conference scheduling problem, Information J. 13 (1975) 308-315.
C191
D. de Werra, Multigraphs with quasiweak odd cycles, J. Combinatorial Theory (B), 23 (1977) 75-82.
C201
D. de Werra, Obstructions for regular colourings, to appear.
Annals of Discrete Mathematics 13 (1982) 139-158 0 North-Holland Publishing Company
EDGE-COLOURING REGULAR BIPARTITE GRAPHS
A.J.W.HILTON and C.A. RODGER University of Reading England
We give necessary and sufficient conditions for a regular bipartite multigraph of degree m to have an edge-colouring with m colours in which two specified edges receive the same colour. Call a cubic bipartite multigraph G with bipartition ( A , B) b k e w if there is a cutset K of six edges whose removal separates G into two subgraphs G1 and G2 such that each edge of K to a vertex of joins a vertex of A n V ( G 1 ) B n V ( G Z ) . We consider the problem of finding necessary and sufficient conditions for a skew cubic bipartite multigraph to have an edgecolouring with three colours in which three specified edges of K receive different colours. This problem arose in an investigation concerning latin squares. We conclude with a conjecture on this problem. is a multigraph, if A and B are disjoint sets of vertices G and if K is a set of edges of G , then we let V ( G ) and E ( G ) denote the vertex set and edge set respectively of G, we let E ( A , B) denote the set of edges joining vertices in A to vertices in B and we let N(K) denote the set of vertices of G which are incident with at least one edge of K. If of
G
An e d g e - c o l o u h i n g
of G is an assignment of colours to the edges of so that no two edges of the same colour have a common vertex. K h i g C51 proved that a bipartite multigraph with maximum degree m can be edge-coloured with m colours.
G
In the first part of this paper we consider some edge-colouring results for regular bipartite multigraphs of general degree m 2 2 . 139
140
A.J. W.Hilton and C.A. Rodger
In the second part we consider some more specific problems concerning skew cubic bipartite multigraphs. In the third part we state a conjecture for which our results provide strong evidence. All the results have the common theme that the graph has to be edgecoloured with a given number of colours with certain specified edges receiving specified colours. 1.
REGULAR BIPARTITE MULTIGRAPHS OF DEGREE m.
Our first object is to prove the following theorem. THEOREM 1. Let G be a regular bipartite multigraph of degree m. Let el and e2 be two edges of G. Then G can be edge-coloured with m colours with el and e2 receiving the same colour if and only if G does not contain a cutset K containing el and e2 with IKI = m which separates G into two disjoint submultigraphs G1 and G2 such that each edge of K joins a vertex Of A n V(G2) to a vertex of B n V(G1) for some bipartition (A, B) of G. The exceptional multigraph of Theorem 1 is illustrated in Figure 1.
Figure 1. The exceptional multigraph of Theorem 1. In the proof of Theorem 1, we make use of the following lemma due to Hlggkvist C 3 1. Lemma (HBggkvist). Let G be a regular bipartite multigraph with bipartition (A, B) , let e E E(G) and let E* 5 E(G) \{el. Then G has a 1-factor which contains e and avoids E* if and only if there do not exist disjoint sets El and E2 of edges such that e E El, E2 5 E*, I El I = I E2 I and El u E2 is a cutset whose removal separates G into two disjoint subgraphs G1 and G2 such that El 5 E(A n V(G1), B n V(G2) and E2 5 E(A n V(G2), B n V(Gl)).. The exceptional multigraph of HBggkvist's lemma is illustrated in Figure 2 . In fact HBggkvist did not make it clear that his lemma applies to multigraphs, but an examination of his proof shows that it does.
141
Edge-colouring regular bipartite graphs
....
.*
.. . . . . . .
....... G1
Figure 2 .
The exceptional multigraph of Hdggkvist's lemma.
Proof of Theorem 1. 1. Necessity. Suppose t h a t G can be edge-coloured w i t h m colours so t h a t two s p e c i f i e d edges, el and e 2, r e c e i v e t h e same colour. Suppose t h a t G c o n t a i n s a c u t s e t K with IKI = m and and G2 el e K which s e p a r a t e s G i n t o two d i s j o i n t subgraphs G1 with each edge of K j o i n i n g a v e r t e x of A n V ( G 2 ) t o a v e r t e x of B n V(G1). Each v e r t e x of B n V ( G Z ) i s joined by m edges t o v e r t i c e s of A n V ( G 2 ) , b u t t h e m edges of K j o i n v e r t i c e s of A n V(G2) t o v e r t i c e s which a r e not i n B n V ( G 2 ) . Therefore IB n V ( G 2 ) I = I A n V ( G 2 ) I - 1. Therefore each colour has t o be used on e x a c t l y one edge of K , and so e2 k K .
Sufficiency. Suppose t h a t G does not contain a c u t s e t K with IKI = m and el, e2 E K which s e p a r a t e s G i n t o two d i s j o i n t subgraphs G1 and G2 such t h a t each edge of K j o i n s a v e r t e x of A n V(G2) t o a v e r t e x of B n V(G1) f o r some b i p a r t i t i o n (A, B) L e t e2 be i n c i d e n t w i t h a v e r t e x v i n A and l e t E* be t h e s e t of a l l t h e o t h e r edges i n c i d e n t w i t h v. Then el k E* f o r o t h e r w i s e E* u {e,) would be a set K of t h e forbidden type with V ( G 2 ) = {v}. 2.
.
I f t h e r e i s a 1 - f a c t o r F c o n t a i n i n g el and c o n t a i n s e2 a l s o . Then G \ F i s a r e g u l a r 1 and so, by Kbnig's theorem, can degree m m 1 c o l o u r s . Thus G can be edge-coloured el and e2 r e c e i v i n g t h e same colour.
-
-
avoiding E* then F b i p a r t i t e graph of be edge-coloured w i t h with m colours with
I f t h e r e i s no 1 - f a c t o r c o n t a i n i n g el and avoiding E* E%ggkvist's lemma, t h e r e a r e sets El and E2 of edges el e El, E2 5 E*, l E l l = l E Z l and El u E2 i s a c u t s e t s e p a r a t e s G i n t o two subgraphs and G$ such t h a t of El j o i n v e r t i c e s of V(Gf) n A and V ( G 5 ) n B and
GI
then,by such t h a t which t h e edges t h e edges
142
A.J. W.Hilton and C A . Rodger
of E2 join vertices of V(Gf) n B and V(G5) n A. Let E3 = (E* u {e,}) \ E 2 . Then E2 u E3 is the set of edges incident with v. The situation is illustrated in Figure 3 .
Figure 3. Let G1 = Gf \ {v] and let G2 be G% with the vertex v and the edges of E2 adjoined. Then the edges of El u E3, which number lEll + (m - IE21) = m and include el and e2, join A n V(G1) to B n V(G2) and no edges join A n V(G2) to B n V(G1). But then these m edges form a cutset K of the forbidden type, a contradiction. This proves Theorem 1. We also deduce two more specialized consequences of HBggkvist's lemma, of which use will be made later. Call a regular bipartite multigraph of degree m ;?. 2 a c h c L c muLtighaph if it has the following form, For some bipartition ( A , B), G has a cutset F with IF1 = m whose removal separates G into two disjoint submultigraphs G1 and G2 such that each edge of F joins B n V(G1) to A n V(G2); G also has a second cutset {el, e2j, which separates G into two disjoint where {el, e2} n F = Cp, submultigraphs H1 and H2; IE(H1) n FI = 1, IE(H2) n FI = m - 1, el joins a vertex of A n V(H1) n V(G1) to a vertex of B n V(H2) n V(G1) and e2 joins a vertex of B n V(H1) n V(G2) to a vertex of A n V(H2) n V(G2). A circle multigraph is illustrated in Figure 4 .
I-
Edge-colo uring regular bipartite graphs
T : Fiqure 4 .
143
'H1
*
/
?$=-
A circle multigraph.
THEOREM 2 . Let G be a regular bipartite multigraph of degree m with a cutset F with the properties that IF1 = m and the removal of F separates G into two disjoint submultigraphs G1 and G2 such that, for some bipartition ( A , B), each edge of F joins a to a vertex of B n V ( G 2 ) . Let el E E(G1) vertex of A n V(G1) and e2 E E ( G 2 ) . Then G can be edge-coloured with m colours with ell e2 receiving different colours if and only if G is not a circle multigraph. Proof. By Hdggkvist's lemma, G has an edge-colouring with m colours so that el and e2 receive different colours if and only if Cell e,) is not a cutset. If {el, e,) is a cutset then it is easy to see by counting arguments that G is a circle multigraph. Now call a regular bipartite multigraph of degree m 2 3 a d o u b l e c i h c e e muLtighaph if it has the following form. For some bipartition (A, B), G has a cutset F with IF1 = m whose removal separates G into two disjoint submultigraphs G1 and G2 such to A n V ( G 2 ) ; G also has that each edge of F joins B n V(G1) a second cutset El u E 2 , where El = E (G1), E2 c E ( G 2 ) and lEll = lE21 = 2 , which separates G into two disjoint subgraphs
A.J. W.Hilton and C.A. Rodger
144
H1 and H2; IE(H1) n FI = 2, IE(H2) n FI = m - 2 , the edges of El join vertices of A n V(H1) n V(G1) to vertices of B n V(H2) nV(Gl) and the edges of E2 join vertices of B n V(H1) n V(G2) to vertices of A n V(H2) n V(G2). A double-circle multigraph would be illustrated by Figure 4 if each of the single edges were replaced by two edges. THEOREM 3 . Let G be a regular bipartite multigraph of degree m with a cutset F with the properties that IF1 = m and the removal of F separates G into two disjoint submultigraphs G1 and G2 such that, for some bipartition (A, B), each edge of F joins a vertex of A n V(G1) to a vertex of B n V(G2). Let G not contain a cutset of two edges, one of which is a member of F. Let el E E(G1), e2 E E(G2) and f E F. Then G can be edge-coloured with m colours with the edges el, e2, f receiving different colours if and only if G is not a double-circle multigraph with el E El, e2 E E2 and f E F n E(H1). Proof.
The necessity is obvious.
We shall prove the sufficiency.
Suppose that G can be edge-coloured with m colours with el, e2 and f receiving different colours. Consider two regular bipartite multigraphs Gi* and GS* of degree m constructed as follows. Let G; and GS be the multigraphs induced by V(G1) u (A n V(G2)) and V(G2) u (B n V(G1)) respectively and let GI* and G;* be formed from Gf and G$ by identifying the vertices of A n V(G2) and B n V(G1) respectively. Let the edges of F be f = fl, f2, . , f
...,
By Hdggkvist's lemma and Kdnig's theorem, for any i, 1 5 i 5 m, Gf* and G$* can be edge-coloured with m colours so that el and fi or e2 and fi respectively receive different colours. Suppose that there is some i, 2 5 i 5 m, such that either GI* or G$* (say Gt* to be specific) has no 1-factor including el or e2 respectively (say e,) and avoiding fl and fi. Then by Hdggkvist's lemma, G$* has a cutset of four edges including e2, fl and fi and say e which separates G$* into two submultigraphs M1 and - M 2 , where fl and fi join vertices of BnV(M1) to vertices of A n V(M2) and e and e2 join vertices of A n V(M1) to vertices of B n V(M2). Also in every edge-colouring of G in which e2 and fl receive different colours, e2 and fi receive the same colour.
Edge-colouring regular bipartite graphs
145
Consequently, for Gi* there is an edge-colouring with el, fl and fi receiving different colours. Therefore, by Hgggkvist's lemma, Gf* cannot have a cutset consisting of four edges including el, fl and fi, so G cannot be a double-circuit multigraph with el E El, e2 E E2 and f E F n E(H1). 2.
SKEW CUBIC BIPARTITE MULTIGRAPHS.
Call a cubic bipartite multigraph bhew if there is a cutset K of six edges whose removal separates G into two disjoint submultigraphs G1 and G2 such that each edge of K joins a vertex of A n V(G1) to a vertex of B n V(G2) for some bipartition (A, B), and call X the bhew c u t b e t . Our second objective, as yet unachieved, is to characterize those skew cubic bipartite multigraphs for which there exists an edge-colouring with three colours such that three specified edges of' X each receive different colours. We feel that the results here provide strong evidence for the conjecture concerning this characterization problem we make in the last Section. The problem arose in connection with a possible method of solving a conjecture C41 concerning the embedding of partial idempotent latin squares. For details of considerable progress using this method, see Ell. The embedding problem has since been resolved a different way C21, but the solution of our characterization problem would nevertheless materially aid our understanding of the embedding problem and would tell us which idempotent latin squares of order n can be embedded in idempotent latin squares of order t s In. We first look at the simpler problem of deciding which skew cubic bipartite multigraphs G possess an edge-colouring with three colours such that two specified edges el and e2 of X receive different colours. By Xdnig's theorem there is such an edgecolouring if and only if G contains a 1-factor containing el and avoiding e2. From HClggkvist's lemma we see that G contains such a 1-factor if and only if {el, e21 does not form a cutset. At this juncture, we define the exceptional multigraphs of Theorem 4. Call a skew cubic bipartite multigraph with skew cutset X I whose removal separates G into two disjoint submultigraphs G1 and G2, a 6iguhe-od-eight multigraph if there is a cutset H c K such that IHI = 2 which separates G into two disjoint submultigraphs whose vertex sets are C and D and which has the following
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A.J. W.Hilton and C.A. Rodger
properties: Each of G1 and G2 contain two disjoint submu tigraphs, those of G1 being induced by V(G1) n C and V(G1) n D and those of G2 being induced by V(G2) n C and V(G2) n D for some bipartition (A, B) of G, two edges of K join B n C nV(G1) to A n C n V(G2), two edges of K join B n D n V(G1) to A n D n V(G2), one edge of K joins B n D n V(G1) to A n C n V(G2) and one edge of K joins B n C n V(G1) to A n D n V(G2). A figure-of-eight multigraph is illustrated in Figure 5 .
'C
.
. ..
c
Figure 5 .
+
c
TK .I
.. *
A figure-of-eight multigraph.
THEOREM 4 . Let G be a skew cubic bipartite multigraph with skew cutset K. Let H c K be a cutset and let /HI = 2. Then G is a figure-of-eight multigraph. Proof. Suppose that H separates G into two disjoint submultigraphs whose vertex sets are C and D. Consider the four submultigraphs of G induced by C n V(G1), C n V(G2), D n V(G1) and D n V(G2). These vertex sets form a partition of V(G). Since H is a cutset, there are no edges between C n V(G1) and D n V(G1)
147
Edge-colouring regular bipartite graphs
or between C n V(G2) and D n V(G2). Suppose there are a , 8 , y, 6 edges between C n V(G1) and C n V(GZ) , between C n V(G1) and D n V(G2), between D n V(G1) and C n V(G2) and between D n V(G1) and D n V(G2) respectively. Since the only edges between V(Gl) and V(G2) are the edges of K we have ci + f3 + y + 6 = 6. Since [HI = 2 we have 8 + y = 2 . Since G is cubic and the only edges of G joining any of the submultigraphs to the rest of the multigraph are in K, by a simple counting argument we have ci + f3, y + 6 , a + y , 8 + 6 :0 (mod 3). It is now an exercise to check that the only solution to this set of equations and congruences is f3 = y = 1, ci = 6 = 2, from which Theorem 4 follows. The following corollaries now follow easily. Corollary 5. Let G be a skew cubic bipartite multigraph with skew cutset K. Let el and e2 be two edges of K. Then G can be edge-coloured with three colours with el and e2 receiving different colours if and only if G is not a figure-of-eight multigraph with a cutset {el, e,) c K. Corollary 5. Let G be a skew bipartite multigraph with skew cutset K and with a cutset H c K, [HI = 2 . Let el, e2, e3 E K. Then G is a figure-of-eight multigraph and can be edge-coloured with three colours with el, e2 and e3 receiving different colours if and and {el, e2, e3} P’ K\H. only if H Z {el, e2, e,) We have a result similar to Theorem 4 in the case when
IHI = 4 .
Theorem 7. Let G be a skew cubic bipartite multigraph with skew cutset K. Let H c K be a cutset and let ]HI = 4 . Then G is a figure-of-eight multigraph (and so contains a cutset K\H consisting of two edges). Proof.
This is very similar to Theorem 4 (but this time
f3
+
y = 4).
We now describe the exceptional multigraphs of Theorem 7. Call a skew cubic bipartite multigraph with skew cutset K whose removal separates G into two disjoint submultigraphs G1 and G2, a beehmug muL.tighaph if there is a cutset H such that IHI = 4 and IH n KI = 3 which separates G into two disjoint submultigraphs, whose vertex sets are, say, C and D, with the following properties: Let {el = H\K. {el is a cutset of one of G1 and G2, say G1, and the removal of e from G1 leaves two disjoint submultigraphs whose vertex sets are V(G1) n C and V(G1) n D; G2 is the
I48
A.J. W. Hilton and C A . Rodger
union of two disjoint submultigraphs whose vertex sets are V(G2) n C and V(G2) n D. For some bipartition ( A , B) of G, two edges of K join B n C n V(G1) to A n C n V(G2), two edges of K join B n C n V(G1) to A n D n V(G2), one edge of K joins B n D n V(G1) to A n C n V(G2), one edge of K joins B n D n V(G1) to A n D n V(G2) and e joins A n C n V(G1) to B n D n V(G1). A beermug multigraph is illustrated in Figure 6.
D
Figure 6.
A beermug multigraph.
Edge-colouring regular bipartite graphs
Theorem 8 . cutset K. IH\KI = 1.
149
Let G be a skew cubic bipartite multigraph with skew Let G contain a minimal cutset H with IHI = 4 and Then G is a beermug multigraph.
Proof. Suppose that H separates G into two disjoint submultigraphs whose vertex sets are C and D. Let e be the edge of H\K. We may suppose without loss of generality that the edges of K join vertices of V(G1) n B to vertices of V(G2) n A , that e E GI and that e joins a vertex of B n D n V(G1) to a vertex of A n C n V(G2). Then, since H is minimal, a simple counting argument shows that the remaining edges of H consist of two edges joining B n C n V(G1) to A n D n V(G2) and one edge joining B n D n V(G1) to A n C n V(G2). A further simple counting argument then shows that there are 4 , 2 , 3 and 3 edges of K respectively incident with vertices of B n C n V(G1), B n D n V(G1), A n C n V(G2) and A n D n V(G2). Since H is a cutset it then follows that G is a beermug multigraph. Corollary 9 . Let G be a skew cubic bipartite multigraph with skew cutset K. Let K contain a cutset J of G with IJI = 3, but no cutset of two edges. Let el, e2, e3 be three edges of K. Then K can be edge-coloured with three colours with el, e2 and e3 receiving different colours if and only if G is not a beermug multigraph with el, e2, e3 being three edges of a cutset H of G with IHI = 4 and IH\K) = 1.
Necessity. If G is a beermug multigraph there are two possibilities for H I and clearly in neither case can G be edge-coloured with three colours so that the edges of H n K receive different colours. Sufficiency. Suppose there is no such edge-colouring with three colours with el’ e2, e3 receiving different colours. Since K contains a cutset J with IJI = 3, an easy counting argument shows that at least one of G1 and G2, say G2, is disconnected and, in fact, that J(= J1) and K\J(= J2) are both cutsets of G with three edges. In any edge-colouring of G with three colours, it is easy to see that the three edges of J1 receive different coloursI as do the three edges of J 2 . Therefore, we may assume that el E J1 and e2, e3 E J2. In any edge-colouring of G with three colours, e2 and e3 will receive different colours; from
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A.J. W.Hilton and C A . Rodger
Hdggkvist's lemma and KBnig's theorem, it is easy to see that el can be made to receive a third colour unless either el and one of Ie,, e31 forms a cutset of G I or unless there is a further edge e such that {e, el} u {e,, e,} forms a cutset of G. The first possibility is excluded by hypothesis, so the second possibility must hold. But e cannot be a member of K , for then, by Theorem 7 , G would be a figure-of-eight graph, and so there would be a a possibility excluded cutset of G consisting of two edges of K , by hypothesis. Therefore e ,t! K, and so, by Theorem 7 , G is a beermug multigraph. This proves Corollary 9. Corollary 10. Let G be a skew cubic bipartite multigraph with skew cutset K. Let el, e2, e3 be three edges of K . If a proper subset of K is a cutset of G then either (i) G is a figure-of-eight multigraph or (ii) G is a beermug multigraph or (iii) G is edge-colourable with three colours with el, e2, e3 receiving different colours. Proof. An easy counting argument shows that no subset of K with either one or five edges can be a minimal cutset of G. Corollary 10 now follows from Theorems 4 and 6 and from Corollary 9. We next describe the last of the special skew cubic bipartite multigraphs we shall need to consider. Call a skew cubic bipartite multigraph with skew cutset K , whose removal separates G into two disjoint subgraphs G1 and G2, a cat'b-chadLe-muLtigaaph if it contains a minimal cutset M with 1 1 1 = 4 and M n K = @ which separates G into two disjoint submultigraphs whose vertex sets are P and Q and which has the following properties: The submultigraph induced by P contains two edges of K and G * , the submultigraph induced by Q contains four edges of K ; G* has a cutset J with IJI = 2 and J c K whose removal separates G* into two disjoint submultigraphs whose vertex sets are C and D; G1 contains a cutset of two edges elC and elD which join vertices of V(G1) n A n P to vertices of V(G1) n B n C and V(G1) n B n D respectively, and there are no further edges between any of V(G1) n P, V(G1) n C and V(G1) n D; similarly G2 contains a cutset of two edges, e2C and e2D, which join vertices of V(G2) n B n P to vertices of V(G2) n A n C and V(G2) n A n D respectively, and there are no further edges between any of V(G2) n PI V(G2) n C and V(G2) n D; two edges of K join vertices
151
Edge-colouring regular bipartite graphs
of
V(G1)
of
K
t o v e r t i c e s of
n B n P
j o i n s e a c h of and
V(G2) n A n C
n A n P;
V(G2)
and
n B n C
V(G1)
f i n a l l y one edge t o each of
n B n D
V(G1)
n A n D.
V(G2)
A c a t ' s cradle multigraph i s i l l u s t r a t e d i n Figure 7 .
Observe t h a t
a c a t ' s c r a d l e m u l t i g r a p h a l s o c o n t a i n s f o u r minimal c u t s e t s ILI = 4 ,
that J
and
IL n MI = 2 ,
(K n E(G*))\J
I L n KI = 2 ,
a r e c u t s e t s of
and a l s o o b s e r v e t h a t b o t h
G*
c o n s i s t i n g of two e d g e s .
I n t h e discussion we s h a l l r e f e r t o these a s Theorem 11. cutset
J1
and
J2.
b e a skew c u b i c b i p a r t i t e m u l t i g r a p h w i t h skew
G
no p r o p e r s u b s e t of which i s a c u t s e t , and l e t
K,
s u b s e t of
Let
with
G
and
IHI = 4
rl
two d i s j o i n t s u b m u l t i g r a p h s t h r e e edges of
r2,
and
el, e 2 , e3
Let
K.
4
H n K =
{ e l , e 2 , e3}
and
G
w i t h t h e s u b m u l t i g r a p h induced by
into
G
e a c h of which c o n t a i n s
be t h r e e edges of
c e i v i n g d i f f e r e n t c o l o u r s u n l e s s one of
be a
H
which s e p a r a t e s
can b e edge-coloured w i t h t h r e e c o l o u r s w i t h c i s e l y one o f
such
L
rl
e l , e2 and r 2
Then G e3 rec o n t a i n s preK.
and
i s t h e c a t ' s c r a d l e multigraph P c o n t a i n i n g e x a c t l y one of
el, e 2 , e3
and e i t h e r J1 c { e l , e 2 , e3] o r J2 c { e l , e 2, e 3 } . r e f e r t o t h e d e f i n i t i o n of t h e c a t ' s c r a d l e m u l t i P I J1, J2
[Here
graph]. Proof. that
Let
edge of
H
= E l V ( T 2 ) = F.
V(T1)
I
IH n E(G1)
= 2
and
n B n F, V(G1)
n A n F
V(G2)
n B n F,
V(G2)
and
V ( G 1) n A n E
by an edge o f on edges of
I
and t h a t t h e r e i s one
= 2,
j o i n i n g e a c h of t h e p a i r s of v e r t i c e s
V(G1)
v e r t e x of
A s i m p l e c o u n t i n g argument shows
IH n E ( G 2 )
V(G1)
and
n B n E l V(G2) V(G2)
n A n F.
vlAE,
and
n A n E
L e t the nBnF
V(G1)
and l e t t h e o t h e r v e r t i c e s
b e d e n o t e d i n an o b v i o u s s i m i l a r way.
I t i s c o n v e n i e n t t o c o n s t r u c t from
{r;,
rl
and
T2
two s e t s
TiJ} and { r f * , I'iJ*} of m u l t i g r a p h s a s f o l l o w s . L e t r f o b t a i n e d from rl b y a d j o i n i n g t h e two e d g e s vlAE v~~~ and vlBE v~~~ and l e t riJ be o b t a i n e d from r 2 by a d j o i n i n g
vlAF v~~~ and vlBF v2AF. L e t j o i n i n g t h e two edges vlAE vlBE o b t a i n e d from
and
n A n E
which i s j o i n e d t o a v e r t e x o f
be d e n o t e d by
H H
and
n B n E
V(G1)
r2
by a d j o i n i n g
rf* and
be o b t a i n e d from rl v~~~ v~~~ and l e t
vlAF vlBF
and
v~~~ v
be
by adbe
r;* ~
~
~
.
A.J. W.Hilton and C.A. Rodger
152
I
\ .c
Figure 7.
A cat.’s cradle multigraph.
Edge-colouring regular bipartite graphs
153
Suppose el, e2, e3 E E(T1). A simple counting argument shows that it is not possible for r f to contain a cutset of three edges which includes both vlAE v~~~ and vlBE v ~ Similarly ~ ~ . r; cannot contain a cutset of three edges which contains both vlAF v~~~ and vlBF v ~ Therefore, ~ ~ . by Theorem 1, we may colour the edges of rf u r$ with three colours giving all four adjoined edges the same colour, say a . Then elf e2, e3 will each receive different colaurs. We can now reconstruct G with all edges receiving the colours they had in r i u r$ and with the four edges of H each being coloured a. Similarly if el, e2, e3 E E (r2). From now we may suppose that both r l and r2 contain edges of {elf e2, e31. Without loss of generality suppose that el, e2 eE(rl) and e3 E E ( T 2 ) . Let f3 be the third edge of K in rl and let fl, f2 be the other two edges of K in r 2 . Suppose that there is no minimal cutset of three edges of rp containing f3 and vlBE v~~~ and similarly that there is no minimal ~ Then, ~ ~ cutset of r; of three edges containing e3 and vlBF v by Theorem 1, we may colour r$ u I'$ so that the four edges all receive the same colour, say a. f3, vlBE v ~ e3, ~ vlBF ~ v~~~ , Then the other two adjoined edges also receive the colour a and the edges el, e2 receive different colours, not a. As before we may reconstruct G with all edges receiving the colours they had in I'f u r$, with all four edges of H being coloured a . Similarly, for i = 1,2, suppose there is no minimal cutset of three edges of r'l containing vlBE v~~~ and ei and likewise, for j = 1,2, there is no minimal cutset of r$ of three edges containing SO VIBF V2AF and fj* Then, by Theorem 1, we may colour rf u r j that vlBE v~~~~ VlBF V2AFf ei and fj receive the same colour, say a. Then the other two adjoined edges also receive the colour a and we can arrange that el, e2 and e3 all receive different colours. Then, as before, we may reconstruct G with all edges receiving the colours they had in rf u r$, with all four edges of H being coloured a . of three edges conNow suppose there is a minimal cutset of r; taining e3 and vlBF vZAF, and also, for each of j = 1,2, there of three edges containing vlBF V2AF is a minimal cutset of r; and fj. Then it is easy to see that r$* has a cutset consisting of two edges of K, say kl and k2 , such that the removal of {kl, k2) leaves two disjoint submultigraphs M1 and M2 with the
.
A.J. W.Hilton and C.A. Rodger
154
properties that kl joins a vertex of V(M1) n B to a vertex of V(M2) n A, k2 joins a vertex of V(M1) n A to a vertex of V(M2) n B, vlBF v~~~ and the third edge k3 of K n E(r3) join vertices of V(M1) n B to vertices of V(M2) n A and vlAF v~~~ joins a vertex of V(M2) n A to a vertex of V(M2) n B. This and are illustrated in Figure 8. the corresponding r:*
II/
%I--
r.2
,r[y 1-,
A
B
I
.
.
.
*
\ \I
'
I
I
I
. . '..
1.
\
.
' \
' \
' *
*-
'. ' . ./ . .
a '
-0
/
Q
Figure 8.
or (B) {klI kZ1 = {e3,fi 1, k3 = fj, where suppose that kl = e3, k2 = fl, k3 = f21.
{i, jl = 11, 2 ) .
[We may
It is easy to check that I'S cannot be edge-coloured with three colours with the two adjoined edges receiving the same colours, and that the only way of edge-colouring r$* (aside from permutation of the colours) with three colours is shown in Figure 8. We shall suppose that r$* is coloured as in Figure 8.
If e3 = k3 we may edge-colour r$* so that the adjoined edges vlAE vlBE and v~~~ v~~~ are coloured y and B respectively and the edges el and e2 are coloured B and y in some order (so that f 3 is colsured a) and thence obtain the required edge-colouring
Edge-colouring regular bipartite graphs
155
of
G, except, by Hdggkvist's lemma and Theorems 2 and 3, if either r f * contains a cutset of two edges consisting of one adjoined edge and one edge of K or r?* is a double-circle multigraph with I vlAE vlBE E El, v~~~ v~~~ E E2 and f E F n E(H1) [El, E2, F and H1 refer to the description earlier of the double-circle multigraph]. The first case is not possible, for it implies that G has a Cutset consisting of three edges of K. In the second case G is a cat's cradle multigraph and G cannot be edge-coloured in the required way. If kl = e3 we may colour r;?* so that the adjoined edges vlAE vlBE and v~~~ vZAE are coloured y and B respectively and the edges el and e2 are coloured ci and 7.f in some order ( s o that f3 is coloured y) and thence obtain the required edgecolouring of G unless, by Hbggkvist's lemma, {f,, v~~~ v ~ is ~ ~ a cutset of r;*. But this case does not arise, for it implies that G has a cutset consisting of three edges of K. Now consider the case when there is a minimal cutset of r f consisting of three edges which contains fj and vlBE v~~~ and also, for each of i = 1,2, there is a minimal cutset of r; consisting of three edges which contains vlBE v~~~ and el. If we notice that, in an edge-colouring of G with three colours, to require that el, e2, e3 receive different colours is equivalent to requiring that fl, f2, f3 receive different colours, then we see that this case is essentially a duplicate of the previous case. Thus we do not need to consider it separately. We are left with two cases to consider. The first case is when I'q contains: a minimal cutset of three edges that includes vlBF v~~~ and fl; a minimal cutset of three edges that includes vlBF v~~~ and f2; and no minimal cutset of three edges that includes vlBF v~~~ and e3; and simultaneously, r;? contains: a minimal cutset of three edges that includes vlBE v~~~ and f3; and at least one of el and e2 (say el) does not occur in a minimal cutset of three edges with vlBE v ~ Under ~ ~ these . conditions, and recalling that no proper subset of K is a cutset of G , it can easily be verified that either e3 and fl or e3 and f2 (say e3 and fl) do not occur together in a minimal cutset of three edges in r2, and with the exception of the multigraph illustrated in Figure 9 or the analogue of it in which G1 and G2 are interchanged [we may suppose that if there is an exception, then it is the
)
156
A.J.W. Hilton and C.A. Rodger
multigraph of Figure 91, either el and f3 or e2 and f3 (say el and f3) do not occur together in a minimal cutset of three edges in r f . Then, providing r; is not the multigraph shown in Figure 9, using Theorem 1 we can colour r f u r $ so that e3 and fl receive the same colour, say a, and el and f3 receive the same colour, say 6. A simple counting argument reveals that the edges vlAF v~~~ and vlAE v~~~ must be coloured with a and B respectively, and we can arrange that the edges vlBF v ~ ~ , and e2 are coloured with B , a and the third colour, V I B E V2AE say y , respectively. As before, we may reconstruct G with all edges receiving the colours they had in rf u r $ , with two of the edges in H being coloured a , the other two being coloured B . Now consider the situation where r l is the graph illustrated in Figure 9 , where we can assume G1 and G2 are as shown. It can be easily verified that since fl and f2 but not e3 occur in minimal cutsets of three edges that contain vlBF v2-, then either r l contains a cutset of two edges consisting of vlAF v~~~ and e3, or r$ is the graph illustrated in Figure 10. In the latter case, we can immediately reconstruct G. We may then verify that G can be coloured in the required way, remembering that no proper subset of K is a cutset of G. In the former case, removal of vlAl? V2BF and e3 separates r$ into two disjoint submultigraphs, one of which contains no edges joining G1 to G 2 . This submultigraph must be contained within G2 for otherwise K would contain a cutset of less than six edges. Now consider rq*. By Hdggkvist's lemma we can edge-colour r$* with three colours so that e3 and vlAF v~~~ receive different colours unless they form a cutset of two edges. This exception is impossible, for then e3, e2 and f3 form a cutset in G. Clearly e3 and v~~~ v~~~ receive the same colour since they form a cutset of two edges. Now consider r f * . By Theorem 1 we can edge-colour r l * with three colours so that f3 and v~~~ v~~~ receive the same colour unless they occur together in a minimal cutset of three edges. This exception is also impossible as G would then contain a cutset consisting of fl, f2 and e2. Now we can arrange for r f * u r$* to be edge-coloured with three colours go that e3, v~~~ v~~~ and v~~~ v~~~ receive the same colour, say a , vlAE vlBE, el and vlAF vlBF receive the same colour, say 8 , and e2 receives the third colour. So, as before, we can reconstruct G so that all edges receive the colours they had in r f * u r$* with two edges of H being coloured
Edge-colouring regular bipartite graphs
Figure 9.
157
A.J. W. Hilton and C.A. Rodger
158
a,
the other two being coloured
B
The final case is the same as the case just considered if we replace ei by fi and fi by ei for i = 1,2, and 3. However, this case is essentially a duplicate of the previous case, since requiring el, e2 and e3 to receive different colours is equivalent to requiring fl, f2 and f3 to receive different colours in an edgecolouring of G. This proves Theorem 11. 3.
A CONJECTURE.
We conjecture that Theorem 11 holds without the hypothesis concerning the set H. More specifically: Conjecture. Let G be a skew bipartite multigraph with skew cutset K, no proper subset of which is a cutset, and let G not be a cat's cradle multigraph. Let el, e2, e3 be three edges of K. Then G can be edge-coloured with three colours with el, e2 and e3 receiving different colours. REFERENCES. C11
L.D. Andersen, Latin squares and their generalizations. Ph.D. thesis, Reading, 1979.
[2]
L.D. Andersen, A.J.W. Hilton and C.A. Rodger, A solution to the embedding problem for partial idempotent latin squares. To appear.
[3]
R. Hdggkvist, A solution to the Evans conjecture for latin squares of large size. Combinatorics, Proc. Conf. on Combinatorics. Kezthely (Hungary) 1976. Pub. by Jdnos Bolyai Math. SOC. and North Holland Pub. Cp. 1978, 495-513.
C4]
A.J.W. Hilton, Embedding an incomplete diagonal latin square in a complete diagonal latin square. J. Combinatorial Theory 15 (19731, 121-128.
[5]
D. Kdnig, Theorie der endlichen und unendlichen Graphen. Chelsea, New York, 1950.
Annals of Discrete Mathematics 13 (1982) 159-170
0 North-Holland Publishing Company
SOME COLOURING PROBLEMS AND THEIR COMPLEXITY A.J. MANSFIELD and D.J.A. WELSH Mat hematical Institute Oxford, England
§I. INTRODUCTION We consider graph colouring problems at different levels of the complexity hierarchy and use them to illustrate various complexity levels within the languages of graphs. These are the problems: (1) (2) (3)
Is a graph bipartite? Find the chromatic polynomial of a graph. Is a graph 3 colourable?
The complexity terminology is fairly standard and follows closely that used by Garey and Johnson C21; though as will be seen in the next section a certain amount of care needs to be taken when considering space requirements of nondeterministic machines as described in C21. The first problem, that of testing whether a graph is 2-colourable is introduced principally to clarify the notion of space in nondeterministic machines. Our second graph. We associated polynomial
set of problems concern the number of colourings of a have created distinct recognition (= decision) problems with the well known problem of finding the chromatic of a graph.
Section 5 concerns the relationship between the longstanding complexity conjecture that coNP # NP and a graph theorem of Haj6s [ 4 1 . This section was motivated by conversations with Francois Jaeger based on complexity problems raised in Ell]. We no longer believe that this approach will settle the NP = coNP question but it does pose some seemingly intractible graph theoretic complexity problems.
-
92.
THE COMPUTATIONAL MODEL
our model of computation we use a Turing machine M only input tape and a read-write work tape.
As
with a read
A.J. Mansfield and D.J.A. Welsh
160
If M is a Turing machine which accepts a language over its input alphabet C , then for x E C * , t(x) denotes the t i m e ( = number of transitions) taken by M to halt when x is the input. The Apace s(x) taken by M to accept x is the number of squares of the work tape which are used by M in accepting x. If f (n) is any function on the integers we say M accep-tb L in time f(n) (space f(n)) if t(x) < f(n) (s(x) < f(n)) for all inputs x of length
n. L is in the classP = P-TIME if there is a machine M and polynomial p such that M accepts L in time p(n). P-SPACE and LOG-SPACE are defined analogously. It is well known that LOG-SPACE
z
P 5 P-SPACE
.
A nondetehminiAtic machine (NDTM) consists of a Turing machine together with an additional read only guebb t a p e with a restriction that each square of the guess tape can only be read once, that is a "debthuctiue head" guebb t a p e . The machine operates as follows:for a given input x a c e h t i d i c a t e c(x) (a string from C " ) appears (magically?) on the guess tape and henceforth the machine operates as an ordinary deterministic machine using the certificate c(x) to verify that x is a member of the language in question. The time and space taken by such a nondeterministic machine is the time and space used by its deterministic part. Example 1. Let L be the language of 3-colourable graphs. .Then for a given input graph G, the certificate c(G) could be a 3-colouring of G. This can be checked in time n2 (where n is the number of vertices of G) and in space O(n). Example 2. Let be the language of graphs which are not 3-colourable. No one yet knows a certificate for which would result in a nondeterministic polynomial time algorithm to recognise L. Define NP to be the class of language recognisable in polynomial time by nondeterministic machines. It is well known that P 5 NP 5 P-SPACE, but it is conjectured that P # NP # P-SPACE
Some colouring problems and their complexity
161
Define the class co-NP to be the set of languages L such that the complementary language ( X * \ L) is a member of NP. Again P 5 CO-NP 5 P-SPACE. but it is conjectured that NP # CO-NP. Examples 1 and 2 highlight this problem. Similarly we define NLOG-SPACE (NL) to be the class of languages recognisable in log space by a non-deterministic machine. Again L 5 NL 5 P, L 5 co-NL 5 P, and it is conjectured that the inclusions are proper, and that NL # CO-NL. Cook's fundamental theorem C13 shows the existence of a class of hardest problems in NP called NP-compeete phoblemb. These have the property that if any one of them could be shown to be a member of P then NP = P. Finally a problem 7 1 , not necessarily a language recognition problem, is NP-hahd, if it is provably as difficult as some NP complete problem. Hardest (i.e. complete) problems also exist in NL, P and P-SPACE but they do not concern us here. 93.
2-COLOURABILITY
Testing whether or not a graph is 2-colourable is one of the easiest algorithmic problems in graph theory. Call the language of 2colourable graphs BIPARTITE (otherwise NONBIPARTITE). (3.1)
BIPARTITE
Proof.
For an
(3.2)
E
P.
n-graph there is an
NONBIPARTITE
E
O(n2)
algorithm.
NL
Proof, Guess an odd circuit on the guess tape, call it The deterministic verification consists of:
vl,...,vm.
A .J. Mansfield and D.J.A . Welsh
162
(a) Store v1 on the work tape. (b) Read 'vi check adjacency to vi-l. (c) Store vi and the parity of the number of vertices we have read so far on the work tape, and wipe out vi-l. (d) Check that vm = v1 and that parity is odd. Problem. As far as we know there is no nondeterministic logspace algorithm for BIPARTITE. In fact we conjecture BIPARTITE
(3.3)
1
NL,
Note. If we did not stipulate that the guess tape in our nondeterministic machine could only be read once then it would be easy to show that BIPARTITE is a member of NL. All we would do is guess the 2-colouring and check that for each edge of the given graph its endpoints had different colours. We can certainly do this in logspace if we are allowed to read symbols on the guess tape more than once! Indeed this nondeterministic algorithm would also work for 3-COLOURABILITY. Thus we have: (3.4) If the guess tape in a NDTM is read nondestructively then such a machine can accept in log space any language in NP. 14.
THE CHROMATIC POLYNOMIAL
Consider the problem of computing the chromatic polynomial of a graph G. Clearly this is an NP-hard problem since it contains the problem of finding the chromatic number of G which is known to be NP-complete. We can formulate this as the following decision problems. CHROMPOLY INPUT:
Graph G
QUESTION:
.
and integers al,.. ,an
Is the chromatic polynomial n P(G; A ) = 1 aiA i? i=l
P(G; A )
of
G
given by
NUMCOL INPUT:
Graph
QUESTION:
G,
integers x
and
y
Is P(G; x) = y?
Others, notably Gill C31, Simon C81 and Valiant ClOI have considered classes of counting problems similar to the above. A s defined here
Some colouring problems and their complexity
16.1
such counting problems can be considered within the hierarchy of L, PI P-SPACE etc. (4.1) Proposition. CHROMPOLY Proof. Let CHROMPOLY.
A
NP <=>
E
NUMCOL
E
NP
be a nondeterministic polynomial time algorithm for
Then for a given input (G,x,y) to nondeterministic algorithm
NUMCOL consider the following
(i) Guess al,...,an (ii) Verify (using A ) that P(G; h ) = laiXi (iii) Verify that laix' = y. This is a polynomial length nondeterministic algorithm for NUMCOL. Conversely suppose that NUMCOL
E
NP.
Consider an instance < G ; al,...,an>
of CHROMPOLY.
The following is a nondeterministic algorithm for checking the membership of in CHROMPOLY. (i) (ii)
,...
Guess (bl ,bn) Verify (in polynomial time) that P(G; k) = bk for lsksn. (iii) Solve the linear equations laiki = bk to check in polynomial time that is a member of CHROMPOLY. (4.2) Corollary.
CHROMPOLY
E
NP =>
NP = co-NP.
Proof. The complement of the NP complete problem 3-COLOURABILITY consists of those G for which P(G; 3) = 0 which is a special case of NUMCOL. This is evidence that CHROMPOLY 4 NP. It seems equally unlikely that CHROMPOLY E co-NP but we have no formal evidence for this, analogous to the above corollary. Indeed we make the further conjectures about the status of CHROMPOLY (and NUMCOL) (4.3) It does not belong to any level of the Meyer-Stockmeyer hierarchy (see C2l) (4.4) It is not P-SPACE complete, (though it obviously is a member Of P-SPACE).
A.J. Mansfield and D.J.A . Welsh
164
We remark that if CHROMPOLY is a member of any level of this hierarchy then the whole class of decision problems based on the class #P (studied by Valiant [lo]) is contained in this same level. 15.
THE CURIOSITY OF HAJOS' THEOREM
C4l which gives necessary and suffiConsider the theorem of Ha]& cient conditions for a graph to be k-colourable. In the notation of Ore C61 if and el = (al, bl)
,
G1
and
G2
are two disjoint graphs
e2 = (a2, b2)
are edges in G1 and G2 respectively we construct the c o n j u n c t i o n (or b e h i e b c o n n e c t i o n or tlajo'b j o i n ) of G1 and G2 by identifying the vertices al and a2, to a single vertex, by deleting the edges el and e2 and introducing a new edge (bl, b2). If G1 and G2 are disjoint graphs a mehgeh of GI and G2 is obtained by first performing a conjunction of G1 and G2 and then selecting equicardinal sets A1 = {al, at} 5 V1, and A2 = {bl, bt) 2 V2 and identifying the vertex ai with bi for each i.
...,
...,
(5.1) Theorem. (Ha]&) If G is not (k-1)-colourable then there exists a subgraph of G which is obtainable from copies of Kk by a finite sequence of mergers.
Using this theorem, a graph G can be shown to be not (k-1)-colourable by the following constructive 'proof'. Construct a sequence of graphs Gi (1 5 i i m) where G1 = Kk and each Gi(2 5 i 5 m) is obtainable from some Gj , Gk (j,k < i) by a merger and where Gm is a subgraph of G. We will call any such sequence a phood s e q u e n c e for G, and its L e n g t h the integer m. By Haj6s' theorem such a proof sequence always exists (and is finite). Define hk(G) to be the minimum length of a proof sequence for G when G is not k-1 colourable and to be zero otherwise. alternative way of looking at this proof technique is to regard such a proof of the non-colourability of G in k-1 colours as a rooted directed binary tree with root a subgraph H of G which is not (k-1)-colourable, with leaves copies of Kk and with each vertex, a non-(k-1)-colourable graph Gi obtained by a merger of the two
An
165
Some colouring problems and their complexity
graphs Gjl Gk which directly precede Gi in the tree. We call such a tree a pa006 t a e e for G and for each integer k define. tk(G) to be the minimum number of vertices in such a proof tree where, by convention, we define tk(G) to be 0 when G is (k-1) colourable. Clearly the complexities of colourability as measured by either a 'tree proof' of a 'sequence proof' are closely related in pretty much the same way as formula size and circuit size are related in standard complexity (see for example Savage C 7 1)
.
The following remarks about proof trees and proof sequences are easily verified. (5.2) In any minimal proof sequence are non-isomorphic. (5.3) Any proof tree using a tree of sequence of length i m.
...,Gm>
m
the graphs
Gi
vertices gives a proof
(5.4) Any proof sequence of length m gives a tree proof of size 2m - 1 and this upper bound is best possible.
5
Example.
Consider the sequence G1 = K k + l '
G2 = G1 V G1, G3 = G2 V G2, where
V
represents the operation of a merger.
This sequence forms a minimal proof that G3 is not k-colourable but the equivalent tree proof needs a tree with 7 vertices. This idea can be generalised to arbitrary m.
A corollary of (5.3) and ( 5 . 4 ) is (5.5)
hk(G)
5
tk(G)
5
2
hk (G)
When k = 3, we are looking for proofs of non 2-colourability. Such a proof will basically consist of constructing an odd circuit which is a subgraph of G. Hence we have the easy result: (5.6)
If the smallest odd circuit in
G
has cardinality
2c+l
then
A .J. Mansfield and D.J.A . Welsh
166
For
k
h3(G)
5
t3(G)
S
4
t
+ 1,
[log(2c-l)1 2C-1.
however we have no polynomial upper bound.
This probably reflects the fact that whereas 2-colourability can be decided in polynomial time 3-colourability is NP-hard. Although we have so far been concerned with k-colourability it is a well known fact that the following restricted problem is NP-complete. INPUT:
Planar graph
QUESTION:
Is G
G
with no vertex degree exceeding 4.
3-colourable?
Henceforth therefore we will concern ourselves primarily with the case of 3-colourability and accordingly define the Haj64 numberr h(G) of a graph G to be the minimum length of a proof sequence that G is not 3-colourable (if G is not 3 colourable) and to be zero if G is 3-colourable, in other words h(G) = h4 (G)
.
The following result is almost immediate. (5.7) Theorem. If NP # co-NP then given any polynomial p there exists an infinite sequence (ni : 1 s i s 0 1 ) with ni < ni+l and such that for each i there exists a graph Gi with (a)
IV(Gi)l
5
ni
(b) Gi
is planar, with vertex degrees 3 or 4
(c) Gi
is edge critically 4-chromatic
(d) the Haj6s numbers
h(Gi)
satisfy h(Gi) > p(ni).
Based on Ha]&' proof of his theorem it is straightforward to get the following upper bound on the Haj6s number of a graph, (5.8) Proposition.
Otherwise
For a graph
G
with less than
nL/3
edges,
h(G) = 1.
Proof. When IE(G) I 2 n2/3 then by Turbn's theorem G must contain K4 as a subgraph and hence h(G) = 1. For fixed n let Go be a counterexample with a maximum number of edges. Since Go
167
Some colouring problems and their complexity
cannot contain K4 there must exist distinct vertices a,b,c with the edges e = (a,b), f = (a,c) not present in Go but with the edge (b,c) in Go. Consider Go + e. By assumption h(Go+e)
5
2
n2/3
-
IE(G,)
I -
1.
Let GA + f’ be an isomorphic copy of Go with the edge f’ inserted joining the vertices (a’, c’). Conjunct Go + e with the graph GA + f’ in the obvious way (i.e. identifying a with a’) and removing the edges e, f’. Now identifying all the vertices x’, of GA + f’ with their mates x of Go + e we arrive at the graph Go. Hence we can form a proof sequence for Go consisting of a proof sequence for Go + e l a proof sequence for Gi + f’ together with a final merger of Go + e with GA + f’. This gives
which is a contradiction. Another way of looking at this problem is as follows.
For each positive integer n, define h(n) to be the supremum of h(G) over all n-vertex graphs G. Then by the foregoing: (5.9)
If h(n) is polynomially bounded
coNP = NP.
Since it is thought likely‘that NP # coNP it is unlikely that we should be able to find such a polynomial bound for h(n) What is perhaps more surprising is that we have been unable to find a sequence of graphs which would show that h(n) was not polynomially bounded. Indeed we know very little about h(n).
.
The following results are obvious. (5.10)
h(1) = h(2) = h(3) = 0
(5.11)
h(4) = h(5) = 1,
(5.12)
h(6) = 3 ,
(5.13) h(n) < h(n+l).
A.J. Mansfield and D.J.A. Welsh
168
while from proposition (5.8) it follows that (5.14) h(n)
2
5
2" 3'
In the terminology of Lov6sz C51, Ha]&' theorem does not give a "good characterisation" of non 3-colourability (see C5 1 p.58) and accordingly we could expect there to exist a sequence of "bad graphs" whose Haj6s numbers were exponentially large. A natural candidate sequence is the sequence of critically 4-chromatic graphs constructed by Toft C91 with 4n vertices and at least n2 edges.
The Toft Graph on 12 vertices With some difficulty we were able to show that a graph in this sequence with 4n vertices has Haj6s' number bounded by Alogn + B for suitable constants A and B. 16.
THE COMPLEXITY OF THE HAJOS NUMBER
Consider the problem of finding the Haj6s number of a graph. example if Wn denotes the wheel with n spokes
(6.1)
h(Wn) =
[log (n-111 0 n
For
n odd
even.
However in general we have found this problem exceptionally difficult apart from a few other special cases. The difficult is demonstrated for example by the Grdtsch graph whose Haj6s number we still do not know.
Some colouring problems and their complexity
169
The Gretsch Graph This is edge critically 4-chromatic and hence has Hajds number h(G) > 0. The difficulty we had dealing with this specific graph prompted us to consider the status of the problem: HAJOS NUMBER INPUT:
Graph
QUESTION:
G
and positive integer
Is the Hajds number
t
(in binary form)
h(G) s t?
It is obvious that HAJ6S NUMBER is an NP-hard problem for when t = 0 it becomes 3-COLOURABILITY. However, the problem does not seem to be in NP, indeed at the moment we have no proof that it is in P-SPACE. The difficulty in resolving the status of HAJ6S NUMBER is that it does seem to depend heavily on the value of t given in the input. If t is exponential in the order of the graph G then naively it seems there is no way to avoid having to store at least at ( a > 0 ) of the graphs Gi in a Ha]& sequence for G. Thus it seems that it may be difficult to settle: (6.2)
Problem.
Is HAJ6S NUMBER a member of P-SPACE?
It is however easy to prove: (6.3) If h(n) only belongs to
is bounded by a polynomial then HAJ6S NUMBER not P-SPACE but also to NP.
A.J. Mansfield and D.J.A. Welsh
170
Acknowledgement. We would like to acknowledge several stimulating conversations with theorem. Francois Jaeger on the subject of Ha]&' REFERENCES
e.
c11
COOK, S.A., The complexity of theorem proving procedures, 3rd Ann. ACM Symp. on Theory of Computing, Association for Computing Machinery, New York, (19711, 151-158.
c21
GAREY M.R. and JOHNSON D.S., Computers and Intractibility W.H. Freeman (San Francisco) (1979).
C31
GILL J.T. I11 Computational complexity of probabilistic Turing machines, SIAM J. Comput. (1977) 6 , 675-695.
C41
HAJ6S G. ifber eine Konstruktion nicht n-ftlrbarer GraDhen Wiss. Zeitschr. Martin Luther Univ. Halle-Wittenberg i1961) A 10, 116-117.
C51
LOViSZ L. (1979).
C61
ORE 0. The Four-Color Problem, Academic Press, New York and London (1967).
C71
SAVAGE J.E. The Complexity of Computinq, John Wiley New York, (1976).
C81
SIMON J. On the difference between one and many, Springer Lecture Notes in Computer Science 52 (1977) 480-491.
C91
TOFT B. graphs.
.
Combinatorial Problems and Exercises, North Holland
S .
Sons,
On the maximal number of edges of critical k-chromatic Studia Sci. Math. Hung. 5 (1970) 461-470.
ClOl VALIANT L.G. The complexity of enumeration and reliability problems SIAM J. Computinq 8 (19791, 410-421.
C113 WELSH D.J.A. Matroids and combinatorial optimisation, Lectures: Proc. of Varenna Conf. 1980 (ed. A . Barlotti) C.I.M.E. (to be published).
Annalsof DiscreteMathematics 13 (1982) 171-178 @ North-Holland Publishing Company
A HAMILTONIAN GAME ALEXIS PAPAIOANNOU University of Cambridge England
.
P be a p r o p e r t y of g r a p h s of o r d e r n W e consider the following game, c a l l e d Game P , played on a boahd V c o n s i s t i n g There are two p l a y e r s : Green and Red, and t h e y of n v e h t i c e b . move a l t e r n a t e l y , s t a r t i n g with Green. A move of Green i s j o i n i n g by a gheen edge two v e r t i c e s of V t h a t a r e non-adjacent a t t h a t s t a g e , and a move of Red i s j o i n i n g two v e r t i c e s by a hed e d g e . A f t e r r # ( g )1 moves o f Green and 1 moves of Red a l l p a i r s of V a r e edges, some g r e e n , some r e d . Green wins Game P i f t h e graph formed by t h e r # 1 g r e e n edges has p r o p e r t y P , otherwise it i s a win f o r Red. [ I n f a c t t h e game may end w e l l b e f o r e a l l t h e v e r t i c e s of V a r e j o i n e d by some edge, namely Green wins a s soon a s a l l g r a p h s of s i z e r # 1 on V t h a t c o n t a i n t h e g r e e n edges have p r o p e r t y P and Red wins when no graph of s i z e r # 1 on V t h a t c o n t a i n s no r e d edge h a s p r o p e r t y P . 1 This v a r i a n t of an Achievement Game of Harary C 4 1 was s u g g e s t e d by Bollobds. Let
L$(i)
(2)
(t)
I n t h i s n o t e w e examine G a m e P f o r two p r o p e r t i e s : t h a t of having a Hamilton p a t h and t h a t of having a Hamilton c y c l e . The f i r s t i s t h e Hamilton Path Game and t h e second i s t h e Hamilton C y c l e Game. The Hamilton p a t h game i s e a s i l y a n a l y s e d . For n s 3 Gheen W i n b trivially. However, f o r n = 4 Red had a w i n n i n g b t h a t e g y by c o u n t e r i n g each move of Green w i t h t h e unique edge independent of t h e l a s t move of Green. I n t h i s way t h e game ends i n one of t h e two c o l o u r i n g s of K4 shown i n F i g u r e 1, n e i t h e r of which c o n t a i n s a Green p a t h of l e n g t h 3 .
F i g u r e 1.
The end of t h e game on f o u r v e r t i c e s 171
112
A. Papaioannou
Theorem 1. For n Hamilton Path Game. Proof.
L
5
Green h a s a winning s t r a t e g y f o r t h e
W e s h a l l a p p l y i n d u c t i o n on
n
,
g o i n g from
The main d i f f i c u l t y i s t h e proof of t h e theorem f o r We g i v e t h e d e t a i l s o n l y f o r n = 5. Suppose
V = {1,2,3,4,5).
n
to
n+2.
n = 5
and
6.
W e may assume t h a t a f t e r two moves of
Green and one move of Red, 1 2 and 45 a r e t h e g r e e n edges and 23 i s t h e r e d edge (see F i g u r e 2 a ) . f e r e n t moves:
Red can make f o u r e s s e n t i a l l y d i f -
1 3 , 34, 1 5 and 25.
I n e a c h o f t h e s e c a s e s Green
c o u n t e r s w i t h 35 (see F i g u r e s 2 b , 2 c I 2 d and 2 e ) .
A f t e r t h e s e moves
...........
Figure 2 .
green red
The g r a p h s a f t e r f i v e moves
Green h a s a d o u b l e t h r e a t t o j o i n up t h e p a t h s 354 and 1 2 t o a Hamilton p a t h :
4 1 and 4 2 .
As t h e n e x t move o f Red e l i m i n a t e s a t
most one of t h e s e p o s s i b i l i t i e s , Green c a n win a f t e r h i s f o u r t h move. The c a s e
n = 6
c a n be d e a l t w i t h s i m i l a r l y , b u t t h e r e w e have t o
c o n s i d e r t e n e s s e n t i a l l y d i f f e r e n t sequences of moves. Suppose now t h a t
n t 7
and t h e theorem h o l d s f o r b o a r d s w i t h n-2
vertices. Green can make s u r e t h a t a f t e r two moves o f Green and one move of Red t h e g r e e n edges a r e ala2 and yz and t h e r e d edge alx
, where
w=v-
{a, I a 2 1
is
al,a2,x,y
and z
are distinct vertices.
From now on Green p l a y s h i s winning s t r a t e g y f o r
n-2
Set
v e r t i c e s on
t h e s e t W , w i t h yz . a s h i s s t a r t i n g move. Thus e v e r y move o f Red t h a t j o i n s t w o v e r t i c e s o f W i s c o u n t e r e d by t h e move of
Green a p p r o p r i a t e f o r t h e Hamilton P a t h Game played on W . If a move of Red j o i n s a i t o a v e r t e x w i n W t h e n Green answers by j o i n i n g a t o w, where ' { i , j l = { 1 , 2 l . j
A Hamiltonian game
173
What i s t h e s i t u a t i o n when a l l t h e edges have been drawn? Green h a s a wl-w2 p a t h P of l e n g t h n-3 t h a t c o n t a i n s a l l v e r t i c e s of W. W e may assume t h a t t h e n o t a t i o n i s chosen so t h a t w1 f x . Then w1 i s j o i n e d by a g r e e n edge t o one o f a l and a 2 , s a y a 2 , so t h a t ala2w1m2 i s a g r e e n p a t h c o n t a i n i n g a l l v e r t i c e s o f V. 0 The Hamilton Cycle G a m e is c o n s i d e r a b l y more cumbersome t o a n a l y s e . I d e a l l y one would l i k e t o prove a r e s u l t s i m i l a r t o Theorem 1. S t r a n g e l y enough one c a n prove t h e i n d u c t i o n s t e p from n t o n+2 b u t t h e r e a r e f a r t o o many c a s e s t o d i s c u s s i n o r d e r t o s t a r t t h e i n d u c t i o n f o r n = 8 and 9 , s a y . Theorem 2 . I f Green h a s a winning s t r a t e g y f o r t h e Hamilton Cycle Game on a board with no v e r t i c e s t h e n he h a s a winning s t r a t e g y f o r n0 + 2 v e r t i c e s . Proof. W e know t h a t Green cannot win t h e game on a board o f f o u r v e r t i c e s s i n c e f o r n = 4 Red wins t h e Hamilton P a t h G a m e . Furthermore, f o r n = 5 Red can win t h e Hamilton Cycle Game by making s u r e t h a t a t t h e end one o f t h e v e r t i c e s i s i n c i d e n t w i t h a t l e a s t t h r e e r e d edges. T h e r e f o r e no 2 6
.
Suppose n = no + 2 5 8 and Green has a winning s t r a t e g y f o r n-2 vertices. A s i n t h e proof of Theorem 1, Green can e a s i l y make s u r e t h a t a f t e r two moves of Green and one move o f r e d a b and yz are g r e e n edges and ax is r e d , where a , b r x , y and z a r e d i s t i n c t vertices. S e t W = V - {a,b) A f t e r t h i s t h e s t r a t e g y of Green i s a s follows. ( i ) On t h e board W Green p l a y s h i s winning s t r a t e g y f o r t h e Hamilton Cycle G a m e , t h a t i s i f Red j o i n s t w o v e r t i c e s o f W t h e n Green j o i n s two v e r t i c e s o f W a s i n t h e Hamilton Cycle Game p l a y e d on W I f a l l adges j o i n i n g v e r t i c e s of W have been used, he p l a y s an a r b i t r a r y move. (ii) A Red move of t h e form a w o r bw (w E W) is answered by j o i n i n g a v e r t e x z E W which i s a d j a c e n t t o a or b by a r e d edge t o a o r b , whenever p o s s i b l e . Otherwise Green p u t s i n a n edge a z o r bz , z E W I f t h a t i s a l s o i m p o s s i b l e , Green j o i n s two v e r t i c e s of W. Given a c h o i c e , Green j o i n s z t o t h a t one of a and b which h a s fewer g r e e n edges j o i n i n g it t o W.
.
.
.
Green p l a y s t h i s s t r a t e g y u n t i l t h e v e r y end, p r o v i d e d h e can make s u r e t h a t when o n l y o n e v e r t e x of W i s n o t j o i n e d t o a and b t h e n both a and b are i n c i d e n t w i t h g r e e n {a,b) - W edges. I f
114
A. Papaioannou
t h i s may n o t be p o s s i b l e t h e n a f t e r a move o f Red t h e r e i s o n l y one v e r t e x of
,
W
say
u
, not
i n c i d e n t w i t h any
r e d e d g e s j o i n a l l o t h e r v e r t i c e s of
W
to
{a,b}
-w
a , say.
edge and
Then Green
d e v i a t e s from t h e s t r a t e g y g i v e n i n ( i i ) t o p l a y a n e x c e p t i o n a L m o v e : au
.
Note t h a t when t h e e x c e p t i o n a l move i s made t h e
{arb}
-
W
edges are d i s t r i b u t e d i n one o f two e s s e n t i a l l y d i f f e r e n t ways, shown i n F i g u r e 3a, 3b.
From h e r e on Green c o n t i n u e s t o answer
(a) F i g u r e 3. each
{arb}
-W
The e x c e p t i o n a l move edge by one o f t h e same k i n d , g i v i n g p r e f e r e n c e t o
.
an edge i n c i d e n t w i t h
a
What c o l o u r i n g o f
s h a l l w e o b t a i n when a l l moves a r e completed?
(A)
Kn
Suppose Green i s n o t f o r c e d t o p l a y h i s e x c e p t i o n a l move.
w1 , As Green p l a y e d
Then when t h e game i s o v e r , W h a s e x a c t l y one v e r t e x , s a y which i s j o i n e d t o b o t h
a
and
h i s winning s t r a t e g y on
W,
t h e r e i s a green cycle
c o n t a i n i n g a l l v e r t i c e s o f W. joined t o one of
a
a r e a d j a c e n t t o some
and
wi
b
b
by a r e d edge.
1
and
wi+lb
a r e green edges.
Then
i
a
, is a
and
b
T h e r e f o r e w e may assume
by a g r e e n edge.
w i t h o u t loss o f g e n e r a l i t y t h a t f o r some w.a
C = w ~ w ~ . . . w ~ - ~
Each wi, 2 5 i 5 n-2 by a g r e e n edge and b o t h
,
2 s i
5
,
n-3
both
W ~ W ~ - ~ . . . W1 wn-2wn-3' ' .wi+l
i s a g r e e n Hamilton c y c l e , showing t h a t Green h a s won t h e game. (B)
final
Suppose Green h a s t o p l a y h i s e x c e p t i o n a l move.
{a,b)
-W
Then t h e
edges a r e t h e edges shown i n F i g u r e 3 t o g e t h e r
w i t h two more r e d e d g e s and one more g r e e n edge.
T h e r e f o r e one o f
two c a s e s w i l l o c c u r .
i s j o i n e d t o b o t h a and b by r e d e d g e s , e a c h o t h e r v e r t e x of W i s i n c i d e n t w i t h a g r e e n { a , b l W edge and both a and b a r e i n c i d e n t w i t h g r e e n {a,b} W edges. T h i s i s e x a c t l y t h e same a s t h e d i s t r i b u t i o n d e s c r i b e d i n ( A ) so Green wins t h e game. (B1)
One v e r t e x of
W
-
-
A Hamiltonian game
175
(B2) There a r e j u s t two v e r t i c e s i n W , s a y w1 and w2 , which a r e j o i n e d t o b o t h a and b by red edges, t h e r e i s j u s t one v e r t e x i n W j o i n e d t o both a and b by green e d g e s , a i s i n c i d e n t w i t h 2 green edges j o i n i n g i t t o W and b i s i n c i d e n t w i t h (n-2)-3 = 51-5 2 3 g r e e n 'edges j o i n i n g it t o W. Furthermore, a s Green played h i s winning s t r a t e g y on W , t h e r e i s a g r e e n c y c l e C of l e n g t h n-2 c o n t a i n i n g a l l v e r t i c e s of W
.
Suppose C = w1P1w2P2w1 , t h a t i s C i s t h e union of t h e w1-w2 p a t h s P1 and P2 W e may assume t h a t P1 has d i n t i n c t i n t e r n a l v e r t i c e s such t h a t one i s j o i n e d w i t h a g r e e n edge t o a and t h e o t h e r i s j o i n e d w i t h a g r e e n edge t o b S i n c e each i n t e r n a l v e r t e x of P1 i s j o i n e d t o a t l e a s t one of a and b w i t h a g r e e n edge, t h e r e a r e v e r t i c e s z1 and z2 such t h a t t h e y a r e neighbours on P1 and both a z l and bz2 a r e green edges. F i n a l l y , w e may assume t h a t going a l o n g P1 from w1 t o w2 t h e v e r t e x z1 p r e cedes z 2 Then azlP1w1P2w2P1z2b i s a g r e e n Hamilton c y c l e . 0
.
.
.
As w e mentioned e a r l i e r , a t t h e moment w e c a n n o t u s e Theorem 2 t o
prove t h a t t h e Hamilton Cycle Game i s a win f o r Green f o r a l l v a l u e s of n t h a t a r e n o t t o o l a r g e . However, w e make t h e f o l l o w i n g conjecture. Conjecture. Green h a s a winning s t r a t e g y f o r t h e Hamilton Cycle Game on a board of a t l e a s t e i g h t v e r t i c e s .
0
To conclude t h e paper w e prove t h a t t h e game i s a win f o r Green provided t h e board i s s u f f i c i e n t l y l a r g e . Theorem 3. I f n i s s u f f i c i e n t l y l a r g e t h e n Green c a n win t h e Hamilton Cycle Game on a board of n v e r t i c e s . W e need a lemma s t a t i n g t h a t Green can win a v a r i a n t of Proof. t h e game i f he s t a r t s w i t h a c e r t a i n advantage. The proof is r a t h e r cumbersome and w i l l be g i v e n elsewhere.
Lemma 4 . Suppose t h e f i r s t move of Green i s p u t t i n g i n t h r e e independent edges and from t h e n on Red and Green a l t e r n a t e a s before. Then Green c a n c o n s t r u c t a Hamilton p a t h j o i n i n g two preassigned v e r t i c e s .
0
Now t o prove t h e theorem assume t h a t n L 6 0 0 , s a y . W e s h a l l desc r i b e b r i e f l y a winning s t r a t e g y € o r Green. (1) The f i r s t 2 0 0 moves of Green a r e independent edges. S i n c e
A . Papaioannou
176
(lZo)
> 200
,this
is possible.
Green u s e s t h e n e x t 50 e d g e s t o make s u r e t h a t he h a s a
(2)
t o w50 , o f l e n g t h a t most 50 ( p o s s i b l y p a t h P , s a y from w1 much less) such t h a t a f t e r t h e f i r s t 250 moves of Red e v e r y v e r t e x i n c i d e n t w i t h a t l e a s t 25 r e d e d g e s i s a n i n n e r vertex o f P To
.
o b t a i n such a p a t h w e make u s e of t h e f a c t t h a t i f a g r e e n p a t h ends
i n x and none o f wx , wy adding xw and t h e n one o f
and wz i s a r e d edge t h e n f i r s t wy o r wz , t h e g r e e n p a t h c a n be
y
or
z.
xuvy
or
xvy
c o n t i n u e d t o end i n
i n x , y i s a g i v e n v e r t e x and f i r s t a d d i n g ux and t h e n vy
by a d d i n g e i t h e r
Furthermore, i f a green p a t h ends
u,v a r e isolated vertices , the , o u r p a t h c a n be e x t e n d e d t o y
.
What do we know a b o u t t h e g r a p h t h a t w i l l a r i s e a f t e r t h e f i r s t 250 moves o f Green and t h e f i r s t 250 moves o f Red? There are a t l e a s t 200-51 independent g r e e n e d g e s having no v e r t e x i n common w i t h t h e g r e e n w1-w50 p a t h P. I f xy i s s u c h a g r e e n edge t h e n x and y t o g e t h e r a r e i n c i d e n t w i t h a t most 48 red e d g e s . By a theorem o f C o r r d d i and H a j n a l C21 (see a l s o C31 and C1;Ch.VI) we f i n d t h a t t h e s e independent edges c a n be p a r t i t i o n e d i n t o 4 9 c l a s s e s a s e q u a l as p o s s i b l e such t h a t no r e d edge j o i n s e n d v e r t i c e s o f t h e e d g e s bel o n g i n g t o t h e same c l a s s . To see t h i s w e a p p l y the Corrtidi-Hajnal theorem t o t h e g r a p h o b t a i n e d from t h e g r a p h o f r e d e d g e s spanned by t h e i n d e p e n d e n t g r e e n edges i n which e a c h o f o u r g r e e n e d g e s have been c o n t r a c t e d t o a v e r t e x . I n t h i s c o n t r a c t e d graph every v e r t e x h a s d e g r e e a t most
2.24 = 48
,
hence t h e 49 c l a s s e s .
a r e a s e q u a l a s p o s s i b l e , e a c h o f them h a s a t l e a s t edges. Let
As t h e c l a s s e s L149/48J = 3
be t h e s e t o f v e r t i c e s o f t h e e d g e s i n t h e i t h c l a s s ,
W!
..., 4 9 . Note t h a t I W i I 6 . S i n c e w1 i s i n c i d e n t w i t h a t most 2 4 red e d g e s , we may assume t h a t no r e d edge joins w1 t o Wi . S i m i l a r l y w e may assume t h a t no r e d edge joins w50 t o Wag . A f t e r a s u i t a b l e renumbering of t h e sets W i w e c a n 3
i = 1,2,
2
Wi
enlarge
v
t o a set
u
= V(P)
I,,,uwil 49
Wi-1
n
w 3.
wi
n
49
u wi ,
i=l
nV(P) =
wi
such t h a t
Wi
=
= p
tWiI
,
tw1,w50~,
,
2
5
w1
c
w1 , Wso
i s 49
,
1 s j < i s 4 9 .
c
w49 ,
177
A Hamiltonian game
Figure 4 .
The sets
Wi
and some of t h e g r e e n e d g e s
To f i n d such sets Wi n o t e t h a t i f a v e r t e x w is n o t j o i n e d t o either W i or Wk t h e n w c a n p l a y t h e r o l e o f wi i f W i is renumbered W;-l and W d is renumbered W i Furthermore, t h e r e a r e a t l e a s t 600-2.250-49 = 51 v e r t i c e s t h a t a r e n o t i n n e r v e r t i c e s o f P and a r e i n c i d e n t w i t h no r e d e d g e s . Having found
.
distinct vertices W;
= {wl,w21 u
w1,Wi
W ~ , W ~ , . . . , W ~E ~ V - V ( P )
= ' i w 2 , w 3 1 u w2
,...,wig
s u c h t h a t t h e sets = ~ w ~ ~ u, wqg w ~ a~r e I
independent i n t h e g r a p h spanned by t h e r e d e d g e s , one by one we enlarge t h e
W;
s
so t h a t t h e y s a t i s f y t h e remaining c o n d i t i o n s .
T h i s is e a s i l y done s i n c e a v e r t e x i n most
24 (3)
W1,W2,...IW49
is i n c i d e n t w i t h a t
Having d e c i d e d on an a p p r o p r i a t e c h o i c e o f t h e sets , Green is concerned o n l y w i t h t h e r e d e d g e s j o i n i n g
v e r t i c e s of t h e same s e t board
V-V(P)
r e d edges.
Wi
v e r t e x set
.
H e p l a y s h i s s t r a t e g y t h a t on e a c h Wi g u a r a n t e e s him a g r e e n p a t h from wi-l t o wi w i t h Wi
.
The union o f
P
and t h e s e
49
p a t h s is a
Hamilton c y c l e on t h e e n t i r e b o a r d .
17
With more c a r e t h e bound 600 can be lowered b u t o u r method is u n l i k e l y t o p r o v i d e a bound t h a t is c l o s e t o b e i n g b e s t p o s s i b l e . REFERENCES
C11
C21
B. B o l l o b d s , Exthemat Ghaph T h e o h g , London Math. SOC. Monographs No.11, Academic P r e s s , London and New York, 1978. K. C o r r d d i and A. H a j n a l , On t h e maximal number of i n d e p e n d e n t c i r c u i t s i n a g r a p h , A c t a Math. Acad. S c i . Hungar. 2 (1963) 423-439.
178
A . Papaioannou
[31
A. Hajnal and E. SzemerBdi, Proof of a conjecture of P.ErdBs, in C o m b i n a t o 4 i a . t Theohy a n d i t n A p p L i c a t i o n . 5 , vol.11 (P.Erdbs, A. Rgnyi, V. S ~ S ,eds.), Collog. Math. SOC. J. Bolyai 4, North Holland, Amsterdam-London, 1970, 601-623.
C41
F. Harary, Achievement and avoidance games for graphs, in Glraph Theohy (B. Bollobbs, ed.), Annals of Discr. Math. 1982.
Annals of Discrete Mathematics I3 (1 982) 179-190 0 North-Holland Publishing Company
FINITE RAMSEY THEORY AND STRONGLY REGULAR GRAPHS J. SHEEHAN University of Aberdeen Scot land
We discuss some connections C51, C61, C91, C l O l between strongly regular graphs and finite Ramsey theory. All graphs in this paper are both finite and simple. Let G1 and G2 be graphs. Then t h e Ramdeq numbeh, r(G1, G 2 ), 0 6 G1 and G2 is the smallest integer n such that in any 2-colouring (El, E2) of the edges of Kn either (El> 2 G1 or 'E2> 1. G2. So, thinking of El and E2 as being "red" and "blue" edges respectively, if the edges of Kn are coloured red and blue then there exists either a red G1 or a blue G2. Furthermore, since n is minimal, there must exist a graph G on n-1 vertices so that G G1 and its complement G2. A AthOngLq h e g u t a h ghaph w i t h pahametehd (v,k,X,u) (or more briefly we say a (v,k,X,p)-graph) is a graph which is regular of degree k on v vertices and is such that there exist exactly X ( p ) vertices mutually adjacent to any two distinct adjacent (non-adjacent) vertices. Excellent elementary introductory articles on strongly regular graphs by Cameron and Seidel appear in C11 and C21 respectively. Notice that if G is a (v,k,X,p)-graph then is a (v, v-1-k, v-2k+p-2, ~-2k+A)-graph. Now let Bn (n 2 1) denote the graph K2 + fn (see C81 for notation). Then the interaction between strongly regular graphs and Ramsey theory which we wish to discuss is made formally by the following observation.
i
OBSERVATION.
If there exists a
i
(v,k,X,p)-graph G
then
This follows since G BX+l, Bv-2k+p-l and G has exactly v vertices. Now we can consider this inequality from two viewpoints. If a particular (v,k,X,p)-graph exists then this determines a lower bound for the corresponding Ramsey number. We give an example of this approach in section 1. On the other hand if we can independently determine an upper bound f o r a particular Ramsey number this 179
J. Sheehan
180
gives some information as to the non-existence of strongly regular graphs with the appropriate parameters (and usually of course to the non-existence of a much larger class of graphs). We take this viewpoint in section 2. In section 3 , which might simply be regarded as an appendix, we give a proof of one of the theorems quoted in section 2. We do not recommend the reader to pursue all the details of the proof but simply to note its elementary character. In the final section we make two conjectures. Section 1 We prove in C91 the following theorem and corollary:Theorem 1. If 2 (m+n) + 1 > (n-m)2/3 then r(Bm,Bn) 5 2(m+n+l). By refinement, r(Bn-llBn) 5 4n - 1 and, if n 2 (mod 31, r(Bn-21Bn) L 4n - 3 . Corollary. If 4n + 1 is a prime power, then r(BnlBn) = 4n + 2. If 4n + 1 cannot be expressed as the sum of two integer squares, then r(Bn,Bn) 5 4n + 1. In the first example of the latter, r(B5,B5) = 21. We can indicate the main idea of the proof of the theorem by directly proving the Corollary. This proof illustrates the first vie-oint on the observation made in the introduction. Proof
(of the Corollary).
Let p,n 2 1. Suppose Kp +> (BnlBn). Let (E1,E2) be a 2colouring of the edges of K so that <EL> B, (i = 1,2). Let M P be the number of monochromatic triangles produced by this colouring. Then a classical result of Goodman C71 gives
On the other hand since on each red (blue) edge there exist at most n-1 red (blue) triangles
From (1) and (21, p 5 4n + 1. Hence r(BnlBn) L 4n + 2. Suppose p = 4n + 1. Then equality holds in (1) and (2). Write G = <El>. Goodman's result also tells us that since equality holds in (11, G is regular of degree 2n. Equality in (2) implies that on each edge of G there are exactly n 1 triangles and on each edge of
-
181
Finite Ramsey theory and strongly regular graphs
there are exactly n-1 triangles. So G is a (4n+l, 2n, (n-l,n)graph. Hence, using our observation, r(BnIBn) = 4n + 2 if and only if there exists a (4n+l), 2n, n-1, n)-graph. Such graphs C41 are called conference graphs and are well known to exist if 4n + 1 is a prime power. No such graph exists if 4n + 1 cannot be expressed as the sum of two integer squares. In the first example of the latter r(B5,B5) = 21. This is proved by giving C91 a direct construction of a graph on 20 vertices with the required properties. 0 Section 2 We have proved C61, [911 Theorem 2. (i) r(BIIBn) = 2n + 3
(ii) 2n
Corollary.
+
3
5
r(B2,Bn)
5
I
(n
6
(2
2n
+ +
5
(12
2n
+
4
(22 5 n 137
2n
+
3
(n
2n
r(B2,Bn) = 2n + 6, n
Theorem 3. Suppose 1 i k n 2 (k-1)(16k3 + 16k2 24k
-
i
-
2)
2
=
n
5 5
2
5
11)
n 521) )
38).
2,5,11.
n. Then r(Bk,Bn) = 2n + 3 for 10) + 1.
Coment. We give a proof of Theorem 3 in the next section. Otherwise all proofs (see C61) are omitted. We simply interpret these results (or at least some of them) in the context of our viewpoint that upper bounds for Ramsey numbers provide information as to the existence of certain strongly regular graphs.
-
Let t 2 1. Write n = 3t 1 and let G(n) denote (if it exists) a (6t+3, 2t+2, 1, t+l)-graph. Then if G(n) exists r(B2,Bn) 2 2n + 6 and so by Theorem 2 (ii), r(B2,Bn) = 2n + 6. Now someone (see Cameron E l ] ) with knowledge of the theory of strongly regular graphs would proceed as follows. If G(n) exists then (using the so-called i n t e g h a t i t y condition) (V /(p
is an integer, where
-
-
1) ( p
-
A)
X I 2 + 3(k
k = 2t + 2, X
=
-
2k
-p
1, p
= t
+ 1 and v
=
6t + 3 .
182
J. Sheehan
Hence t = 1,2,4,10. The line graph L(K3,3) of K 3 , 3 1 the complement of L ( K 6 ) and the line graph of the 27 lines on a cubic surface show respectively that G(2)I G(5) and G(11) exist. However as yet we have not determined whether G(29) exists. The only other general necessary condition for the existence of a (v,k,A,u)-graph are the so-called K h e i n conditions (see Seidel C21). This states that if r+s=X-uand
r s = p - k
( r > s )
then (i) (ii)
(r + 1) (k + r (s
+ 1) (k +
s
+
+ (k +
2rs) s (k
+ 2rs)
s
r) ( s
+
s ) (r
+
2 1) 1)2
.
In our case r + s = -10, rs = -11 and so r = 1, s = -11. We see that these values of r, s and k = 22 do not satisfy the second Krein condition. Hence no G(29) exists. Now suppose we know nothing about the theory of the existence of strongly regular graphs. Then, by Theorem 2(ii), no G(n) exists for n 2 12. So in particular G(29) does not exist. Hence we do not require the Krein conditions to prove the non-existence of G(29). However against this the existence of G(8) is undecided by our Ramsey theory, whereas the integrality test shows that no G(8) exists. Generally we may interpret Theorems 2 and 3 as follows:"if r(Bm,Bn) < N then there exists no (vIk,m-l,n+2k-v+l) -graph with v ;? N". Of course a much stronger there exists no graph G of G there are at most there.are at most n
-
statement is also true viz., for v 2 N on v vertices such that (i) on each edge 1 triangles and (ii) on each edge of m 1 triangles.
-
Section 3. Almost all of our notation in this section will be standard (C31, C81). There is one exception. Let G be a graph and x E V(G). Then N(x) denotes the neighbourhood of G and for any subset Y 5 V(G) we write Y(x) = : N(x) n Y.
183
Finite Ramsey theory and strongly regular graphs
Furthermore if
x1,x2 Y(xl
E
V(G)
we write
x,) =
:
Y(xl)
Y(Xl u x,) =
:
Y(Xl) u Y(x2).
n
n
Y(x2)
and
This is simply a notational device to restrict the number of symbols used. In the proof of the theorem below we shall need to consider a 2colouring (ElIE2) cf the edges of K2n+3 (n 2 1). As above we call the edges of El red and those of E2 blue. In general a suffix i (i = 1,2) will refer to the i-th colour. For example if v E V ( G ) , Nl(v) is the red neighbourhood of v and if Y V(K2n+3) , Y2(v) is the blue neighbourhood of v contained in Y. Again if Y 5 V(K2n+3) , l is the subgraph of K2n+3 with vertex set Y and edge set consisting of all red edges with both end vertices in Y. We abuse this notation very slightly when we present and use the next lemma but it is only in this context that we shall do this and no confusion should arise.
Lemma.
Let A1, A , ' . . . ' Am (m 2 2 ) be subsets of a finite set A. Suppose 6 and 1.1 are integers such that for all i , j E Ilr2r.. , m I r i # j, IAi u A . 1 2 6 and [Ai n A.1 2 1.1. Then
.
3
3
- m(m - 2)u 1'2,. ..,mI i # j. Then = [Ai u A + I A ~ n A.I.3 j 2
Proof. Choose
I Ai
m6
Summing over all possible such pairs
i
and
j we obtain
But
The lemma now follows from ( 3 1 , ( 4 ) and the definitions of
6 and 1.1 .O
Theorem. Suppose k Then
5
and
n
are integers such that
1
5
k
n.
J. Sheehan
184
r(BkIBn) = 2n provided
n
2
(k
-
+
1) (16k3 + 16k2
3
-
24k
-
10)
+ 1.
Proof. We may in fact suppose k > 1 since the case k = 1 is proved in C91. Since Kn+l,n+l does not contain a Bk and its complement does not contain a Bn we have r(Bk,Bn)
2
2n
+
3.
(5)
Unfortunately to prove equality is not so straightforward. Suppose (BkIBn), n 2 (k 1) (16k3 + 16k2 24k - 10) + 1. Choose K2n+3 +> (E1,E2) to be a 2-colouring of the edges of K2n+3 so that (El> Bk and <E2> Bn. Choose a , $ E V(K2n+3) so that af3 E E2 and IN1(") n N1(B) I is as large as possible. Write D = Nl(a) n N1(f3) , A = N2(a) n N2(B), B = N2(a)\A, C = N2(B)\A and H = B u C u D (see Figure 1, the broken lines indicate red edges). We assume t
-
IBI
IcI.
FIGURE 1
-
185
Finite Ramsey theory and strongly regular graphs
We emphasise the choice, in particular the maximality, of IDI. will play a prominent role throughout the subsequent arguments. proceed by a series of propositions. Proposition 1. (1)
I A ~5
n
1
(2)
[HI
-
(2n
+
(3)
IBl(b)( s k
-
1 (b
E
B), /C,(C)~ 5 k
IDl(d) I
5
-
1 (d
E
D)
(4)
IH1(X) I
5
(5)
IH,(d)
(6)
IH2(x)I
=
k
(k
I
-
1)
-
2(k
n t 2
1) + ID1 (X
-
(IHI
2
I A ~t
-
1) 1)
-
(d
E
B u C)
E
D)
-
It We
1 ( C . EC),
max{2(k-l),(k-l) + (Dl1 (x
E
H)
Proof. This proof follows directly from the various definitions. For example Proposition 1 (4) (which we abbreviate to P.1.4) is proved by using P.1.3 and the maximality of IDI. 0 When the reader is in doubt he should refer back to this proposition which will not always be quoted. Proposition 2.
2
K3.
Proof. Write G = 2. Then the minimal degree, 6 ( G ) , of satisfies, using P.1.3 6(G) t
If
G
i K3
IBI
-
G
k.
then, by Turan's theorem C31, BI 21 4 .
IE(G)I
Hence, from (6) and (7), IBI
ID/
=
(2n + 1)
2
(n
+ 2)
-
-
5
2k.
( A1 +
Therefore, using P.1.1, IB/ + ICo
4k.
We now use the same argument for K = : 2. Again if K K3, ID1 s 2k. So from ( 8 1 , n s 6k + 2. This is a contradiction. Proposition 3.
ID1
2
2k2 + 1.
J. Sheehan
186
Proof. Let 0 = max{2(k
-
v1,v2,v3 be the vertices of a triangle in 2. Write l), k - 1 + ID/). Then, from P.1.6, for i,j E {1r2r3Ir
i # j
Since
<E~$ > B ~ ,from ( 9 1 ,
By the maximality of ID1 , for all pairs IA1(vi n vj) I 5 ID[. Hence IA2(Vi u Vj)I Write m = 3.
2
i
and
j
above,
/ A /- IDI.
(11)
= A ~ ( v ~(i ) = 1,2,3), 1-1 = - n + 28, 6 = Then from the lemma, (10), (11) and P.l.l
IA~
I A ~ - IDI
and
Now suppose 0 = 2(k - 1). Then ID1 5 k - 1 and from (12), n 2 15k - 17, which is not true. On the other hand if 0 = (k - 1) + ID1 then ID1
t
(n
-
6k
+
8)/9
(13)
and so from (13) and the magnitude of
n, ID1
Proposition 4.
+
l has at least
2k
1
2
0
2k2 + 1.
independent vertices.
Proof. Let G = l. Then, from P.1.3 and P.3, G is a graph with maximal degree at most k - 1 and G has at least 2k2 + 1 vertices. The result now follows as an elementary exercise in graph theory. Proposition 5.
ID1
2
((2k
-
1)(AI)/ (2k
+ 1)
- (8k2 -
14k
+
0
3).
Proof. Let V ~ , V ~ , . . . , V ~ be ~+~ distinct independent vertices of l. Let m = 2k + 1 and write Ai = A2 (vi) (i = 1,2,. ,m) Now vi's not with minor modifications (allowing for the fact that the only belong to H but also to D) we repeat the argument of P.3. Let i,j E {lr2r...rmIr i # j. From P.1.5
.. .
Finite Ramsey theory and strongly regular graphs
Hence, since <E2>
Bn
I A ~ n A . I3
and using P.1.2,
(n 4k -
s (n s 5
1) -
I H ~ ( vn ~v.11
1)
-
IHI+ 4 ( k
3
1) + 2 (14
5.
Again, by the maximality of ID1 and since IAl(vi n vj)I 5 I D 1 - 2. Hence ]Ai
UAjI
187
2
IAl
-
ID1
+
a,B
N1(vi) n N (v ) 1 j
E
2.
(15)
Write 6 = [ A ]- I D 1 + 2, p = 4k - 5. Then the result follows from U the lemma using m = 2k + 1 and equations ( 1 4 ) and (15). Proposition 6.
(1) If
x1,x2
E
and
D
IH2(x1 n x2)/ 2
n
-
IHZ(xl n x2)I
2
(IH
(2)
IAl
4(k
x1x2
-
2
n
-
1).
E
then
E2
4(k
-
1)
4(k
-
1)
Proof. (i) From P.1.5.
= (2n
-
2) 1)
-
-
[A1
-
4(k
-
1).
The result now follows from P.l 1. (ii) Since ID1 2 2k2 + 1 least one blue edge x1x2 with follows from (16). Proposition 7.
and l $ Bk there exists at xi E D , i = 1,2. The result now 0
l does not contain 2 independent edges.
Proof. Suppose < D > l does contain 2 indeDendent edges. Then we may choose xi,yi E D (i = 1,2) so that xlYl and x2yz are red edges and such that X1X2, X1y2, y1x2, yly2 are all blue edges. for i = 1,2, since a,B E N1(xi n yi) and since <El> 3 Bkl IA1(xi n yi) I s k Hence IA2(XI u yi) I
2
IAl
-
3.
-
(k
-
3).
Therefore, with no loss of generality, we may suppose that
Then,
J. Sheehan
188
NOW, from (171,
-
IA2(X1) \A2 (x2 u y2) I s k
(19)
3.
Therefore from (19), again with no loss of generality, we may suppose that IA2(x1 n x,) Hence, from P . 6 . 1 ,
-
n
i.e.,
n s 23k
Proposition 8 . Then
2
(IA2(x1)I
-
-
(k
3))/2.
2
IA2(x1 n x2) I + IH2(x1 n x2) I
2
(n
-
(20)
(181, ( 2 0 ) and since <E2>
P.6.2,
1
-
I
7k
+ 13)/4 + (n
-
4(k
Bn
- 1)) 0
33 which is a contradiction. X = B u C
Write ID*(
2
ID1
-
(2k2
and let D* = Ed
-
6k
+
E
D :IX1 (d)l
11.
2
7).
Proof. Firstly notice that at most one element of D is isolated in l i.e. IHl(d)l 2 1 for all but at most one element of D. Otherwise choose two such elements dl,d2 E D, dl # d2. Then, using P.1.2
-
n
1
2
IH2(dl n d2)I
2
\HI - 2
2
n.
Now write G = l. Then G is a graph with at least 2 k 2 + 1 vertices and maximal degree at most, using P.1.3, k - 1. Since G has at most one independent edge xy an elementary exercise in graph theory (see Figure 2) shows that G contains at least 2 ID/ (2 + ( k 2) + 2 ( k 2) ) isolated vertices.
-
-
0
0
0 0 . .
-
..
..
0
0 0
0 . .
. . ..
0
0
isolates 5
2(k
- 2)2
5
2(k
-
= 2
FIGURE 2
2)
189
Finite Ramsey theory and strongly regular graphs
This, together with the opening remark, proves the proposition. ID1 s ((2n + 1
Proposition 9 .
-
IAl)(k - 1) + 2k2
-
6k + 7))/k.
Proof. Let E1(B u C, D) denote the set of red edges with one end vertex in B u C and the other end vertex in D. Then, from P.8,
Since
<El>
Bk
if
x
E
B u C, ID1(x)I
IE1(B u C,D) I
5
IB
u
= (2n
CI (k
+
1
-
5
k
-
1)
-
IAI
-
1. Hence
lDl)(k
-
1).
(22)
The proposition follows from (21) and (22). Proof of Theorem 3. k(2k
-
1) I A l
-
From Propositions 5 and 9
k(2k + 1) (8k2- 14k + 3)
2
+
(2k + 1) (2n + 1 (2k
+
-
( A ( ) (k-1)
1)(2kL + 6k + 7).
Now use P.6.2 to obtain a contradiction to the magnitude of n. Hence K2n+3 + (Bk,Bn) and so, from (51, r(Bk,Bn) = 2n + 3.
0
A similar C61, but very much more delicate, analysis proves Theorem 2(ii). Theorem 2(i) is proved in C91. Section 4 We have conjectured in C 9 I . Conjecture 1.
There exists a constant
A
> 0 such that
Our theorems support this conjecture although of course they are a very long way from giving the whole picture. The well-known (253, 112, 36, 60)-graph shows that r(B37,B88) 2 254 and so A 2 2. This conjecture would imply, if true, the truth of:Conjecture 2. For any (A 2 2) such that where
a = k
-h -
(v,k,h,p)-graph there exists a constant
2(a + 8 ) - V 1 and 6 = k
2
-
A
A p.
0
190
J. Sheehan
Comment. This conjecture is true for conference graphs. It is also true when A = p. For readers not familiar with the parameters c1 and $ it is worthwhile recalling in this context that if we write R = v - k - 1 then since k(k - A - 1) = R u ,
Finally we would like to mention that if instead of discussing r(Bm,Bn) we consider the Ramsey number r(Km + En) then in C51 and [lo1 conference graphs are used to provide lower bounds. In El01 conference graphs are used to provide lower bounds. In [lo] especially the asymptotic lower bounds are discussed in some depth. REFERENCES c11 Beineke, L.W. and Wilson, R.J., Selected topics in graph theory, Academic Press, London (1978). c21
Bollob6s, B., Surveys in combinatorics, (Proceedings of the 7th British Combinatorial Conference), C.U.P., Cambridge (1979).
131
Bondy, J.A. and Murty, U.S.R., Graph theory with applications, Macmillan, London (1976).
141 Cameron, P.J. and Van Lindt, J.H., Graph theory, coding theory and block designs, C.U.P. , Cambridge (1975). C51
Evans, R.J., Pulham, J. and Sheehan, J., On the number of complete subgraphs contained in certain graphs, J.C.T. (to appear).
C61
Faudree, R.J., Rousseau, C.C. and Sheehan, J. Strongly regular graphs and finite Ramsey theory, (submitted).
L7 1
Goodman, A.W. On sets of acquaintances and strangers at any party, Amer. Math. Monthly 66 (1959) p.778-793.
181
Harary, F.
C91
Rousseau, C .C. and Sheehan, J. J.G.T. 2 (1978) p.77-87.
Graph theory, Addison-Wesley, Reading, Mass. (1969).
[lo] Thomason, A.G.
On Ramsey numbers for books,
Ph.D. Thesis, Cambridge (1979).
Annals of Discrete Mathematics 13 (1982) 191-202 0 North-Holland Publishing Company
THE CONNECTIVITIESOF A GRAPH AND ITS
COMPLEMENT^
RALPH TINDELL Department of Pure and Applied Mathematics Stevens Institute of Technology Hoboken, N.J. 07030, U.S.A.
I t i s an elementary observation t h a t t h e
complement o f a d i s c o n n e c t e d g r a p h must b e T h i s f a c t may b e viewed a s a
connected.
n e c e s s a r y c o n d i t i o n for a p a i r k , E o f nonnegative i n t e g e r s t o be realizable a s t h e p o i n t - c o n n e c t i v i t i e s of a g r a p h and i t s comp l e m e n t : k+E must b e p o s i t i v e .
This observation
l e a d s n a t u r a l l y t o t h e problem of determining t h e p a i r s of nonnegative i n t e g e r s k,E f o r which t h e r e e x i s t s a p-point connectivity
k
graph having p o i n t -
and whose complement h a s
E.
point-connectivity
I n t h e p r e s e n t paper
w e show t h a t s u c h p a i r s are c o m p l e t e l y d e t e r mined by t h e f o l l o w i n g c o n d i t i o n s : (1) 0 < k + E 2 p-1; ( 2 ) i f k+E = p-1 t h e n a t l e a s t o n e o f E l k i s e v e n ( 3 ) i f one o f k,; i s p-2, t h e n t h e o t h e r i s z e r o . 1.
INTRODUCTION
of a disconnected graph
S i n c e t h e complement
by a c o m p l e t e b i p a r t i t e g r a p h i t must b e c o n n e c t e d .
G
i s spanned
I n the notation
o f t h e book [ 4 1 by H a r a r y , which w e h e n c e f o r t h assume, t h i s may b e restated as
K
(G)
+
K
(G) .>
0.
p a i r s of nonnegative i n t e g e r s
T h i s l e a d s t o t h e q u e s t i o n o f which k ,E
o c c u r as t h e p o i n t - c o n n e c t i v i t i e s
As w e s h a l l s e e , k+E > 0 i s b o t h n e c e s s a r y a n d s u f f i c i e n t i f t h e number p o f p o i n t s of t h e g r a p h i s unrestricted. T h i s w i l l be a p p a r e n t from o u r s o l u t i o n of t h e more o f a g r a p h a n d i t s complement.
d i f f i c u l t v e r s i o n o f t h e p r o b l e m where t h e number o f p o i n t s i s D e d i c a t e d t o P r o f e s s o r Frank H a r a r y on t h e o c c a s i o n o f h i s s i x t i e t h birthday.
191
R. Tindell
192
s p e c i f i e d i n advance.
K
(prkrk) is
L e t us say t h a t a t r i p l e
r e a l i z a b l e ' i f t h e r e i s a p-point graph G K ( Z ) = E; i f t h e g r a p h G a l s o s a t i s f i e s
with
K(G)
+
K(G) = 6 ( G )
k and and
(G)
= 6 ( G )I w e w i l l s a y t h a t t h e t r i p l e i s & - r e a l i z a b l e . The p u r p o s e o f t h e p r e s e n t p a p e r i s t o p r o v e t h e f o l l o w i n g
We shall w r i t e a 2 a ' , b 2 b ' , and c 2 c ' .
c h a r a c t e r i z a t i o n of r e a l i z a b l e t r i p l e s . (a,b,c) 2 (a',b',c')
when
Theorem. An i n t e g e r t r i p l e ( p , k , E ) 2 (3,0,0) i s r e a l i z a b l e i f and o n l y i f t h e f o l l o w i n g t h r e e c o n d i t i o n s are s a t i s f i e d . (1)
(2)
(3)
0 < k
+ E
< p-1.
If k + I f one o f k,E i s p-2
= p-1 t h e n one o f k,E i s e v e n .
t h e n t h e o t h e r i s zero.
I n s e c t i o n 2 w e e s t a b l i s h t h e n e c e s s i t y of conditions ( 1 1 ,
(21,
and ( 3 ) f o r r e a l i z a b i l i t y and show t h a t any p - p o i n t g r a p h G w i t h K ( G ) + K ( Z ) = p-1 must b e r e g u l a r and have maximum c o n n e c t i v i t y ,
which i s t o sag7 t h a t K ( G ) = 6 (GI , and t h a t t h e same h o l d s f o r i t s complement. I n s e c t i o n 3 w e s t a t e and p r o v e an e l e g a n t theorem of 2 Watkins [ 5 ] c o n c e r n i n g p o i n t - t r a n s i t i v e g r a p h s W e also introduce an i m p o r t a n t c l a s s o f point-symmetric
graphs
-
.
circulants
-
and
a p p l y W a t k i n ' s r e s u l t t o show t h a t s p e c i f i c examples o f t h e s e g r a p h s have maximum c o n n e c t i v i t y . These examples are used i n s e c t i o n 4 t o e s t a b l i s h t h e s u f f i c i e n c y of c o n d i t i o n s (11, ( 2 1 , and ( 3 ) f o r r e a l i z a b i l i t y ( i n f a c t , f o r 6 - r e a l i z a b i l i t y ) i n t h e cases where k + = p-1. The r e s t o f s e c t i o n 4 i s d e v o t e d t o show how t h e examples f o r t h e e x t r e m a l case may be m o d i f i e d t o y i e l d r e a l i z a t i o n s i n t h e r e m a i n i n g cases. 2.
NECESSITY
T h a t t h e c o n d i t i o n 0 < k+E i s n e c e s s a r y f o r r e a l i z a b i l i t y o f t h e t r i p l e ( p , k , E ) w a s n o t e d above. R e c a l l t h a t 6 ( Z ) = p-l-A(G) , where A ( G ) i s t h e maximum d e g r e e o f G.
I n t h e usage of Boesch and S u f f e l ( [ 2 1 ,[ 3 1 ) , w e s h o u l d p e r h a p s s p e a k o f ( p , ~ , i ? ) - r e a l i z a b i l i t y , where ?t i s a n o t a t i o n f o r K ( G ) . 4.
The p r o o f w e p r e s e n t i s s h o r t e r and more a c c e s s i b l e t h a n t h a t given i n [ S ]
.
193
The connectivities of a graph and its complement Thus K(G)
+ K(E)
< 6(G)
+
6 ( G ) = ( p - l ) - ( A ( G ) - & ( G I ) < p-1
Moreover, w e c a n
which e s t a b l i s h e s t h e n e c e s s i t y o f c o n d i t i o n (1).
a l s o n o t e t h a t i f K ( G ) + K (G) = p-1 t h e n i s r e g u l a r a n d a l s o t h a t K ( G ) = 6 ( G ) and
, 6 (GI.
A(G) = 6(G) K
(G) =
so t h a t
G
We g a t h e r
t h e s e o b s e r v a t i o n s t o g e t h e r as a p r o p o s i t i o n . Proposition 1 L e t K(G)
+ K(G)
K(E)
= 6(G).
= p-1.
G
be a p-point graph w i t h
Then
is regular,
G
p 2 3
From t h e p r e c e d i n g p a r a g r a p h w e n o t e t h a t i f then
and
K(G)
an odd-regular
K(G)
g r a p h o n a n odd number o f p o i n t s . nor
K(G)
K(G)
connected graph on
+ K(G)
c a n n o t b e odd s i n c e i n t h a t case
t h e n e c e s s i t y of condition ( 2 ) . neither
and
K ( G ) = & ( G I , and
K(G) p 2 3
Moreover, when
G
= p-1
would be
This establishes K(G)
+ K ( G ) = p-1,
c a n b e 1 s i n c e t h e r e i s no 1 - r e g u l a r , points.
This establishes t h e necessity
of ( 3 ) and concludes t h e p r e s e n t s e c t i o n . 3.
POINT-SYMMETRIC GRAPHS
A graph
i s point-symmetric i f f o r any p a i r
of p o i n t s W e now d e f i n e a n d e s t a b l i s h t h e point-symmetry o f a c l a s s o f g r a p h s c a l l e d of
G
G
t h e r e i s a n automorphism
We s h a l l d e n o t e by
circulants.
0
[XI
of
., n r
G
with
u, v
$(u)=v.
t h e i n t e g e r p a r t of t h e real
w i t h l L n 1 < n 2 . . . t n r-<[ p/2] , t h e p - p o i n t c i r c u l a n t w i t h jump s i z e s n l , . . . , n r i s d e f i n e d t o h a v e p o i n t s number
x.
Given i n t e g e r s n l , . .
V ~ , V ~ , . . . , V w~ i t~h~
vi a n d v
a d j a c e n t i f and o n l y i f one of j - i , i - j j t o o n e o f t h e jump s i z e s . Note t h a t t h e a b o v e
i s c o n g r u e n t mod p and h a s d e g r e e g r a p h i s r e g u l a r , a n d h a s d e g r e e 2 r - 1 i f nr=p/2 otherwise. I t i s s i m p l e t o show t h a t f o r any i n t e g e r m t h e rotation
pm
d e f i n e d by
~ , ( v ~ ) = v ~a + l l ~i ,n d i c e s r e d u c e d mod
i s a n automorphism of any p - p o i n t c i r c u l a n t . Thus c i r c u l a n t s are p o i n t - s y m m e t r i c s i n c e p o i n t
vi
2r p,
may b e mapped t o
point
v b y a p p l y i n g t h e r o t a t i o n pj-i. j I t i s somewhat s u r p r i s i n g t o l e a r n t h a t p o i n t - s y m m e t r i c
graphs can f a i l t o h a v e maximum p o i n t - c o n n e c t i v i t y , i n t h a t t h e c o n n e c t i v i t y c a n 1 The c i r c u l a n t on 1 2 p o i n t s w i t h b e less t h a n t h e minimum d e g r e e
.
One o f t h e e a r l i e s t e x a m p l e s , a c i r c u l a n t o f 15 p o i n t s , was g i v e n by Boesch and F e l z e r [ l l .
R. Tindell
194
jump s i z e s 1,3,4,5 i s a n example i n t h a t i t i s r e g u l a r of d e g r e e 8 y e t h a s p o i n t - c o n n e c t i v i t y 6 . Watkins [ 5 ] s t u d i e d t h e p o i n t - s y m m e t r i c graphs
G
with
K(G)
6(G)
insight into their structure.
and h i s r e s u l t s p r o v i d e s i g n i f i c a n t Moreover h i s r e s u l t s may be u s e d t o
e s t a b l i s h t h a t c e r t a i n s p e c i f i c p o i n t - t r a n s i t i v e g r a p h s do h a v e maximum c o n n e c t i v i t y .
W e now p r e s e n t some o f Watkins r e s u l t s .
I f U i s a s e t o f p o i n t s o f a g r a p h G w i t h G-U d i s c o n n e c t e d , w e c a l l U a d i s c o n n e c t i n g s e t o f G ; i f t h e number I U I o f p o i n t s i n U e q u a l s K ( G ) , t h e n U i s a minimum d i s c o n n e c t i n g s e t o f G. A subgraph B of G i s a K-part o f G i f B i s a component o f G-U f o r some minimum d i s c o n n e c t i n g s e t U o f G. Note that i n this N* ( B ) , c a s e U must b e w h a t w e s h a l l c a l l t h e d e l e t e d n e i g h b o r h o o d which i s d e f i n e d f o r a r b i t r a r y s u b g r a p h s B o f G t o b e t h e set of p o i n t s o f G which a r e n o t i n B b u t are a d j a c e n t i n G t o a t l e a s t one p o i n t o f B. The a t o m i c number o f G i s d e f i n e d as a ( G ) = min { I B I
:
i s a K-part o f G I .
Recall t h a t IBI d e n o t e s t h e number o f p o i n t s i n
graph
having e x a c t l y
G
A K-part
B.
of a
a ( G ) p o i n t s i s c a l l e d a n atomic p a r t o f
G.
The f o l l o w i n g i s o b v i o u s . Proposition 2
F o r any g r a p h
G I K ( G ) = 6 ( G ) if a n d o n l y i f a ( G )
= 1.
The n e x t r e s u l t i s a v e r y s l i g h t e x t e n s i o n of one o f Watkins main results. WATKINS THEOREM: L e t connected graph
G.
A
b e a n a t o m i c p a r t and
If A n B i s nonempty t h e n
B A
a Ic-part o f a i s a subgraph of
P r o o f . The p r o o f i s by i n d u c t i o n on t h e number of p o i n t s i n and t h e b a s i s s t e p i s t r i v i a l .
GI
Thus w e assume t h e t h e o r e m t r u e f o r
a l l graphs w i t h fewer then p-points
and l e t
B
and
A
and an atomic p a r t , r e s p e c t i v e l y , of a p-point graph nonempty
B.
.
F i r s t c o n s i d e r t h e case where t h e r e i s a p o i n t
u
b e a K-part G I w i t h AnB
.
i n N* ( A ) n N * (B)
i s a d i s c o n n e c t i n g s e t o f G-u i f and o n l y i f Wu{u} i s a G. Thus B , i s a K-part and A a n atomic p a r t of G-u w i t h AnB nonempty. By o u r i n d u c t i v e a s s u m p t i o n , A i s a subgraph of B. We t h u s assume t h a t N * ( A ) n N * ( B ) i s empty. Now f o r any t w o s u b g r a p h s A , B of G w i t h nonempty i n t e r s e c t i o n , it i s e a s y t o see t h a t N* (AnB) = (N*( A ) n B ) u (N* ( B ) n A ) u ( N * (A) n N * ( B ) )
Then
W
disconnecting set of
.
195
The connectivities of a graph and its complement
Therefore,
N"(Am) =
S,
where
We now note that G-S is disconnected since G-S-(Am) contains
*
G-(N (B)uB) , which cannot be empty as
B
is a K-part of
G.
Moreover, we know that A m must contain a component of G-S. The very definition of atomic part would force +f!B
=
knew that
G,
S were a minimum disconnecting set of
may conclude the proof by showing that IS1
A if we so
we
= K(G).
Since G-S is disconnected we have (i)
IN*(A)nBI + IN*(B)mI
IS(=
Moreover, since IN* (A)I = "*(A) ClBl
K(G).
+ IN*(A)-BI
= K(G)
we have
(ii) I N * ( B ) ~2I IN*(A)-BI.
Now consider the subgraph of H = ( A m *(A))
*
G-N (B)-B given by
- (Bm*(B))
= (
(A-B)-N* (B)) U(N* (A) -B)
.
Then
.
by (ii)
*
\ H I= IA-BI-\N*(BIMA( + IN*(A)-BI5 IA-BI< I A I
Since IG-B-N* (B)I 2
a(G)
, we
see that the subgraph
It is simple to show that
G-(AuN (A)UBUN*(B)) is nonempty.
"
N* (AUB)= (N*(A)-B) (N*(B)-A) and hence that N*(AUB) disconnects G. K(G) 2
Therefore
IN*(A)-BI + IN*(B)-AI 2 I N * ( B ) ~ + I IN*(B)-AI = IN*(B)I = K ( G ) .
The second inequality in the preceding line follows from (ii). It now follows that IS1 =
IN*(A) I
= K
IN*(B) nAl =
IN*(A)
-
BI and hence that
(GI, and the proof 'is complete.
Corollary 1 Distinct atomic parts of a graph Corollary 2
If A
is an atomic part of G
automorphism of then $(A)=A.
G
G
and
are disjoint.
4
is an
with 0 (A) fl A nonempty,
R. Tindell
196
Corollary 3
The atomic parts of a point-symmetric graph are themselves point-symmetric graphs.
Corollary 4
Each point of a connected, point-transitive graph G
lies in a unique atomic part of
G
a(G) divides the number of points
and hence
of G.
We now apply Watkins results to obtain useful structural properties of circulants. mod
p.
Lemma 1
Let
let
G
be a connected circulant on the points
v P-1 I with p
vo, Vl,..., G,
Recall that all indices are reduced
Let a be the atomic number of
be the atomic part of
A
m = p/a.
4.
G
containing vo
I
and let
Then (i) vertex vi is in
if and only if
A
m
divides
i; and (ii) for each i, o < i < a-1, there is a i m < j < (i+l) m such that v no point of Proof.
j
j
with
is adjacent to
A.
First we note that when a=l conclusion (i) is
immediate and conclusion (ii) follows from the assumption that G
is not complete.
s
be the least positive index with vs in
rotation Theorem.
p,
For each i vis
is in
Moreover
2
if vb is in
j
Since the
with j a multiple of
if and only if
s
s
and is < b <(i+l)s, then
A
~ is- in ~ A ~since pis
this violates our choice of A
Let
1 the i-fold application of ps will
and hence every v
o < b-is < s and v in
A.
p/2.
maps v o to vs , p s (A)=A by corollary 2 to Watkins
map vo to A.
Thus assume a > 1 so that m
( v ~ - ~ ~vb. ) = Since
we may conclude that v
s divides
j.
is j It then follows that
a = p/s and hence that s=m, and conclusion (i) is established.
197
The connectivities of a graph and its complement
To prove conclusion (ii) suppose to the contrary that for some i, o 5 i 5 a-1, and sverqr v. with i m < j < (i+l)m, 3
vj is adjacent to some point of A. If vk is not in then there is an integer d, o 5 d 2 a-1, with d m < k
A,
< (d+l)m.
where j = k + (i-d)m. and maps vk to v i-d)m j' Thus i m < j < (i+l)m so that v is adjacent to some point j of A. Since P ( ~ - ~maps ) ~ A onto itself, vk is adjacent
The rotation
'(
to some point of G
A.
Thus we have shown that every point of
is either in the atomic part
hood.
A
or in its deleted neighbor-
This contradiction establishes the lemma. We now use lemma 1 to establish that for certain cir-
culants
G,
both
G
and
have maximum connectivity.
proposition 2 Let r, p be integers with p 24 and Let
G
and, if
15r
< p/2.
be the p-point circulant with jump sizes 1,2,...,r is even, let G' be the p-point circulant with
p
jump sizes 1,2,...,r, (i)
p/2.
and
G
> (ii) If p -
(iii) if
p
GI
Then both have atomic number 1;
4r then
has atomic number 1; and
is even and p 2 4r+3, then
has atomic
number 1. Proof.
Recall that p-point circulants have points
v~,v~,...,v and indices are always assumed reduced mod p. P-1 For any circulant under discussion we shall denote its atomic number by a, the atomic part containing vo by
A,
and set
Since a(K ) = 1 we may assume that none of the graphs P considered is complete-that is, not all possible jump sizes
m=p/a.
have been used.
For each of the graphs
G,G'
, E , and G?
our basic method of proof will be to assume a > 1 and derive a contradiction to conclusion (ii) of lemma 2, which is to say we will show that for some b, o
b 5 a-1, and every j with
R. Tindell
198
bm <j < (b+l)m, v
is adjacent to some point of
j
The proof for case (i) is quite simple.
A.
Since G
not complete, r < [p/21 and thus, assuming a > 1, v
must
0
be adjacent to some vim with 0 < tion of
By the defini-
im < p/2.
G, im 2 r and thus vo is adjacent to every v
(i-1) m < j < im, a contradiction. GI, if a
is
Thus a(G) = 1.
3 the preceding argument is valid.
j
As
with for
If a=2, so
that m=p/2, note that every point, save vp/2, which is adjacent to vo is in N* (A); there are
6(G')-1
such points.
More-
is not complete, k < (p/2)-1 and thus vo is
over, since GI
adjacent to neither v
nor v Since both of (P/21+1 (P/21 -1. these points are adjacent to v (r > 1) they are in N* (A), P/2 so 1 2 6 +1, a contradiction.
IN*(A)
The graph
r+2,
...,[p/2].
r < j < p-r.
Q
is the circulant with jump sizes r + l ,
Thus vo is adjacent to v
if and only if j The argument given above for GI in the case
a=2 is valid for
e,
so we assume a
2 3. Let b be the
least positive integer with bm > r; we claim that (b+l)m 5 p-r with bm < j < (b+l)m. j In the case where m 2 r, note that (b+l)m= (b-1) m+2m 5 3r
and hence that vo is adjacent to every v
since (b-l)m < r. is attained.
Since 3r 5 p-r, the desired contradiction
If m > r then p-m=(a-l)m < p-r; since a 2 3,
we have justified our claim and thus have established that has atomic number 1. Now suppose p
is even and consider
GI,
circulant with jump sizes r + l , r+2,...,(p/2)-1.
which is the Since v0 is
a+2. Thus we assume a 2 3 . With b not adjacent to v P/2 ? defined to be the least positive integer with r 5 bm, the argument of the previous paragraph establish that every point
199
The connectivities of a graph and its complement
v with bm
< j < (b+l)m is adjacent to vo unless j=p/2. If j a 2 4 , so that m 5 ~1'4,the point v ( ~ - of ~ ) A~ is adjacent
-
to vpI2 in
G',
since (a-l)m-p/2 = p/2-m and r < p/4
The proof is thus complete in this case. thus the point of
.<
If a=3 then m=p/3 and
are vo 'vPl3, and v2p/3.
A
p/Z-m
the above, the only point not known to be in N
*
Moreover, by (A)
are their
of the points adjacent to vo
antipodes vp/6' vp/2' and
5/P6
*
G?
only v and v are not in N ( A ) . Since p 2 4r+3 2P/3 P/3 and r L 1, p L 7 and thus vl, v are not in A and are P-1 also not one of the antipodes v vsPl6 of points in A. in
*
Thus there are at least two points of vo and hence a=3.
I
G'
Thus
I
N*(A)
6
N (A) not adjacent to
, contradicting
our assumption that
has atomic number 1 and the proof of proposi-
tion 2 is complete. 4.
SUFFICIENCY
The following lemma and its corollary are the final ingredients in proving the sufficiency for 6-realizability of conditions (1), (2) and ( 3 ) of our main theorem. I
Lemma 2 2 <
K
4 and let
Suppose p
(G)= 6 (G) and
such that K(G-e)=
K
( E l = 6 (z)
.
G
be a p-point graph with
Then there is a line of
6 (G-e)= K(G)-~and
6 (CY)= 6 ( G ) .
K(=)=
Proof.
Let u
be a point having minimum degree in G,
let w
be a point having minimum degree in
That such a point degree in
G,
then
z
w
exists if
is clear: if
&(GI=
u
u
E with
G
and
are regular graphs,
will do.
we may choose a line
incident with
cident with
w.
G
W~U.
does not have minimum
and any point distinct from u of
and
does have minimum degree in
A(G) and hence
e
G
It then follows that
Since u
4(G)
22
but not in-
8(G-e)= 6(G)-1 and
p/2.
R. Tindell
200
u
is a point of minimum degree in G-e; the degree In
u
is unchanged by the addition of
=
K(=)
to
Morever K ( C + ~ )2 K ( E ) =
&(G-e)= - & ( G I . thus
e
of
and hence and
I :
K(G).
Clearly (G-e) 5
6(G-e)=
6(G)-l=~(G)-l. Thus let W
be a minimum disconnecting set of G-e. in W
E
E
since G-e-u=G-u
and
I
W
I
Clearly u
< K(G).
If u
is not is isolated
in (G-e)-W, then ic(G-e)= IWI =&(G-e)=K(G)-l and the result follows.
If u
is not isolated in (G-e)-W, then (G-e)-
w-u is still disconnected. disconnecting set for G
Since G-e
- w = G-w, Wu{u]
is a
so that ~(G-e)+l= IWu{u,l 2 K(G),
and the proof is complete. Corollary
If (p,ktE) > (3,0,0) is 6-realizable and 2 < k < i;,
then (p,k-l,E) and (p,k,R-l) are also &-realizable. We are now ready to prove the sufficiency of conditions (l), ( 2 ) , and (3), for 6-realizability. That (pro,p-1) is
&-realizable may be seen by considering G=K Thus, by repeatP' ed application of the corollary to lemma 2, every triple (p,o,E) with 1
If (p,l,E) sat-
isfies conditions (l), (2), and ( 3 ) , then
The triple
(p,2,p-3) is seen to be &-realizable by considering the p-point circulant with jump size 1 (i.e., Cp); by the corollary to lemma 2 ,
(p,l, p-3)
is &-realizable, and thus repeated app-
lication of that corollary implies that ( p , l , R ) is realizable.
All that remains is to consider the triples (p,k,E) satifying the conditions, where 2 k (
LE
9-3.
say k=2r, then the p-point circulant G 1,2,...,r
satisfies K(G)=G(G)=~ and
If k
is even,
with jump sizes
w(E) =G(Z)=p-l-k, and
hence (p,k,p-1-k) is &-realizable; It then follows by repeat-
The connectivities of a graph and its complement
20 1
ed applications of the COrOllarY that (p,k,E) is 4-realizable. If k
is odd, say k=2r+l, and
circulant
G1
6(G1)=k and
p
is even, then the p-point
with jump sizes 1,2,...,r,
~ ( g S(g)= ) = p-1-k.
p/2 satisfies K(G@)=
Thus (p,k,p-1-k) is &-real-
izable and hence the same holds for (p?k,ii). and
p
If k
is odd
is odd, then k
is 6-realizable and hence so is (p,k,E), and the proof of the main theorem is complete. We conclude by observing that, due to the sufficiency of our conditions €or 6-realizability, the conditions are also necessary and sufficient for the existence of a p-point graph G
with line-connectivity k
and with
having line
connectivity ii.
REFERENCES F. Boesch and A. Felzer, A general class of invulnerable graphs, Networks 2 (1972) 261-0283.
F. Boesch and C. Suffel, Realizability of p-point graphs with prescribed minimum degree, maximum degree, and line connectivity, J. Graph Theory (1980) 363-370. F. Boesch and C. Suffel, Realizability of p-point graphs with prescribed minimum degree, maximum degree, and point connectivity, Discrete Appl. Math. 3 (1981) 9-18. F. Harary, Graph Theory, Addison Wesley, Reading, Mass (1969).
M.E. Watkins, Connectivity of transitive graphs, J. Combinatorial Theory 8 (1970) 23-29.
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