GRAPH THEORY
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Peter L. HAMMER, University of Waterloo, Ont., Canada Advisory Editors
C. BERGE, Univer...
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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
Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributors for the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE, NEW YORK, NY 10017
Library of Congress Calaloging in Publication Data
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
-
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
This Page Intentionally Left Blank
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
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
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
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.
This Page Intentionally Left Blank
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 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
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-