Graph Theory and Combinatorics, 1988: Proceedings: Cambridge Combinatorial Proceedings in Honour of Paul Erdos

GRAPH THEORY AND COMBINATORICS 1988 ANNALS OF DISCRETE MATHEMATICS General Editor: Peter L. HAMMER Rutgers Universit...

Author: Bela Bollobas

17 downloads 758 Views 17MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

GRAPH THEORY AND COMBINATORICS 1988

ANNALS OF DISCRETE MATHEMATICS

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

Advisory Editors: C. BERGE, Universitb 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-AMSTEROAM

NEW YORK

OXFORD

TOKYO

43

GRAPH THEORY AND COMBINATORICS 1988 Proceedings of the Cambridge Cornbinatorial Conference in Honour of Paul Erdos B.

BOLLOBAS

Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge, England

1989

-

NORTH-HOWND AMSTERDAM

NEW YORK

OXFORD

TOKYO

0 Elsevier Science Publishers 8.V.,

1989

All rights reserved. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission o f the copyright owner. No responsibility is assumed by the Publisher for any injury andlor damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein.

ISBN: 0 444 87329 5

Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 103 lo00 AC Amsterdam The Netherlands

Sole distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010 U.S.A.

Reprinted from the Journal Discrete Mathematics, Volume 75, Nos. 1-3, 1989

Ukuy of Congress Cataloging-in-Publication Data Cambridge Cornbinatorial Conference (1988 : Trinity College) Graph theory and combinatorics, 1988 : proceedings of the Cambridge Cornbinatorial Conference in honour of Paul ErdBs. (Annals of discrete mathematics ; 43) Includes bibliographies and index. 1. Graph theory-congresses. 2. Cornbinatorial analysidongresses. 3. ErdiZs, Paul, 1913Congresses. I. Erdbs, Paul, 1913. 11. BollobBs, BBla. Ill. Title. IV. Series. QA166.C35 1988 511’5 89-9300 ISBN 0444-87329-5

-

PRINTED IN NORTHERN IRELAND

GRAPH THEORY AND COMBINATORICS 1988 Proceedings of the Cambridge Combinatorial Conference in Honour of Paul Erd6s Guest Editor: B. BOLLOBAS

CONTENTS B. BOLLOBAS, Preface B. BOLLOBAS, Paul Erd6s at Seventy-Five J. AKIYAMA, F. NAKADA and S.TOKUNAGA, Packing smaller graphs into a graph 1. ALGOR and N. ALON, The star arboricity of graphs N. ALON and B. BOLLOBAS, Graphs with a small number of distinct induced subgraphs J.-C. BERMOND, K. BERRADA and J. BOND, Extensions of networks with given diameter N. BIGGS, Confluence of some presentations associated with graphs B. BOLLOBAS and G. BRIGHTWELL, Long cycles in graphs with no subgraphs of minimal degree 3 B. BOLLOBAS and S.RASMUSSEN, First cycles in random directed graph processes J.A. BONDY, Trigraphs E. BOROS and P.L. HAMMER, On clustering problems with connected optima in Euclidean spaces P.J. CAMERON, Some sequences of integers A.G. C H E W N D and A.J.W. HILTON, 1-Factorizing regular graphs of high degree - An improved bound F.R.K. CHUNG and P.D. SEYMOUR, Graphs with small bandwidth and cutwidth R. DIESTEL, Simplicia1decompositions of graphs: A survey of applications P. ERD6S and A. HAJNAL, On the number of distinct induced subgraphs of a graph P. ERD6S, J.L. NICOLAS and A. SARK6ZY, On the number of partitions of n without a given subsum (1) P. FLAJOLET, D.E. KNUTH and B. PITEL, The first cycles in an evolving graph 2. FUREDI, Covering the complete graph by partitions H. FURSTENBERG and Y. KATZNELSON, A density version of the Hales-Jewett theorem for k = 3 R. HAGGKVIST, On the path-complete bipartite Ramsey number R. HAGGKVIST, Towards a solution of the Dinitz problem? R. HAGGKVIST, A note on Latin squares with restricted support J. HAVILAND and A. THOMASON, Pseudo-random hypergraphs M. LAURENT and M. DEZA, Bouquets of geometric lattices: Some algebraic and topological aspects 1. LEADER, A short proof of a theorem of VBmos on matroid representations L. LOVASZ, M. SAKS and W.T. TROTTER, An on-line graph coloring algorithm with sublinear performance ratio

1 3 7 11 23 31 41 47 55 69 81 89

103 113 121 145 155 167 217 227 243 247 253 255 279 315 319

L.A. SEKELY and N.C. WORMALD, Bounds on the measurable chromatic number of 03" A. THOMASON, A simple linear expected time algorithm for finding a hamilton path A. THOMASON, Dense expanders and pseudo-random bipartite graphs D.R. WOODALL. Forbidden graphs for degree and neighbourhood conditions

327 335 343 373 381 387

List of Contributors

405

Author Index

409

J. NESEaIL and V. RODL, The partite construction and Ramsey set systems P. ROSENSTIEHL, Scaffold permutations

Professor Paul Erdds and some participantsof the conference on the stairs of the Wren Library in Trinity College, Cambridge.

Professor Paul ErdBs giving the closing lecture at the conference in the Old Combination Room of Trinity College, Cambridge.

Discrete Mathematics 75 (1989) 1 North-Holland

1

PREFACE The 1988 Cambridge Combinatorial Conference was held at Trinity College, Cambridge, from 21 to 25 March 1988, under the auspices of the London Mathematical Society, the Department of Pure Mathematics and Mathematical Statistics of Cambridge University and Trinity College, Cambridge. The financial support from these institutions is gratefully acknowledged. The conference was in honour of Professor Paul Erdiis on the occasion of his seventy-fifth birthday. Thirty leading combinatorialists accepted the invitation to give talks at the meeting. This volume consists of most of the papers they presented together with some additional articles on closely related topics. In organising the meeting, I received invaluable help from many people, especially Graham Brightwell, Hugh Hind, Yoshiharu Kohayakawa, Imre Leader, Jamie Radcliffe, Andrew Thomason, Jurek Wojciechowski and above all my wife, Gabriella. It would have been impossible to run the conference without their enthusiastic assistance. Combinatorics has not been an established branch of mathematics for very long: the last quarter of a century has seen an explosive growth in the subject. This growth has been due in large part to the doyen of combinatorialists, Paul Erdiis, who through his penetrating insight and insatiable curiosity has provided a huge stimulus for the workers in the field, a great many of whom have collaborated with him. All combinatorialists are vastly indebted to Paul Erd& there is hardly any part of combinatorics that has not been greatly enriched by his ideas. It is a pleasure to dedicate this volume to Paul Erd6s on the occasion of his seventy-fifth birthday. Cambridge, 23rd October 1988

0012-365X/89/$3.500 1989, Elsevier Science Publishers B.V. (North-Holland)

B. Bollobiis

This Page Intentionally Left Blank

Discrete Mathematics 75 (1989) 3-5 North-Holland

3

PAUL ERDdS AT SEVENTY-FIVE* We are here to celebrate Paul ErdBs on his 75th birthday. This is a delightful occasion for all of us, with Paul himself, perhaps, being the only dissenting voice. We are happy to see him here in such good form, and to benefit from his rich experience and tremendous insight. In fact, we are a little early with the celebration: Paul was born on the 26th March 1913. His father, Louis, was a school master of unusual calibre. His mother, Anna, was a wonderful and remarkable lady, who helped her son in every way. Some of us, including myself, were fortunate to have known her in later years, and were impressed that, even into her nineties, she travelled around the world with her son, enjoying her status as “queen mother”. As you would expect of Paul, he was a mathematical prodigy: at the age of six he could multiply six-digit numbers. He was a sensitive child: to save him from severe discipline, his parents took him out of school and educated him privately. In 1930, after hesitating between medicine and mathematics, he started to read mathematics at the PhmBny PCter University, in Budapest, where Leopold FejCr held the chair of mathematics. Paul proved his first famous theorem as a first year student: he gave an elementary proof of Chebyshev’s theorem, foreshadowing his later work with Selberg on the Prime Number Theorem. At the university, he was the leader of a small group of enthusiastic and highly talented mathematicians, including Paul TurBn, Tibor Gallai, George Szekeres, and Esther Klein. He received his doctorate in Budapest in September 1934, and on the 1st of October he arrived in Cambridge, to be greeted by Harold Davenport and Richard Rado. Originally Paul had intended to go to Germany, to Gottingen, but when Hitler came to power in 1933 he decided to come to England instead. He wrote to Louis Mordell and, by way of introduction, sent his paper on abundant numbers. It did not take long for Mordell to recognize the unusual talent of the author, and he secured for Paul a Fellowship in Manchester. Paul spent four happy and very productive years in that city. In 1938 he sailed for America. After a year at the Institute in Princeton, the tempo quickened: he went to Philadelphia, then to Stanford, then to Notre Dame, and so on. The now familiar pattern was set: Paul ErdBs has not stopped moving ever since; he is the most peripatetic scientist the world has ever seen. At times, he seems to be everywhere dense in the world. He has never had a * A transcript, with some slight modifications, of a speech given at the Banquet for the 75th birthday of Professor Paul E r a s held in the Old Kitchens of Trinity College, Cambridge, on 24th March 1988.

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

4

B. Bollob69

permanent job. Indeed, it is often said that he has not slept in the same bed for more than seven consecutive nights, and this is only a slight exaggeration. Paul Erdos is also the most prolific mathematician: he has published more than 1200 papers, and is in fact the only mathematician with over loo0 publications. He has about 250 coauthors: many more than the total number of papers most of us will ever write. His most frequent collaborators are Hajnal and SBrkozi, with about fifty papers each. He had many papers with TurBn and RCnyi as well. Among the ladies, or “bosses” as he would say, Vera S6s and Fan Chung are the leaders. Another of Paul’s collaborators, Mark Kac, wrote about his work with Paul. Let me quote briefly from his autobiography. “In March 1939 I journeyed from Baltimore to Princeton to give a talk. Erdos, who was spending the year at the Institute for Advanced Study, was in the audience but half-dozed through most of my lecture; the subject matter was too far removed from his interests. Toward the end I described briefly my difficulties with the number of prime divisors. At the mention of number theory Erdos perked up and asked me to explain once again what the difficulty was. Within the next few minutes, even before the lecture was over, he interrupted to announce that he had the solution!” After a description of the great Erdos-Kac theorem concerning the number of divisors, Kac continues as follows. “The reader, I hope, will forgive my lack of modesty if I say that it is a beautiful theorem. It marks the entry of the normal law, hitherto the property of gamblers and statisticians, into number theory and it gave birth to a new branch of this ancient discipline. It took what looks now like a miraculous confluence of circumstances to produce our result. Each of us contributed something which was almost routine in our respective areas of competence and neither of us was familiar with the ingredients which the other had in his possession and which were all essential for success. It would not have been enough, certainly not in 1939, to bring a number theorist and a probabilist together. It had to be ErdBs because he was almost unique in his knowledge and understanding of the number theoretic method of Viggo Brun, which was the decisive and, I may add, the deepest of the ingredients, and me because I could see independence and the normal law through the eyes of Steinhaus”. One of the most important of Paul’s many, many contributions to the world of mathematics is precisely his ability to see randomness where there does not seem to be any, enabling him to use the probabilistic method in so many areas. In fact, he worked on some questions of random graphs with Hassler Whitney in the same year, in 1939. Most of us here tonight are combinatorialists and, as has been said many times,

Paul Erd&

at sevenv$ve

5

our debt to Paul is particularly great. He has more or less created a subject where before there was nothing. But Paul is a universal mathematician: he has had an enormous influence on contemporary mathematics. He has proved and conjectured a lot not only in combinatorics but also in set theory, analysis, probability theory, number theory, geometry, and many other areas. He has an uncanny ability to pose problems that look innocent but strike to the heart of the matter. He also has a tremendous feel for what various people can do: like a general, he distributes problems throughout the world, matching people to problems with a remarkable knack and so giving work to an army of mathematicians. Many of us will recall the first time we met this great man how approachable and friendly he was. Unusually for a mathematician of the first rank, his personal warmth puts even young mathematicians at ease, and makes the young feel close to him, as I myself have special cause to remember. I first heard him when I was fourteen: he addressed a club of secondary school students, talking about unsolved problems in elementary number theory. He was in Budapest for a few days, in transit between Israel and Switzerland. During his next visit to Budapest, I was introduced to him and was overwhelmed by his friendliness and the very fact that he was willing to talk to me. I was also overwhelmed by the lavish lunches in the finest hotels in Budapest, where he stayed with “Aunt Annus”, his beloved mother. For a long time, Paul Erdos was the young man of mathematics. Over the years he has matured gracefully into one of the world’s most senior mathematicians, and yet in mathematical spirit he has remained young: his scientific curiosity is as strong as ever. Let us hope that he continues to prove and conjecture at his current phenomenal rate, and that we shall benefit from his incisive mind for many years to come! BCla Bollobis



7

PACKING SMALLER GRAPHS INTO A GRAPH Jin AKIYAMA, Fumi NAKADA Dept, of Mathematics, Tokai University, Hiratsuka, Kanagawa 259-12, Japan

and Sinichi TOKUNAGA Dept. of Pure & Applied Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153, Japan Let G be a graph. Given an integer m < IV(G)l, we obtain a lower bound for the largest number of vertex-disjoint subgraphs of G, each of which has m vertices.

A polyomino is an arrangement of unit squares joined along their edges. A polyomino with rn squares is called an m-ornino. What is the largest number of 3-ominos that can be contained in a polyomino with n squares? The answer to this question is provided by the solution to the rn-packing problem: Let G be a graph, given an integer rn < IV(G)l, find the largest number of vertex-disjoint subgraphs H I , H2, . . . , Hk of G such that each Hi, 1 S i S k, has rn vertices. For a connected graph, the following theorem leads to the solution of the rn-packing problem. Theorem 1. Let G be a connected graph with n vertices and maximum degree A. Then for any integer rn < n, G has a connected subgraph H such that G - H is connected and

rn s IV(H)I d ( m - l)(A - 1) + 1. Proof. It suffices to show that a spanning subtree of G has a connected subgraph satisfying (1) and (2). Let G' be a spanning subtree of G with maximum degree A'. There exists a vertex uo of G with deg,,(vo) = 1. Let W ( e ) denote the number of vertices of the component of G' - e which does not contain vo, where e is an edge of G' . Now consider a path P = uovl * v k in G' satisfying

for any edge e E E ( P ) , W ( e )3 m.

(3)

We may suppose that P = vovl - - vk ( k 3 1) is the longest path with this property. We show that the following inequalities hold:

rn 6 W ( V ~ vk) - ~S , ( m - l)(A - 1) + 1. 0012-365X/89/$3.500 1989, Elsevier Science Publishers B.V. (North-Holland)

J . Akiyama er al.

8

‘ 1 r n - I

vertices

‘. \

,‘ ’ /m-1

Fig. 1

Suppose u E T ( v k )- {vk-,}, then W ( v k u )S m - 1, whence

Let H be a component of G ’ - ( v k - , v k )which does not contain vo. Then H satisfies (1) and (2). 0 The upper bound in Theorem 1 is best possible, since any connected subgraph H of order at least m and at most ( m - l ) ( A - 1) of a graph G in Fig. 1 does not satisfy condition (1) in Theorem 1. Let G be a connected graph. The largest number of vertex disjoint connected subgraphs HI,H2, . . . , Hk of C such that each Hi has m vertices is called the m-packing number of G, and it is denoted by am(G). Corollary 1. Let C be a connected graph with maximum degree A and let

IV(G)l= n. Then

Packing smaller graphs into a graph

v,

9

v2

m- 1

Fig. 2

The lower bound in Corollary 1 is also best possible. Fig. 2 shows a graph G with n-m+l % ( G ) = l ( m - l ) ( A - 1) + 1

1.

The next corollary follows from the previous one. If m = 2, then the 2-packing problem reduces to the problem of finding the largest matching in G. Corollary 2. Let H be a connected graph with maximum degree A and n vertices. Then



11

THE STAR ARBORICITY OF GRAPHS I. ALGOR and N. ALON* Department of Mathematics, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel

A star forest is a forest whose connected components are stars. The star arboricity st(G) of a graph G is the minimum number of star forests whose union covers all edges of G. We show that for every d-regular graph G, i d < st(G) S i d + O(dj(1og d)f), and that there are d-regular graphs G with st(G) > i d S(log d). We also observe that the star arboricity of any planar graph is at most 6 and that there are planar graphs whose star arboricity is at least 5 .

+

1. Introduction All graphs considered here are finite, undirected and simple unless otherwise specified. A star forest is a forest, whose connected components are stars. The star arboricity st(G) of a graph G is the minimum number of star forests in G whose union covers all edges of G. This notion was introduced in [4], where the authors show that the star arboricity of the complete graph on n vertices is [n/2] + 1. In [5], the author determines the star arboricity of every complete multipartite graph with equal color classes G and shows that it does not exceed [d/2] +2, where d is the degree of regularity of G. Notice that by a trivial edge-counting the star arboricity of every d-regular graph is greater than i d , and in view of the results above one may be tempted to suspect that s t ( G ) s i d + 0(1) for every d-regular graph G. This would also resemble the linear arboricity conjecture. A linear forest is a forest whose connected components are paths. The linear arboricity la(G) of a graph G is the minimum number of linear forests in G whose union covers all edges of G. The linear arboricity conjecture, raised in [2], asserts that for every d-regular graph G, la(G) = [(d 1)/2]. This conjecture is proved for d s 6, d = 8 and d = 10 in [2, 3, 9, 16, 17, 11, 121. In [l] it is shown that for every E > O and every d-regular graph G, $d do(&). Here we observe that the star arboricity st(G) of a d-regular graph G can be bigger than i d by more than an additive constant. In fact, we show that there are d-regular graphs G with st(G) 2 i d + B(1og d ) . On the other hand, st(G) cannot be much bigger than i d . Our main result is that the star arboricity of any d-regular graph G does not exceed i d + O(di(1og d ) i ) . This result is proved in

+

* Research supported in part by Allon Fellowship and by a grant from the United States Israel Binational Science Foundation. 0012-365X/89/33.50 @ 1989,Elsevier Science Publishers B.V. (North-Holland)

I. Algor, N. Alon

12

Section 2 by probabilistic arguments, in a method that resembles the one used in [l] but contains some additional ideas. In Section 3 we observe that there are d-regular graphs G with st(G) 3 $d + Q(log d ) . In Section 4 the star arboricity of planar graphs is considered. We observe that for any planar graph G , st(G) 6 6 and construct planar graphs G with st(G) 3 5. The final Section 5 contains some concluding remarks and open problems.

2. An upper bound for the star arboricity of regular graphs In this section we prove the following theorem.

Theorem 2.1. There is a positive constant b such that for every d 3 1 the star arboricity of any d-regular graph does not exceed id + 4 * d3 . (log d); b.

+

Notice that an immediate corollary of this theorem is the following.

Corollary 2.2. For every E > 0 there is a do = do(&)so that for every, d > do the star arboricity of and d-regular G satisfies f d < st(G) < (f+ E)d.

To prove Theorem 2.1, we first need a lemma, occasionally referred to as the L O V ~ SLocal Z Lemma, proved in [lo] (see also, e.g. [14]). Lemma 2.3. Let A*, AZ,. . . , A,, be events in a probability space. A graph T = ( V ( T ) , E ( T ) ) on the set of vertices V ( T ) = (1, 2 , . . . , n } is called a dependency graph for { A ; }if, for all i, the event A; is mutually independent of the system {A,:{i, j } 4 E ( T ) } . Suppose that for all i , Pr(A,) G p and that the maximum degree of a vertex of T is A. If ep(A + 1) < 1 then Pr(n7=l A,) > 0. Using this lemma, we prove the following.

Lemma 2.4. Suppose d 2 100, and let G = ( V , E ) be a d-regular graph. Then there is a (non-proper) coloring of the vertices of G by c = L(d/log d)fJ colors 1,2, . . . , c, so that for each v E V and each color i, 1 6 i 6 c, the number N ( v , i ) of neighbors of v in G whose color is i satisfies

Remark 2.5. All logarithms here and throughout the paper are in the natural base e. The constant 100, as well as the constant 3 in the last inequality, can be easily reduced. We do not make any attempts to optimize the constants here or in the following proofs.

Star arboricity of graphs

13

Proof of Lemma 2.4. Let f :V + { 1 , 2 , . . . , c} be a random vertex coloring of V by c colors, where for each v E V, f(v) E { 1 , 2 , . . . ,c} is chosen according to a uniform distribution. For every vertex v E V and every color i, 1 S i S c, let Au,i be the event that the number N ( v , i ) of neighbors of v in G whose color is i does not satisfy inequality (2.1). Clearly, N ( v , i ) is a Binomial random variable with expectation d / c and standard deviation v ( d / c ) ( l - l / c )< Hence, by the standard estimates for Binomial distribution (see, e.g. [6, p. l l ] ) ,for every v E V and l S i C c

a.

Pr(A,,i) < e - y < l / d 4 .

It is also clear that each event is independent of all the events A,,, for all vertices u E V that do not have a common neighbor with v in G. Therefore, the graph T whose vertices are the events v E V, 1 S i S c } in which two vertices AUeiand A,,j are adjacent iff v and u have a common neighbor in G (including, of course, the case v = u), is a dependency graph for with maximum degree A 2 c - 6. This completes the construction of H, and establishes the claim. To complete the proof of Proposition 2.6 we define H as the union of H , , H,, . . . ,H,. Clearly st(H) 6 c s (d/log d)f, as each HI is a star forest. Also, the degree of each v E V in H is at least 2c - 6 2 2(d/log d); - 8. Indeed, suppose V is colored i. Then in H,, the degree of V is at least c - 5. If its degree in each H,(j # i) is positive, then its total degree in H is at least (c - 5) + (c - 1) = 2c - 6, as needed. Otherwise, there is some j , so that the degree of u in H, is 0. In this case, the degree of u in HI U * U H,-l is at least 2c - 6, and hence, certainly, its degree in H is at least that quantity. This completes the proof of the proposition. 0

1x1

-

--

-

Star arboriciiy of graphs

15

We can now prove the main result of this section.

Proof of Theorem 2.1. Let bl be a constant so that the assertion of Proposition 2.6 holds for every d a b , . Let b2 be a constant so that for every d a b , the following inequality holds. Put d = d - [2(d/log d)f - 81 then &(log d ) f - &(log a)+> 1.

(2.2)

It is not too difficult to check that such a b2 exists. This is because if f (x) = x;(log x)f then, as x tends to infinity 2 logx

f ’ ( x ) = -3 (-x)I

1 logx f + 3xf(log x)’ = (3 - o(l))(-) X . 2

Therefore, by the mean-value theorem, for large d there is some d’, d d d’ s d, so that the left-hand-side of (2.2) is

+

= ($ o(1)) > 1.

Hence b2 indeed exists. Put b = max(bl, b,). We now prove Theorem 2.1 with this b by induction on d. For d S b the theorem is trivial, as it follows from the easy fact that st(G) d d for any d-regular graph G. (To prove this fact observe that any graph contains a spanning star-forest in which all degrees are positive and delete, repeatedly, such star forests from G until it is empty). Suppose the theorem is true for all d’ < d, and let us prove it for d, (d 3 b). Let G = (V, E) be a d-regular graph. Since d 3 b 3 bl, we can apply Proposition 2.6 to conclude that G contains a spanning subgraph H whose star arboricity is at most (dllog d ) f and whose minimum degree is at least [2(d/logd)f-81. Let G‘ be the graph obtained from G by deleting all edges of H . The maximum degree in G’ is at most d = d - [2(d/log d)f - 81. Since we can add, if necessary, vertices and edges to G‘ and embed it in a d-regular graph T to which the induction hypothesis can be applied, we conclude that

d

+

st(G’) d - + 4&(log d)f b. 2 Therefore, st(G) d st(H) + st(G’)

=

d2 + 4 + 4d3(log d)f + b d 2

(d 1)/2 + 1 = [ ( d + 2)/2]. if d is even then there is a column with at least (d/2) 1’s. If it has exactly (d/2) 1’s then since (d/2) < d, This column must have more elements, hence if d is even then st(G) 3 (d/2) + 1 = [(d + 2)/2]. If st(G) = ( d / 2 ) 1 then d is even and there are exactly (d/2) 1’s in each column (otherwise there is a column with at least (d/2) + 1 l’s, but (d/2) + 1 < d , hence st(G) 2 (d/2) 2 > (d/2) 1). The remaining element in each column is d / 2 . Thus every forest is a spanning forest with only Kl,d,2stars which means that ( ( d / 2 )+ 1) IVI. 0

+

+

+

+

I

Next we show that there is a d-regular graph G satisfying st(G)>(d/2)+ sZ(log d). The Paley graph C is defined as follows: (cf. e.g. [6 pp. 315-3231). Let

17


p be a prime, p 5 l(mod4) and put V ( G )= (0, 1, . . . ,p - 1). Two vertices x and y are adjacent in G iff x - y is a square in GF(p). Clearly G is d = ( p - 1)/2 regular. Using some known estimates for character sums it is shown in [8] (see also [13] and [6 p. 3191) that if p > k2 - 22k-2 and A E V ( G ) , IAl= k, then there is a v E V which is not adjacent to any member of A . So if S E V is a dominating set (i.e. every vertex of V\S is adjacent to a vertex of S) then IS1 > k. Let H be a star forest in C and let S = {v E V I deg,(v) = 0 or v is a center of a star in H}. Clearly S is a dominating set so IS( > k. But IE(H)I = p - IS1 < p - k, and k 3 (4 - o(1))logp hence

1 d .,(P+(t-o(l))logp)3~+(Q-o(l))logd.

0

4. The star arboricity of planar graphs The main result of this section is that the star arboricity of any planar graph is at most 6 and that there are planar graphs G with st(G) 3 5. First we show that if G is planar then st(G) G 6. The arboricity of a graph G, A ( G ) , is the minimum number of forests in G whose union covers E ( G ) . Let G be a graph and put qn = max{ IE(H)J: H is a subgraph of G with n vertices}. A well known theorem of Nash Williams [15] states that A ( G ) = maxn { f q n / ( n - 111I*

Lemma 4.1. If G is a forest then st(G) 6 2. Proof. It is clearly sufficient to assume that G is a tree. Fix v E V ( G ) ,and let d(u, v) be the distance of u from v in G. Then the two star forests H, = { ( u , w ) E E ( G ) I d(u, v ) = 2i, d ( w , v) = 2i

+ 1, i 3 0 }

and

H2 = { ( u , w ) E E ( G ) I d(u, v ) = 2i cover all edges of G.

+ 1, d ( w , v ) = 2i + 2, i 3 0 }

0

As a consequence of Lemma 4.1 we conclude that for every graph G: A ( G )6 st(G) 6 2A(G).

If G = (V, E) is planar then qn G 3n - 6 and hence, from Nash Williams theorem we get A ( G ) S 3, which implies that st(G) S 6.

18

I. Algor, N. Alon

It is easy to find a planar graph G such that st(G) = 4, but it becomes more difficult to find one with a bigger star arboricity. We next show how to construct a planar graph G satisfying st(G)>5. Let C = (V, E ) , be a graph on V = {ul, . . . , v,} and let C = ( H I ,. . . , H,) be a star decomposition of G. We say that v E V is a good vertex in the decomposition C if I{i 1 dH,(v)> 1}1 s 1. This means that v is taken as a center of a non-trivial star (i.e. a star with more than one edge) in at most one forest. Equivalently: v’s has at most one element bigger than 1. column in The decomposition C is good if all the vertices are good. Let G be a planar graph. By adding edges and vertices (if necessary) to G , G can be embedded in a planar graph GI with minimum degree 5. Let G2 be the graph consisting of 7 disjoint copies of GI. Finally, let HG be a graph obtained from G2 by triangulating each of its faces. We have thus associated every planar graph G with a (non-unique) planar triangulation H =Hc.

Lemma 4.2. Let G be planar. tf HG has a decomposition into 4 star forests then C has a good decomposition into 4 star forests. Proof. HG. is planar, it has minimum degree 6 = 5 and ( E ( H G ) (= 3 (V(HG)(- 6. Let C = ( H I ,. . . , H4) be a decomposition of HG into star forests. We claim that in C there are at most 6 vertices which are not good ( = bad vertices). Indeed, in A , , , . , 1 3 IEl= 3 IVI - 6 and there are no columns with four l’s, since 6 ( H G )= 5. Thus the number of 1’s in a column is at most 3, and as the total number of 1’s is 1331VI - 6 there are at most 6 columns with less than 3 1’s. Obviously, columns with 3 1’s are good and hence there are at most 6 bad vertices. But HG contains 7 disjoint copies of GI (containing G), hence at least in one of the copies of G , all the vertices are good. Thus, if we restrict C to that G1 we get a good decomposition of G1 (and hence of G), as claimed. 0 Next we show that there exists a planar graph G with no good decomposition into 4 star forests. This implies that HG is planar and st(HG)2 5.

Lemma 4.3. Let C be planar with a good decomposition C = ( H I, . . . , H4). tf u and u are vertices with more than 6 common neighbors in G (see Fig. l), then they musi be taken as centers of non-trivial stars in different forests. Proof. Since C is a good decomposition a vertix v can be taken as a non-trivial center at most once. In the other forests its degree is at most 1. If u and u have r common neighbors ( r 3 7 ) then when u(v) is taken as a non trivial center, the corresponding star must cover at least r - 3 of the common neighbors. Since there are no 2 disjoint sets of r - 3 vertices of the common neighbors (as r a 7), u and u must be taken as non trivial centers in different forests. 0


19

Fig. 1.

Theorem 4.4. There exisi3 a planar graph G such that st(G) 3 5.

Proof. Consider Fig. 2. A dashed line between 2 vertices means that they have 7 common neighbors (like u and v in Fig. 1).A full line between two vertices x and y is like a dashed line with the additional edge (x, y). Suppose the graph in Fig. 2, G, has a good decomposition into 4 star forests. In view of Lemma 4.3 if x and y are connected by a line (dashed or full), they must be taken as non trivial centers in different forests. The 7 vertices in the middle form an odd cycle so they are taken as non trivial centers in 3 forests (or more). u and v are connected to all of them so they are non trivial centers in the 4th forest. But u and v cannot be nontrivial centers in the same forest because they have 7 common neighbors. Thus, G does not have a good decomposition into 4 star forests and hence st(H,)35. 0 We know that for every graph G A(G) s st(G) s 2A(G). A natural question to consider is the determination of the maximum star arboricity of a graph G satisfying A(G)=k. We conclude this section by showing that for k = 2 this maximum is 4, even if we restrict ourselves to planar graphs. Proposition 4.5. There is a planar graph G such that A(G) = 2 and st(G) = 4.

-1

I

I I V

Fig. 2.

I

I. Algor, N. AIon

20

-1 r----I I

r--I

Fig. 3.

Proof. Consider Fig. 3. Clearly it is a planar graph (edges like (u, v ) can be drawn surrounding the graph), and its arboricity is 2 (the full lines form one tree and the dashed lines the other). We refer to this graph as G = (V, E), IV/ = n. We know that 2 =sst(G) S 4, but clearly st(G) 3 3 since it contains a K4. Suppose st(G) = 3, and let C = (H,, H2, H,) be a decomposition into 3 star forests. G has 4 vertices of degree 3 and the rest are of degree 4. Hence 15 .1 = 2n - 2. If deg(v) = 4 its column in AG,Cis (up to a permutation) one of the following four types:

In the first 3 types the number of 1’s is less than half of the sum of the column, and in the fourth type it is precisely half. But in AG,=1 3 4 C:=, Cin,lai,, so if there are columns from the first three types they must be balanced by columns of vertices of degree 3 whose columns are

(9

. Since there are only 4 vertices of

degree 3, there are only a few columns from the first 3 types (no more than 6). Hence we can assume that there are no such bad columns (since if the graph is taken to be long enough, there is a long enough section with no bad vertices of degree 4)., Thus, we may assume that the columns of vertices of degree 4 are all . of type

(i)

(up to a permutation). Clearly there are also very few K l , , stars in

C , because the number of the K1,l stars in C is exactly 1 - IEI = 1 - (2n - 2). If deg(v) = 3 then there are at most 3 1’s in its column and if deg(v) = 4, there are exactly 2 l’s, hence 1 - [El = 1 - (2n - 2) c 12 + 2(n - 4)- (2n - 2) = 8.

We can thus assume that there is a section of G containing four vertices as in

Star arboriciv of graph

21

1

1

Fig. 4.

Fig. 4,where every v of the 4 vertices satisfies:

1. Its column is

C) 1

(up to a permutation).

2. There is no K1,l in C in which v participates. We claim that if u and 21 are adjacent and satisfy 1 and 2 then they are centers of Kl,z stars in different forests. Indeed, otherwise they are centers of Kl,zstars in the same Hi.In this case (u, v ) must be in another forest, but according to property 2 this edge must be a part of some K l V zand its center must be u or v , contradicting 1. Since there is a K4 satisfying 1 and 2 we conclude that st(G) 3 4 and hence st(G) = 4.

5. Concluding remarks and open problems A directed star forest in a directed graph D is a forest whose connected components are stars with edges emanating from the center to the leaves. The directed star arboricity dst(G) of a directed graph G is the minimum number of directed star forests in G whose union covers all edges of G. A directed graph G is d-regular if the indegree and the outdegree of every vertex in it is precisely d . An easy modification of the proof of Theorem 2.1 yields the following result, whose detailed proof is omitted.

Proposition 5.1. There is a positive constant b so that for every d 3 1 the directed star arboricity of any d-regular directed graph G satisfies d < dst(G) < d + 6d$(logd)f

+ b.

Since the edges of any undirected d-regular graph can be oriented so that the indegree and the outdegree of every vertex in the oriented graph will lie between [d/21 and rd/21, this proposition implies Theorem 2.1.

22

I. Algol, N. Alon

Similarly, the construction in Section 3 can be easily modified to produce d-regular directed graphs G with dst(G) > d + S2(log d). In fact, the quadratic tournaments (see, e.g. [13]) have this property. By Corollary 2.2, for any E > 0 and any sufficiently large d , the edges of every graph G with maximum degree d can be covered by less than (4 ~ ) star d forests. Our proof does not supply an efficient algorithm for finding such star forests. It would be interesting to find for some small E > 0 (say E = 0.01) a polynomial time (deterministic or randomized) algorithm for producing the desired star forests. Finally, it would be interesting to determine if the maximum star arboricity of a planar graph is 5 or 6.

+

References [l] N . Alon, The linear arboncity of graphs, Israel J. Math. 62 (1988) 311-325. [2] J. Akiyama, G. Exoo and F. Harary, Covering and packing in graphs 111, cyclic and acyclic invariants, Math. Slovaca 30 (1980) 405-417. (31 J. Akiyama, G. Exoo and F. Harary. Covering and packing in graphs IV, Linear arboricity, Networks 11 (1981) 69-72. [4] J. Akiyama and M. Kano, Path factors of a graph, in: Graph Theory and its Applications (Wiley and Sons, New York, 1984). [5] Y. Aoki, The star arboricity of the complete regular multipartite graphs, preprint. 161 B. Bollobis, Random Graphs (Academic Press, London, 1985). [7] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1976). (81 B. Bollobds and A. Thomason, Graphs which contain all small graphs, Europ. J. Combinatorics 2 (1981) 13-15. [9] H. Enomoto, The linear arboncity of 5-regular graphs, Technical report, Dep. of Information Sci., Univ. of Tokyo, 1981. [lo] P. Erdos and L. Lovhz, Problems and results on 3-chromatic hypergraphs and some related questions, in Infinite and Finite Sets, A. Hajnal et al., Eds, (North-Holland, Amsterdam, 1975) 609-628. [11] H. Enomoto and B. Peroche, The linear arboricity of some regular graphs, J. Graph Theory 8 (1981) 309-324. [12] F. Guldan, The linear arboricity of 10-regular graphs, Math. Slovaca 36 (1986) 225-228. [13] R.L. Graham and J.H. Spencer, A constructive solution to a tournament problem, Canad. Math. Bull. 14 (1971) 45-48. [14] R.L. Graham, B.L. Rothschild and J.H. Spencer, Ramsey Theory (Wiley-Interscience, New York, 1980) 79-80. fl5] C.St.J.A. Nash-Williams, Decomposition of finite graphs into forests, J. London Math. SOC.39 (1964) 12. 116) B. Peroche, On partition of graphs into linear forests and dissections, preprint. 1171 P. Tomasta, Note on linear arboricity, Math. Slovaca 32 (1982) 239-242.


23

GRAPHS WITH A SMALL NUMBER OF DISTINCT INDUCED SUBGRAPHS Noga ALON* Department of Mathematics, Sackler Faculty on Exact Sciences, Tel Aviv University, Tel Aviv, Israel

BCla BOLLOBAS? Department of Mathematics, Cambridge University, Cambridge CB2 lSB, England Let G be a graph on n vertices. We show that if the total number of isomorphism types of induced subgraphs of G is at most En2, where E < then either G or its complement n This settles a problem of Erdiis and contain an independent set on at least (1- 4 ~ ) vertices. Hajnal.

1. Introduction All graphs considered here are finite, simple and undirected. For a graph G, let i(G) denote the total number of isomorphism types of induced subgraphs of G. We call i ( G ) the isomorphism number of G. Note that i(G)= i ( G ) , where G is the complement of G, and that if G has n vertices then i ( G )2 n, as G contains an induced subgraph with m vertices for each m, 1S m G n. An induced subgraph H of G is called trivial if it is either complete or independent. Let t ( G ) denote the maximum number of vertices of a trivial subgraph of G. Note that the complete bipartite graph G with vertex classes of size n / 2 (>1) each has t ( G ) = n / 2 and i(G) = @(n2). The above two estimates hold for a matching of n/2 edges, too. In March 1988, at the Cambridge Combinatorial Conference, And& Hajnal conjectured that if G is a graph on n vertices and i ( G ) = o(n2), then t ( G ) = n - o(n). As the main result of this paper, we shall prove this conjecture. Independently of us, the conjecture was proved in a stronger form by Erdos and Hajnal [Z].

Theorem 1.1. Let G be a graph on n vertices. If i ( C )d En2, where E < t(G)3 (1 - 4 ~ ) n .

then

It is worth noting that both constants and 4 in the theorem above can be improved easily. We make no attempt to optimize the constants here and in the rest of the paper. The proof of Theorem 1.1 is somewhat lengthy, and is presented in the next * Research supported in part by Allon Fellowship, by a Bat-Sheva de Rothschild grant and by the Fund for Basic Research Administered by the Israel Academy of Sciences. t Research supported in part by NSF Grant DMS 8806097. 0012-365X/89/$3.500 1989, Elsevier Science Publishers B.V. (North-Holland)

N. Alon, B. Bollobcis

24

three sections. We first consider, in Section 4, graphs G which contain relatively large trivial subgraphs. Somewhat paradoxically, graphs with no large trivial subgraphs are more difficult to deal with; this case with be discussed in Section 3. In Section 1, Theorem 1.1 is obtained as an easy consequence of the results of Sections 2 and 3. In the final section we present some unsolved problems.

2. Graphs with large trivial subgraphs In this section we prove the following theorem, which implies the assertion of Theorem 1.1 for graphs with relatively large trivial subgraphs.

Theorem 2.1. Let G be u graph on n vertices and put t = t(G). Then

This theorem is an easy consequence of the following lemma.

Lemma 2.2. Let G = ( V , E ) be a graph on n vertices and put t = t(G). If t 3 n / 2 then i ( G )3 t ( n - t ) / 3 .

Proof. By replacing, if necessary, G by its complement, we may assume that there is an independent set T of t vertices. Let H be the bipartite subgraph of G with vertex classes T and V\T whose edges are the edges of G joining a vertex of T to a vertex of V\T. Let M = { a l b l ,a 2 b z ,. . . , a,b,} be a maximal matching in H , where a , , . . . , a, E T and b l , . . . , b, E V\T. Furthermore, set A = {a, , . . . ,a,), B = { b l , . . . , b , } , C = V \ ( T U B ) and r = I C I = n - t - s . Note that by the maximality of M there are no edges from C to T\A. Given 1 and m satisfying 0 S 1 S s , I G m S t and m 3 1, let T ’ be a subset of T\ { u , , a?, , , , , a t } of cardinality m - 1 and let G,,,n be the subgraph of G spanned by the set of vertices { a l , b , , a 2 , b2, . . . , a t , b , } U T ’ . It is easily checked that G,,mhas 1 + m vertices and that its independence number is rn. Therefore no two distinct members of the family { G,,m:0 S 1 S s, 1 G m 6 t, m a 1 ) are isomorphic and hence

Similarly, for each p, 0 s p 6 r and each q , 0 q S t - s, let Hp,qbe the induced subgraph of C on C’ U A U T ’ , where C‘ c C is a subset of C of cardinality p, and T’ c T\A is a subset of T with 1T’I = q. Since in G there are no edges from C to T\A it is easy to check that Hp,q has p + q + s (31) vertices, and that its

Graphs with dhtinct induced subgraphs

independence number is q

i ( G ) a (r + l ) ( t - s

25

+ s. Thus

+ 1) = (n - t - s + l)(t -s + 1).

(2)

We shall make no attempt to obtain the best bound implied by inequalities (1) and (2); we shall prove only the claim of the lemma. Multiplying inequality (1) by two and adding to it inequality (2), we see that

+ 1) - s(s - 1) + (n - t + 1- s)(t + 1- s) = (2t - n)s + 2t - s + (n - t + l)(t + 1)a (n - t + l ) ( t + 1).

3i(G)3 24s

0

Proof of Theorem 2.1. Let G = (V, E) be a graph on n vertices and put t t(G). If t 3 n/2 then the assertion of the theorem follows from Lemma 2.2. Otherwise, let T c V be the set of vertices of a trivial subgraph of G, with IT1 = t. Let U be an arbitrary subset of cardinality t of V \ T, and let H be the induced subgraph of G on T U U. Clearly t ( H ) = t = 4 IT U UI and hence, by Lemma 2.2 i ( G )2 i ( H ) 3 t2/3. This completes the proof. 0 =1

3. Graphs without large trivial subgraphs This section is the heart of the paper; our main aim is to prove the following result.

Theorem 3.1. Let G be a graph on n vertices. I f t ( G ) < n / l O " then i ( G ) a n'/ 10''.

The proof of this result is rather long and is based on two propositions. In turn, in the proofs of these propositions we make use of the following very useful lemma of Erdiis and Lovbz [3] (see also [ l , pp. 20-221) sometimes called the Erdos-LovGsz Local Lemma.

Lemma 3.2. Let A , , . . . ,A, be events in a probability space and let H be a graph of maximal degree d 2 2 on the set (1, 2, . . . ,s}. Suppose that each Ai i~ independent of the system { A j :i is not joined to j in H } and P(Ai)< l/ed. Then the probability that no A ioccurs is positive.

Proposition 3.3. Let G = (V, E ) be a graph of order n and maximal degree A, with 10' s A s 0.9n. Then for every two integers j and 1 that satisfy 0.51A <j < 0.524 < 1 6 0.51

there is an induced subgraph H of G with IV(H)I = 1 vertices and maximal degree A ( H ) = j . I n particular i ( G )3 nA/104.

N. Alon, B. Bollobh

26

Proof. Let f: V + (0, 1) be a random function, i.e. a random two-colouring of V obtained by choosing, for each u E V independently, a colour f(u) E ( 0 , 1) according to a uniform distribution on { 0 , 1 } . For each vertex u E V, let A, be the event that u has more than ( A / 2 ) + 2 l / m neighbours having the same colour. By the standard estimates for the probability in the tail of the binomial distribution (see e.g. [l. p. 13, Theorem 7]), it is easy to check that for every u E V we have P(A,,)S A-8'3. Let H be the square of G, i.e. the graph obrained from G by adding all edges joining vertices at distance 2. Then A(H) zs A(A - 1) < AW3/e,and so the graph H and the events A,, u E V , satisfy the conditions of Lemma 3.1. Therefore, with positive probability no A, occurs. Since (A/2) 2 v m 2 and d > 4 .

+

1. Introduction This article is motivated by some problems in the design of large interconnection networks. It is well known that such a network can be modelled by a graph in which the vertices represent the processors and the edges the links of the interconnection network. Various considerations and parameters play important roles in the design of such networks. For example one wants a small transmission delay in the network, which corresponds to a small diameter. In order to keep the cost low and to facilitate drawing one also wants a small number of links on each node, corresponding to a small maximum degree. Other constraints might be added (see the forthcoming book of D.I. Ameter and Max de Gree [l],or the surveys in [2, 4, 91). Here we emphasize the extendability properties of the network which are important, for example, for local area networks or communication networks. The number of processors in the network is not known in advance and ideally one wants to be able to add a new processor at any time while maintaining the desired properties of the network. For example, the designer might want to keep a small diameter and a small maximum degree. However, a graph with given maximum degree and diameter has a limit on the number of vertices, namely the Moore bound. Therefore it is not possible to add processors indefinitely while keeping the maximum degree and diameter fixed. Different classes of extensions can be considered according to the constraints imposed by the designer (for example * Research partially supported by P.R.C.C3. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

32

I.-C. Bermond et af.

“atomic” extensions if the processors are to be added one by one), and whether “relinkage” is allowed or not. “Relinkage” consists of deleting some links before adding vertices. This operation might be necessary; for example if one wants to keep the degree at most A and the original graph is A-regular. In this case it is impossible to add a vertex and still have a connected graph without relinkage. Finally let us note that the way we handle the problem will differ depending on the presence or absence of a bound on the size of the networks. If an upper bound is given for the maximum number of vertices of the network, it might be better to choose a “good” network having that size (the goal to be attained) and then to delete vertices to obtain different intermediate possible networks. This practical approach has been used in [ 5 ] for some graphs. Here we consider infinite (or unbounded) extensions, where the vertices are to be added one by one (atomic extensions). We do not allow relinkage (strict extension) and want to keep the diameter D constant. We will call such extensions D-admissible extensions. Other extensions have been considered by Bond and Konig [7] (fixed maximum degree) and by Konig [12] (fixed maximum degree and connectivity with or without relinkage). Before stating our results we present some definitions and notation.

2. Definitions and notation The interconnection network will be modelled by an undirected graph G = (V, E) where V is the vertex set and E the edge set. The degree of a vertex is the number of its neighbours and we will denote by AG (A if there is no ambiguity) the maximum degree of G . The distance d&, y ) between two vertices x and y is the length of a shortest path between x and y in G. The diameter DG (or D ) is the maximum of d&, y ) over all pairs of vertices x and y of G. G will always be connected so DG is finite. A (A, D)-graph will denote a graph with maximum degree A and diameter D. Let G and C ’be two connected graphs. We will say that G‘ is a D-admissibfe extension of G or that there exists a D-admissible extension from G to C ’ if there exists a sequence of graphs G = Go,GI, . . . , G,, . . . , Gk = G‘ such that (i) G, is an induced subgraph of G,+l, (ii) lV(G,+,)I = IV(GI)I + 1, (iii) all the (2, have diameter at most D. Property (i) corresponds to adding vertices to G without allowing relinkage. Property (ii) corresponds to adding the vertices one by one and property (iii) to keeping the diameter small during the process. Furthermore, we want C’ to be infinite or at least very large, and the G, to be “good” networks. The problem is that “good” is not well defined. One possibility is to insist that the GI have the smallest possible degree among all graphs with the same number of vertices and diameter D. However, the determination of this

Extensions of networks with given diameter

33

value is an unsolved problem. Another problem is that at some point one extension might be better (in a local sense) than another but so constrains us that we are forced to construct “worse” graphs later. To avoid these difficulties we restrict our attention to extensions such that at some steps the graphs obtained are among the “best” known (at least at the present time). More precisely we choose a family of graphs G A , , D such that G = G A , , D c G A 2 , D c - * * c G A , , D c - . . with A, 6 A2 G - . - s Ai6 * - and we require that there exist D-admissible extensions from G A , , D to G A , + , , D for every i. We first show that this is possible for the family of hypercubes (an interconnection currently used in parallel computers). Then we consider the best known general families of (A, D)-graphs, namely the de Bruijn networks UB(d, D) or the Kautz networks UK(d, D), both of which have maximum degree 2d and diameter D. We show that for D = 2 there exist D-admissible extensions from UB(d, 2) to UB(d 1, 2), and count the number of nonisomorphic D-admissible extensions. Then we prove that for 02.3 and d > 4 there do not exist D-admissible extensions from UB(d, D) to UB(d + 1, D). Similar results are proved for Kautz networks.

+

3. D-admissible extension of hypercubes The hypercube CU(d, D) has as vertices the words of length D on an alphabet of d letters, in other words the D-tuples (al, a 2 , .. . , a,) where a i 6 (0, 1, . . . ,d - l}. Two vertices are joined if their corresponding D-tuples differ in exactly one coordinate. When d = 2, the hypercube CU(2, D ) is the well known Boolean D-cube on 2, vertices. The graph CU(d, D) is a regular graph with dD vertices, diameter D and degree (d - 1)D.Furthermore CU(d, 0)is an This is a family as described above. Note that induced subgraph of CU(d + 1, 0). there exist different definitions for hypercubes, but the one above gives families with constant diameter.

Proposition 1. There exists a D-admissible extension from CU(d, D ) to CU(d + 1, D ) . Proof. It suffices to add the vertices in the lexicographic order. Recall that the vertex ( a l , a2, . . . , aD) is lexicographically before (bl, b2, . . . , b,) if for some i we have ( a l , a2, . . . , ai)= ( b , , b Z ,. . . , bi) and ai+,< bi+,. Suppose that at some step we have constructed a graph Gi, let x be the first (in the lexicographic order) and let Gi+] be the spanning subgraph vertex of CU(d + 1, D) which is not in Gi, of CU(d 1, D) generated by the vertices of Giplus the vertex x. We have only to show that Gi is of diameter D. That follows from the fact that in CU(d + 1, D) between any pair of vertices there exists a monotonic path (in the lexicographic order). For example if ( a , , a 2 , . . . , a,) is before (bl, b2, . . . , b,), the path (up

+

34

I.-C. Bermond er al.

to repetitions of some vertices) is

The hypercubes are interesting because they have nice properties like symmetry, easy routings, and high fault tolerance, but they are not among the best of the known (A, D)-graphs. So it is interesting to study other families.

4. D-admissible extension of de Bruijn graphs The de Bruijn digraph B(d, D ) (defined in [S]) with in- and out-degree d and diameter D is the digraph whose vertices are the words of length D on an alphabet of d letters (0, 1, . . . , d - l}. There is an arc from the vertex x to a vertex y if and only if the last D - 1 letters of x are the same as the first D - 1 letters of y, that is there is an arc from ( a , , a z , . . . , a D ) to the vertices ( a 2 , .. . , a,, A), where A is any letter of the alphabet. This digraph has dD vertices. We will denote by UB(d, D ) the associated undirected de Bruijn graph of maximum degree A = 2 d and diameter D. That is, UB(d, D) is the graph whose vertices are the words of length D on an alphabet of d letters in which the vertex ( a l , . . . , a D ) is adjacent to the vertices ( a z , . . . ,a,, A) and

(A,a , , . .

' I

6 - 1 ) .

Proposition 2. UB(d, D ) is an induced subgraph of UB(d + 1, D).

Proof. It suffices to consider in UB(d + 1, D )the vertices (al, . . . , a D ) such that a,#dfor I S i S D . In view of the proposition above, one can try to construct a D-admissible extension from UB(d, D ) to UB(d 1, D). The case D = 1 is trivial as UB(d, D ) is the complete graph on d vertices. In the case D = 2 we have:

+

Proposition 3. There exists a D-admissible extension from UB(d, 2) to UB(d + 1,2).

Proof. Let G =UB(dl 2). The vertices are the words (al, az) with aiE (0, 1, . . . , d - 1). We must add all of the (al, a2)'s such that at least one ai is d. We can do that in the following way: add first a vertex (a, d) where a E (0, 1, . . . , d - l}, then a vertex (d, b) where b E (0, 1, . . . , d - l}, then add all vertices containing exactly once the letter d (in any order), and finally add ( 4 4. 0


35

In ract as we show in the next proposition, there exist many D-admissible extensions.

Proposition 4. The number of nonkomorphic D-admissible extensions from UB(d, 2) to UB(d + 1,2) is d! N ( d ) where 2d-2 d-1 Proof. We have to add the 2d - 1 vertices of UB(d + 1,2) not in UB(d, 2), that is, the set L U R U {(d, d)}, where L = {(a, d ) I 0 s a s d - 1) and R = { ( d , a ) I 0 s a d d - l}. Note that the vertex (d, d) can be added only if all the vertices of L or all the vertices of R have already been added. Indeed, suppose Gicontains the vertex (d, d) but not the vertices (ao, d) and (d, bo). Then the distance in Gibetween (bo, ao) (which is in UB(d, 2)) and (d, d) is 3, and so Cidoes not have diameter 2. Note also that the first vertex to be added can be chosen to be in L, since UB(d 1,2) admits the symmetry a(a, b) = (b, a ) as an automorphism. Then the second vertex must be in R. Indeed, if not suppose we add first (a, d) and then (b, d). Then in G2 the distance between (a, d) and (b, d) is 3. Counting the number of nonisomorphic D-admissible extensions corresponds to counting the number of distinct ways of adding the vertices such that each step Giis of diameter 2. Two ways are distinct if there exists no automorphism of UB(d + 1, 2) mapping one extension on to the other. Let S(d) be the set of words of length 2d + 1 on the alphabet { I , r, d} containing the character I d times, the character r d times and exactly one d, such that each word begins with lr and such that either all the 1’s or all the r’s (or both) appear to the left of character d. To each D-admissible way of adding the vertices is associated a word of S(d), built by putting an 1, r or d in the ith position depending on whether the ith added vertex is in L, R or is (d, d). Conversely, each sequence corresponds to d! nonisomorphic ways of adding the vertices. Indeed we can decide that the vertices of L are added in the order (0, d), (1, d ) , . . . , (d - 1, d), since any permutation on the letters is an automorphism of UB(d 1,2). But then there are d! choices to associate to the characters r vertices of R which give nonisomorphic ways of adding the vertices. Now it suffices to show that there are N ( d ) words in S(d). First let us count the number of words where the character d appears after all the 1’s. That corresponds to choosing the positions of the d - 1 occurrences r among the last 2 d - 1 characters of the word. There are ( ? I : ) such choices. Similarly there are ( ? I : ) words in which the character d appears after all the r’s. Furthermore there are (7:;)words ending in d, thus

+

+

/.-C. Bermond et 01.

36

The case D > 2 In view of the propositions above we thought that there were also many D-admissible extension from U B ( d , D) to U B ( d + 1, D ) for D > 2. We tried different strategies to add vertices: for example, to add to G, a new vertex having the maximum number of neighbours in Gi. If there are many such vertices, add the one which is the smallest in the lexicographic order. Unfortunately none of the strategies worked. By using a branch and bound method we found a D-admissible extension from UB(2, 3) to UB(3,3 ). We give below an order in which to add the 19 new vertices containing the letter 2. (o,072 ) , (2,0,0), (0, 1,2), (2907 11, (1,2, o), (07 2,0), (1,0,2), (1, 1,2), (2, 1,o), (2, 1, 11, (0,2, I), (172, 1 1 7 (2,0,2), (0,2,2), (27 2,0)7 (1,2,2), (2,1,2)*(2,27 I), (2,292) Then we proved that at most 6 vertices can be added to UB(d, 3) without increasing the diameter if d > 2. The proof can be found in [5]. Finally we proved the following.

Theorem 5 . There does not exist a D-admissible extension from U B ( d , D ) to UB(d + 1, D), where D 2 3 and d > 4. Proof. Let G,, = U B ( d , D ) and suppose that there exists a D-admissible extension from U B ( d , D) to U B ( d + 1, D ) , where D 5 3 and d > 4. We will denote by R ( x ) = {(A, a,, . . . , u + ~ ) } the neighbours of x = ( u l , az, . . . , a D ) obtained by a right shift and by L ( x ) = { ( a 2 , . . . ,a,, p)} the neighbours of (al, a2, . . . , a D ) obtained by a left shift. L ( x ) corresponds to the successors of x in the digraph B(d, D) and R ( x ) to its predecessors. The proof will be split into two cases according to the parity of D.

+

D odd, D = 2k 1 (k 3 1). We will use the following proposition from [ 3 , 61:

Case 1.

Proposition 6. Let a = (u1a2.. - a D )be a vertex of U B ( d , D).Let the sets &(a) be defined for 1 = 1 , . . . , D as follows: bi # u D + ~ - , , u D - ~ for i < 1 (bl bz * . * bD): b, # U D + ~ - / Z/(U= ) bj f U g + ~ - j , a ~ + 1 - , f o r ] > 1

{

1-

Let Z ( a ) =Up=, Z,(a). Then, for every b E Z(a), the distance between a and b is D,

We will use the following corollary.


37

Corollary 7. Let S be a subset of the neighbours of a vertex s in UB(d + 1, D ) , such that IS nR(s)l d d - 3 and IS n L ( s ) l S d - 3. Then there exists a vertex t = ( t , , . . . , t D )such that ti # d and d(x, t ) = D for every x in S. Proof. Let ri = (Ai, sl,. . . ,sD-,) be the vertices of S n R ( s ) and li = (sz, . . . , sD, pi) be the vertices of S fl L (s) (i = 1, . . . ,d - 3). We show that 2= Zl(ri))f (l n f Z , ZD(Zi)) contains a vertex t = ( t , , . , . , t D ) such that ti# k . Note that

(nf:;

Thus we have at most d - 1 constraints on the entries of the vertices in 2, so if we add the additional constraint t i # d the set is still not empty (we are in U B ( d + l , D). 0 This proposition implies that before adding s we must have already added either d - 2 elements of R ( s ) or of L(s). Now let us choose s to be the first vertex added with a letter d in the middle: s = (a,, . . . ,ak, d, b l , . . . , bk). Without loss of generality suppose that when we add s at least d - 2 vertices of R ( s ) have been added (the case L(s) is identical) and let us denote these vertices si= (ai, a,, . . . , ak, d, b,, . . . , bk-,), with 1 6 i 4 d - 2. Let C,be the graph just before the adjunction of s. We will show that in G, there are two vertices si and sj at distance greater than D. First of all, note that the distance between si and any vertex of Go is at least k. Indeed, to reach a vertex of Go from si we have to do at least k right-shifts, aslhere does not exist in G, any vertex having d in the middle. Now consider a shortest path in G, between si and sj. As mi# ai and ak # d and there is no vertex with d in the middle, this path is of the form si - - - zi - ti - - - si, where ziand zi are in Go and zi = (*, . . . ,*, ai, a,, . . . ,a&), zj = (*, . . . , *, aj,a,, . . . ,a&). A s d - 2>2 we can find ai and aj such that ai # a,, aj# a , , and therefore the distance between zi and zi is at least 2. So the distance in G, between si and sj is at least 2k + 2, a contradiction.

-

Case2. D e v e n D = 2 k + 2 ( k a l ) . The proof is similar to case 1, but we use the following corollary of Proposition 6.

Corollary 8. Let S be a subset of the neighbours of a vertex s in UB(d + 1, D ) , such that IS1 d d - 2. If there exist a vertex s such that S c R ( s ) or S c L(s ) then there exists a vertex t such that ti# d and d(x, t ) = D for every x in S.

38

J.-C. Bemwnd el al.

Proof. Without loss of generality let us suppose S c R ( s ) . Let ri = (Ai,sl, . . . ,sD-,) be the vertices of S (i = 1, . . . , d - 2). We show that 2 = ni”=;’ ZD(ri)contains a vertex t = ( f l , . . . , f D ) such that ti# k. Note that bj f (b1b2 . * bD):

S ~ - j ,S ~ - j - 1

bD-1#

Ai, ~1

bD# A i

for 1S j S D - 2 forlsisd-2 forlsisd-2

I

.

This way we have at most d - 1 constraints on the entries of the vertices in 2, so if we add the additional constraint ti# d, the set is still not empty (we are in UB(d + 1, D ) ) . 0 This proposition implies that before adding s we must have added already either d - 1 elements of R ( s ) or L(s). Now let us choose s to be the first vertex added with a letter d in one of the two middle positions: S = (a,,. .

. , i l k , X , d , b1, . . . , bk) Or S = ( a , , . . . , U k , d , X , b , , . . . ,bk), with X # d. loss of generality suppose that s = (al, . . . , ak, X , d , b l , . . . , b k ) , so

Without that when we add s at least d - 1 vertices of R ( s ) have been added (the case L ( s ) IS identical). Let these vertices be s, = (aI, a t , . . . , ak,x , d, b l , . . . ,b k - , ) , with 1s i 6 d - 2. Let G, be the graph just before the adjunction of s. We will show that in G, there are two vertices s, and s, at distance greater than D. First of all the distance between s, and any vertex of Go is at least k. Indeed to reach a vertex of Go from s, we have to do at least k right-shifts, as there does not exist in G, any vertex having d in the middle. Now consider a shortest path in G, between s, and s,. As aI# a,and ak # d and there is no vertex with d in the middle, this path is of the form s, . . z, . zI . s,, where z, and z, are in Go and

-

2,

=

( * j

-

+

. . . *,

a,, . ..

9

1

( I k , X),

ZI = ( * I

. . . , *,

a,,

. . . ak, X ) . 7

As d - 1> 3 we can find aIand m, such that aI# a l , a,# a , , a,f a 2 , a,# a 2 , and therefore the distance between z, and z, is greater than 2. So the distance in G, between s, and sl is greater than 2k + 2, a contradiction. 0 Remarks (1) The proof does not give information on the number of vertices which can be added to UB(d, D) without increasing the diameter. It can be proved that this number is at least 2dD-3 4(d - l)diDnl 2d. ( 2 ) An interesting problem is to give an extension from UB(d, D) to UB(d + 1, D) such that all the Gihave the smallest possible diameter, that is, to determine the smallest D’ such that there exists a sequence UB(d, D), G I , . . . , G,, . . . , UB(d + 1, D), with Gian induced subgraph of Gi+l, (VG,+,( = I V J + 1 and all the Gihaving diameter at most D’. It is easy to prove that D‘6 3 D / 2 , but perhaps even D’ s D + c holds for some constant c.

+

+


39

5. D-admissible extension of Kautz graphs Since the results are very similar for the other family of networks introducea by Kautz [ll], we will not give proofs. The undirected Kauh network UK(d, D) is the induced subgraph of the de Bruijn network UB(d + 1, D) spanned by the vertices without two consecutive identical letters. So the vertices are labelled with words (al, a2, . . . ,aD), where ai belongs to an alphabet of d + 1 letters and ai # u ~ +for ~ 1S i S D - 1. The vertex (al, a2, . . . ,a D ) is joined to the 2d vertices (A, al, . . . ,aD-1) with A # a l and (a2, . . . ,uD, p ) with p # U D . Thus UK(d, D) has dD + dD-' vertices, maximum degree 2d and diameter D. Proposition 9. UK(d, D) is an induced subgraph of UK(d + 1, D). Therefore the UK(d, D) also form a good family of graphs to be considered for D-admissible extensions. They give rise to the following results, which are analogous to those for de Bruijn graphs. Proposition 10. There exists a D-admissible extension from UK(d, 2) to UK(d + 1,2). In fact a proof similar to the one of Proposition 4 shows that there are (d + l)! (7)such extensions. Theorem ll. There does not exist a D-admissible extension from UK(d, D) to UK(d 1, D), where D > 2 and d > 4.

+

Acknowledgement We thank C. Delorme and R. Kerjouan for stimulating discussions on this problem, and A.L. Liestman for his careful reading.

References D.I. Ameter and Max D. Gree, Graphs and Interconnection Networks, forthcoming book. J.-C. Bermond, J. Bond, M. Paoli and C. Peyrat, Graphs and interconnection networks: Diameter and vulnerability, in Surveys in Combinatorics, Invited Papers for the Ninth British Combinatorial Conf., 1-30, London Math. SOC.Lec. Note Ser. 82 (Cambridge University Press, 1983). [3] J.-C. Bermond, J. Bond, S. Rudich, M. Santha and W.F. de la Vega, The radius of graphs on alphabets, submitted. [4] J.-C. Bermond, C. Delorme and J.-J. Quisquater, Strategies for interconnection networks: some methods from graph theory, J. Parallel and Distributed Computing, 3, (1986) 433-449.

40

J.-C. Bemond

et

al.

[5] K. Berrada, Extension de rtseaux d’interconnexion, Thtse, Universitt de Pans Sud (1986). (61 J. Bond, Grands r6seaw d‘interconnexion, These d’Etat, Universitt de Pans Sud (1987). f?] J. Bond and J.-C. Konig, Extension de rtseaux de degrt maximum donnte, manuscript. [8] N.G.de Bruijn, A combinatorial problem, Koninklijke Nederlandse Academie van Wetenxhappen Proc., Ser A49 (1946) 758-764. 191 F.R.K. Chung, Diameter of communication networks, AMS short course on the mathematics of information processing, Proceedings of Symposia in Applied Mathematics, 34, (AMS, Providence, 1986) 1-18. [lo] M.A. Fiol, J.L.A. Yebra and I. Alegre, Line digraph iterations and the (d, k ) digraph problem, IEEE Trans. on Computers, vol. C-33, (1984) 400-403. [ l l ] W.H. Kautz, Bounds on directed (d, k ) graphs, in Theory of Cellular Logic Networks and Machines, 20-28, AFCRL-68-0668, SRI Project 7528, Final report (1968). I121 J.-C. Konig, Extensions de rtseaw de connexitt donnte, Proc. Coll. Combinatoire et Inforrnatique (Montreal, May 1987) to appear.


41

CONFLUENCE OF SOME PRESENTATIONS ASSOCIATED WITH GRAPHS Norman BIGGS Department of Mathematics, London School of Economics, Houghton Street, London WC2A .ME, U.K.

1. Translations on an infinite tree It is a standard result [l l ] that an automorphism g of a tree which fixes no vertex and no edge (considered as an unordered set of two vertices) must be a translation. That is, there is a doubly-infinite path (which we shall refer to as the auk) such that g moves each vertex a constant distance along it, in one direction. Throughout the rest of this paper, the word translation will denote such an automorphism, with the additional property that the constant distance is one. The tree will always be the infinite cubic tree, which we shall denote by T. Our concern will be with groups G of finite type acting on T : that is, those for which the stabilizer of a vertex or an edge is finite. We shall assume that G is generated by two translations u and b which are independent, in the sense that they are not conjugate in G, and that the axes of a and b have a finite, non-empty, intersection. It follows that the intersection is a path, of length s 2 1, which we shall refer to as the fundamental s-arc F. We shall assume that a and b are chosen so that s is maximal with respect to the intersection of axes of translations in G. The vertices of F will be denoted by fo, fi, . . . ,fs; the vertex adjacent to f;: which is not f;:-l or J.+lwill be denoted by ei (1 S i S s - l), and we shall write b ( L )= bl. The preceding formulation is just another way of looking at the “classical” theory of symmetry in finite cubic graphs [7, 121. The aim here is to present the theory in the context of the infinite cubic tree, and to relate this approach to some hitherto unremarked properties of the groups involved. Specifically, we shall show that these groups have finite, confluent, terminating presentations, so that they are automatic, in the sense of [4]. This observation provides a new way of showing that certain quotients of the groups are infinite. a(&) = alp

2. Regularity of the group action A group is said to act regularly on a set of objects if it is transitive and the stabilizer of any object is trivial. In this section we show that our assumptions 0012-365X/89/$3.50 @ 1989, Elsevier Science Publishers B.V. (North-Holland)

N. Biggs

42

about G force it to act regularly on the set of s-arcs of T , where an s-arc is an (oriented) path of length s. Because some parts of the proof are closely related to the standard arguments in the finite case [l],we do not give all the details.

Lemma 1. Let d denote the usual metric in T and let B, denote the set of vertices p of T which satkfy d ( p , fs) = r and d ( p , fs-l) = r + 1 ( r 5 0). There is a bijection from B, to the set W, of words w of length r in a and b, such that p and w correspond when p = w(fs).

Proof. The result is true when r = 1, since B, = { a , , b , } , where

a , = a(fs) and

6, = 6 ( f , ) .Suppose it is true when r = k and consider a vertex p in B,,,. Let p , be the vertex in B,, which is adjacent to p , and let p 2 be the vertex in Bk-l which is adjacent to p , . By the induction hypothesis we have p , = w(fs) for some w in W,.Since w is an automorphism it takes the three vertices adjacent to fs to the three vertices adjacent to p l . But p 2 = w(fs-,), s o p is either w ( a l ) or w ( b , ) ; that is, p is wa(fs) or w b ( f , ) , as required. Since the sets Bk+,and w k + l both have 2,,+' members, the correspondence is a bijection. 0

Lemma 2. The group G contains elements following properties. ti(&) = f , (0 s ; 9, [;(&+I) = ei; z ( J )=fr-, (0 s;s s ) .

ti

(1 6 i S s - 1) and z with the

Proof. It is trivial to check that b-'a' satisfies the conditions for ti. For z , consider ~ . automorphism q fixes fs and takes fo to a vertex in B,. Hence, first q = b s ~ - The by Lemma 1 there is a word u in a and 6, of length s, such that q(f0)= u(fs). It follows that u-'q interchanges fo and fs, and so we may take z = u-'q. 0 Theorem 1. The group G acts regularly on the set of s-arcs of T.

Proof. Let Q be the orbit containing F, in the action of G on the set of s-arcs of T. Suppose that K is in 9,and that L is a successor of K (that is, the initial s - 1 vertices of L are the final s - 1 vertices of K ) . Then k = g ( F ) , for some g in G , and g-'(L) is a successor of F. The successors of F are a ( F ) and 6 ( F ) , so L is either g u ( F ) or g b ( F ) , and L is in the orbit 9. Similarly we can show that 9 is closed under the operations of taking a predecessor (a-' or b-'), branching (t,), and reversing (2). Thus Q contains all the s-arcs in T. Suppose that g is an element of G fixing each vertex of the standard s-arc F. If g is not the identity then, by conjugating with a and b as necessary, we may assume that g switches the vertices a , and 61. Now b and gag-' are translations whose axes intersect in a path of length s 1, contrary to our definition of s. Hence g is the identity. 0

+

The regularity of the action of G on the set of s-arcs implies that certain reiations are satisfied by the automorphisms ti and z specified in the lemma. For

Confluence of presentations associated with graphs

43

example, suppose s = 2 and we have u = ab in the definition of z (Lemma 2). Since z2 fixes F it follows that z 2 = 1 in G. Similarly it can be seen that zaz is either a d z or b-', and it turns out that only the first possibility is consistent with our choice of z. In this way we obtain the relations in the following set. 2 2 -- t 2l = 1, zaz = a-', zbz = b-', btl = a, abza2= b2. The most famous result in this field goes back to the original paper [12] of Tutte in 1947: there are exactly seven non-trivial groups of this kind.

3. Normal forms One way of looking at the relations for a group is to think of them as "rewriting rules". Thus, in the example above, any word in a, a-', b, b-', tl, and z which contains two adjacent occurrences of z may be rewritten by deleting them, and so on. In general the rewriting process is not well-behaved, but we shall explain why our groups have good properties in this respect. In the following theorem W denotes the set of all words in a and b only, that is, the union of the sets W, (r 2 0); and B denotes the corresponding union of the sets B, (r 2 0).

Theorem 2. Let H = (ho,h l , . . . ,h,) be any s-arc in T. Then there is a finite set X of elements of G such that H = xwy (F), where w is a uniquely determined element of W , and x and y belong to X . Proof. We shall assume for simplicity that the distance between F and H is at least s. There is no loss of generality in this assumption since only a finite number of s-arcs fail to satisfy it, and their existence does not affect the conclusion of theorem. Let f;: and hi be the (unique) pair of vertices such that d(F, H) = d($, hi). We show first that we need only a finite number of automorphisms x to ensure that x-'(hj) lies in B. If f;: =fs then we may take x = 1, while iff;: =fo then we may take x = z. For any other A the vertex ei lies on the path from 5 to h,, and we consider the action of t;'. Since t;l(ei)=f;:+lit follows that the vertex of F nearest to t;'(ei) is 5, where i + 1d r S s. If r = s then we take x = ti; if not we repeat the argument with t;' replaced by t;'t;'. Eventually we must find an x = titr - - with the required property. Let x - l ( H ) = M = (mo,m l , . . . ,ms);we have to find suitable automorphism w and y such that M = wy(F). All the vertices of M are in B, and mj = x-'(hi) is the nearest of them to fs. If j = 0 then we choose w to be the unique word such that m, = w(fs). In this case we have M = w(F), and so we can take y = 1. If j > 0 then we choose w to be the unique word such that mo = w(fs). Here we have mk = ~ ( f s - ~=)wz(fk), provided that 0 s k G j. If j = s then M = wz(F), and so w e ta k e y = z. I f O < j < s t h e n

-

mi+' = w(es-j)= wz(ej)= wztj(i.+l).

N. Biggs

44

Now we have mk = wztj(fk), provided that 0 S k S j + 1. It may happen that = wzt,(F), in which case we take y = zrj. If not, there is a least value of r for which m, = wzt,(e,), and we repeat the argument with t, replaced by fjt,, and so on. Eventually we obtain the solution y = ztitr * . . . 17

M

If g is any element of G the theorem implies that g ( F ) = x w y ( F ) , and so g = xwy by Theorem 1. We shall say that this is the normal form for g. In practice the proof of the theorem provides rather more information about the normal forms than the generality of its statement. For the group G2presented at the end of Section 2 there are just nine kinds of normal form: W

zw

tW

WZ

ZWZ

twz

WZt

zwzt

twzt.

These correspond to the nine possible “configurations” of a 2-arc H in T, relative to F: the nine choices for the vertices A and hj (in the notation of the theorem) each give one kind of normal form.

4. Confluence and its applications The rewriting rules derived from the relations for an infinite group may or may not have two desirable properties: confluence and termination. (The reference [9] contains a detailed treatment of this material.) Roughly speaking, confluence means that when two terms are obtained by rewriting a given term in different ways, then these terms may themselves be rewritten so that a common term is obtained. Termination means that the rules cannot be applied infinitely often, starting from any given term. Given a set of relations, an algorithm due to Knuth and Bendix [8] will produce a “good” (that is, finite, confluent, and terminating) set of rules, if one exists. However, because the word problem for groups is known to be unsolvable in general, the algorithm cannot be guaranteed to work. Indeed it appears to be unusual for it to succeed. In the case of the groups under discussion here, the success of the KnuthBendix algorithm is guaranteed by Theorem 2. The following is a good set of thirteen rules derived from the relations displayed at the end of Section 2.

z z + null word at--, b tt 4 null word bt --, a ura + z tza- zb

-

uzb + zt bza zt bzb -+ z ztz +tzt

tzb +za zta + tb ztb + ta

Confluence of presentations associated with graphs

45

As predicted by the theorem, the rules allow any group element to be rewritten in its normal form xwy, as specified in the list displayed at the end of the previous section. Similar results apply to other groups of this kind, and they have been verified in practice. These results can be applied to an old problem in the theory of cubic graphs. A group G of the kind we have been considering is universal, in a sense made precise by Djokovic and Miller [7], for a symmetry type of cubic graphs. Additional relations of the form w = 1 (for some w in W ) define quotients of T i n which cycles corresponding to w must exist. The question of whether such a quotient is finite or infinite was considered in [2] using the method of coset enumeration. When this method is successful it establishes that a particular group is finite; but, on the other hand, it can never show that a group is infinite. In a few cases special constructions have been used to prove that certain groups are infinite [3, 6, 101, but no general methods are available. To see how groups may be proved to be infinite by the techniques discussed here, let G2 be the group defined by the relations at the end of Section 2, for which we have exhibited a good set of 13 derived rules. Let G2(a2n+')be the quotient of G2 obtained by adjoining the relation a&+' = 1. Application of the Knuth-Bendix algorithm yields the result that there is a good set of rules for every group G2(ah+l)and, somewhat surprisingly, the number of rules involved is independent of n. Specifically there is a set of 59 rules (including the 13 derived from G2), and the 46 new ones apply to certain terms containing a"+', an, and , but no small powers of a. Thus any word reduced by the first 13 rules to a normal form in which w contains n - 2 consecutive a's cannot be further reduced by the new rules. Since there are infinitely many such normal forms when n 3 3, we conclude that G2(a2"+l) is infinite for all such values of n. It mst be admitted that the preceding result is not new, since it can be deduced from a classical result [5, p. 541 that G2(a") is infinite whenever m 2 6. But the prospect of applying similar techniques to other groups remains open.

Acknowledgement I am grateful to Derek Holt for allowing me to use his implementation of the Knuth-Bendix algorithm, and to the Science and Engineering Research Council for providing Ursula Martin with the machines on which to run it.

References 11) N.L. Biggs, Algebraic Graph Theory (Cambridge University Press, 1974) vii + 170pp.

[2] N.L. Biggs, Presentations for cubic graphs, in: Computational Group Theory, ed. M.D. Atkinson (Academic Press, London, 1984) 57-63. [3] N.L. Biggs, Homological coverings of graphs, J. London Math. SOC. (2) 30 (1984) 1-14.

46

N. Biggs

[4] J.W. Cannon, D.B.A. Epstein, D.F. Holt, M.S. Paterson and W.P. Thurston, Word-processing and group theory, to be published. [ 5 ] H.S.M. Coxeter and W.O.J. Moser, Generators and Relations for Discrete Groups (Second edition) (Springer Verlag, Berlin, 1965) ix 161 pp. 161 A. Delgado and R. Weiss, On certain coverings of generalized polygons, to be published. 171 D.Z. Djokovic and G.L. Miller, Regular groups of automorphisms of cubic graphs, J. Combin. Theory Ser. B, 29 (1980) 195-230.

+

[S] D.E. Knuth and P.B. Bendix, Simple word problems in universal algebras, in: Computational Problems in Abstract Algebra, ed. J. Leech (Pergamon Press, Oxford, 1970) 263-297. 191

K.Madlener and F. Otto, About the descriptive power of certain classes of finite string-rewriting

systems, J. Symbolic Computation, to appear. 1101 R.C. Miller, The trivalent symmetric graphs of girth at most six, J. Combin. Theory 10 (1971) 163-182. 1111 J.P. Sene, Trees (Springer Verlag, Berlin, 1980) ix + 142 pp. [12] W.T. Tutte, A family of cubic graphs, Proc. Cambridge Philos. SOC. 43 (1947) 459-474.

Discrete Mathematics75 (1989) 47-53 North-Holland

47

LONG CYCLES IN GRAPHS WITH NO SUBGRAPHS OF MINIMAL DEGREE 3 BCla BOLLOBAS* and Graham BRIGHTWELL Dept. of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 ISB, U.K .

If a graph G has n vertices and 2n - 1 edges, it must contain some proper subgraph of minimal degree 3. If G has one edge fewer and contains no such subgraph, then, as proved by ErdGs, Faudree, Gy6rfA.s and Schelp, it contains a cycle of length at least [log n ] . Our aim in this note is to prove an essentially best possible result, namely that such a graph must contain a cycle of length at least 4 log n + O(1og log n ) .

There has recently been a certain amount of interest in graphs all of whose small subgraphs have a vertex of degree at most 2. It is trivial to show that if a graph G, which will always be taken to have n vertices, has at least 2n - 1 edges, then there is some proper subgraph H of G with minimal degree d(H) = 3. ErdBs conjectured that there is in fact always a subgraph H with d(H) = 3 and IHI s (1 - ~ ) n for , some absolute constant E > 0. Some progress has been made on this problem: in [2], Erdiis, Faudree, Rousseau and Schelp proved that if G has 2 n - 1 edges then there is always a subgraph H with 6 ( H ) = 3 and l G W l 3 cnf. Similarly, if G has at least n(k - 1 ) - ( $ ) + 2 edges, it has a subgraph of minimal degree k. We say that a graph G of order n is degree k-critical if it has + 1 edges and no proper subgraph of minimal degree k. We exactly n(k - 1) shall be mainly concerned with the case k = 3. Examples of degree 3-critical graphs include the wheel graphs. In connection with the problem mentioned in the previous paragraph, BollobAs asked whether every degree 3-critical graph contains a long cycle. We define f k ( n ) to be the minimum, over all degree k-critical graphs of order n, of the Iength of the longest cycle. Erdiis, Faudree, Gyairfais and Schelp [l] investigated the cycle structure of such graphs G (and also those with 2n - c edges for general c). One of their results was that [log n] s f 3 ( n ) s c n f , for some constant c, where here and throughout the note all logarithms are taken to base 2. In this paper, we prove the following result, giving the precise asymptotic value of f3(n). Our method of proof, which is based on that of Erdiis, Faudree, GyArfAs and Schelp, extends to the case of general k.

(t)

* Research supported in part by NSF Grant DMS 8806097. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

48

B. Bollobds, G. Brightwell

Theorem. Fur n E N, f 3 ( n )= 4 log n

+ O(1og log n ) .

Proof. First we construct a degree 3-critical graph with n vertices and no cycle of length greater than 4 log n + O( 1). Let G be the graph shown in Fig. 1, which is essentially a tree of degree 3 together with two extra vertices joined to each other and to all of the vertices in the outside "layer". If G has m layers, counting the central vertex as the innermost layer (so in Fig. 1 we have taken m = 5 ) , IGl= 3(2"-' - 1)+ 1+ 2 = 3 . 2n--'.There are 3(2"-' - 1) + 1 vertices of degree 3 and two vertices of degree 3.2""' + 1. So the total number of edges is 2(3 * 2m-1) - 2, as required. The length of a longest cycle, as shown in Fig. 1 , is 4rn - 2 = 4 log(n/3)+ 2. Now we proceed to prove the result in the other direction, namely that every degree 3-critical graph G of order n has a cycle of length at least 4logn + @log log n). The proof consists of several stages. (i) The graph G has a vertex x , of degree at most 3. When we remove this vertex, there is a vertex x 2 of degree 2, otherwise G - x , has minimal degree 3, contrary to hypothesis. Now we remove x 2 and continue in this way until we are left with just two vertices. The total number of edges of G is thus at most 3 + 2(n - 3) + 1 = 2n - 2, with equality iff x , has degree

U Fig. 1 . A degree kritical graph G with no cycle longer than 4 log n

+ c.

Long cycles in graphs

x1

x2

x3

49

x4 Xn-3 Xn-2 Fig. 2. Decomposition of G.

Xn-1

Xn

3, every vertex x 2 , . . . , x,-~ has “forward degree” exactly 2 and the last two vertices x , - ~ and x, are joined by an edge. Since G does have 2n - 2 edges, it is of this form (see Fig. 2). Note also that every vertex has backward degree at least 1, since otherwise that vertex would have degree 2 in G . (ii) We form a partial order < on V ( G ) by setting xi < x i iff there is a “forward” path xi,xiz* * xi, in G with i = il < i2< * * < ik = j . We shall first eliminate the case where there is a maximal chain (=forward path) of length at least 4logn in ( V ( G ) , some zk to A such that 2 log n - 4 log log n (Fig. 4). Suppose that there is no such path. How long can a forward path from zk to a member of A be? No longer than k - 1, since otherwise this path concatenated with Z,Z,-~ - . . zk is longer than Po. Also not longer than 2 log n - 4 log log n - k , by hypothesis. If the longest forward path from Zk to A has length j , then the number of vertices u of A such that v > z k but v>z, for 1 < k is at most 2’. Thus the number of vertices of A which dominate some z k is at most

+

-

Ice),

c

log n-2 log log n

2k-2+

c

22logn-4loglogn-k

k =log n - 2 log log n + 1

k=2

.

< 21ogn-2loglogn+1

-

2n (log n)’

’

But every vertex of A dominates some .?&, and hence IA( rn. Again by minimality of m, these paths are disjoint from PI and P2 except in x, and possibly x,,, which may equal x,~. (xi) We now complete the construction of the cycle as follows. Without loss of generality j , S j z . We form a forward x,, - . x, path, and a path from x,? continued until it hits this path in v say, forming a cycle as shown. This

-

Fig. 8. The complete cycle.

Long cycles in graphs

53

cycle includes P , Q, R and S, and so has length at least 4 log n 10 log log n, as desired (Fig. 8). 0 Identical techniques can be used to prove that f k ( n )= (2k - 2)log,-, n O(1og log n) for each fixed k.

+

References [l] P. ErdBs, R. Faudree, A. Gy6rf6.s and R. Schelp, Cycles in graphs without proper subgraphs of minimum degree 3, Ars Combinatoria 25B (1988) 195-201. [2] P. Erdiis, R. Faudree, C. Rousseau and R. Schelp, Graphs with proper subgraphs of fixed minimum degree, submitted.


Discrete Mathematics 75 (1989)55-68 North-Holland

55

FIRST CYCLES IN RANDOM DIRECTED GRAPH PROCESSES BCla BOLLOBAS* Department of Mathemath, LSU, Baton Rouge, LA 70803, U.S.A. and Department of Pure Mathematics and Mathematical Statistics University of Cambridge, England

Steen RASMUSSEN Physics Laboratory III, The Technical University of Denmark, Lyngby, Denmark

0. Introduction

The complexity we meet in modern lifeforms is immense. Even the basic informational and functional units in living organisms such as the genetic code and the protein synthesis machinery are extremely complex. The origin of these units is impossible to explain by simple random events [12, 131. Combinatorial arguments show that the chance of generating the information accumulated even in the simplest protein in such a system is so small that the evolution of these elements would take more than 10’Oo times the lifetime of the universe. The “frozen accident hypotheses” thus makes the evolution of life an extraordinarily improbable event. However, an alternative and much more plausible explanation of the origin of life is based on the idea of self-organization [2,3,4,5]. For the evolution of the first genes this means a formation of more complex molecules as a result of cooperation between simpler molecules. The crucial steps in order to create such a cooperative structure is the appearance of positive (catalytic) feedback loops ~ ~ 7 1 . A simple model for such a process is a random graph where the vertices represent a vast number of relatively short selfreplicating ribonucleotide strands (RNA molecules), and where the directed edges represent catalytic interactions between the different RNA molecules [14, 151. Given the physico-chemical conditions on how and with what frequency the catalytic formations are made, we want to know when the first catalytic feedbacks appear, and how many different RNA molecules they involve. To be a little more precise, let V be a fixed set of n vertices. At time 1 a random vertex becomes active: it sends out k directed edges at random, with each of the ([t) choices being equally likely. At time 2 a random vertex from the remaining n - 1 vertices becomes active and sends k directed edges at random. * Research supported in part by NSF Grant 8806097. 0012-365X/89/$3.50 0 1989,Elsevier Science Publishers B.V. (North-Holland)

56

B. Bollobh. S. Rasmussen

At time 3 a random vertex from the remaining n - 2 vertices becomes active, etc. For what values of t is it likely that at time t our graph contains an r-cycle? What is the expected time of emergence of a cycle? What is the probability that the first cycle is an r-cycle‘? The main aim of this note is to examine questions like these. The analogous questions for the standard random graph process were studied by Janson [ll], Flajolet, Knuth and Pittel [9] and Bollob6s [4]. Models similar to the one described above can be used to give quantitative estimates of the time of emergence of “real” genes. This is due to the fact that, as a random graph evolves, monotone properties (such as containing cycles) appear rather suddenly. The assumed physico-chemical conditions for the prebiotic environment then define which random graph model we have to choose [14].

1. Preliminary results As customary, a directed graph is a pair (V, E) where E c V x V. Here V is the vertex set and E is the set of directed edges or arcs. Note that we allow loops but we do not allow multiple edges. However, a directed graph may have an arc ab from a to b and an arc ba from b to a. A directed graph process on V = [ n ] = { 1, 2, . . . , n } with parameter k is a sequence D = (D,):of directed graphs such that D o c D,c * * c D,, in D, precisely t of the vertices have outdegree k and every other vertex has outdegree 0. Thus 0,has precisely kt directed edges, including loops. Let G(”(n) be the set of all n! ( : ) n directed graph processes with parameter k. As usual, we turn .$k’(n) into a probability space by endowing it with the normalized counting measure. We shall study random directed graph processes, i.e. random elements of G(‘)(n). E @ k ) ( n ) the directed graph d, is the state of the (directed) graph For = (0,): be the probability space whose random element is process at time t . Let ’3d,(A)(n) the state of a random graph process at time t. Thus q: @ ‘ ) ( r ~ ) + @ ~ ) ( n ) given , by q ( G ) = D,, is a measure preserving map. Note that whenever we take a directed graph process (D,):,we may assume that in each D, the vertices 1,2, . . . , j have outdegree k and the others have outdegree 0. Similarly, having stopped the process at time t , we may assume that for j s t , in the graph D, the vertices 1 , 2, . . . , j have outdegree k. However, each of the vertices t + 1, t + 2, . . . , n has the same probability of becoming the new vertex in D,+l with outdegree k . We are interested in the length of the first cycle (to be precise, in the length of a first cycle) and in the time when this first cycle appears, as n +m. In particular, in our estimates we may and shall assume that n is sufficiently large. Furthermore, the quantities o(l), 0(1), etc., are with respect to n-m. An r-cycle in a directed graph D is a subgraph of D with vertex set { x , , . . . , x,} and arc set {xIxz, ~ 2 x 3 ., . . , x,-,x, and x,xx,}. Thus a 1-cycle is a

-

Random directed graph processes

57

loop aa (together with the vertex a ) and a 2-cycle is a pair of arcs ab, ba (together with the vertices a and b ) . The existence of cycle in random graphs was first studied by Erdos and RCnyi [8]. Recently Janson [ l l ] and Flajolet, Knuth and Pittel [9] proved some deep results about the distribution of the length of the first cycle in a random graph process, and in [4] some of these results were proved by a different method, based on martingales. In this note we shall follow the latter approach. Let us start with a simple result, corresponding to the classical result of Erdos and RCnyi about cycles in random graphs. Denote by X j ( t )= X , ( t ) ( B ) the number of j-cycles in 0 , .

Theorem 1. Let k , 1 and a>O be fixed and let ktln-, (Y (as n - w ) . Then X,(t), X2(t),. . . , X l ( t ) are asymptotically independent Poisson random variables with means A,, A2, . . . , AI where A, = d / r .

Proof. Clearly 1 E(X,(t))= - (t),(k/n)' r

- a r / r = A,.

Furthermore, it is easily checked that every joint factorial moment of X l ( t ) , . . . ,X l ( t ) tends to the appropriate factorial moment of independent Poisson random variables with means A,, . . . , ill. This implies the result (see [2, Theorem 21, p. 231). 0 One should remark that the result above can also be read out of some general results of Whittle [16; Formulae (Sl)]. In the proof of our main results, we shall make use of the following immediate consequence of Azuma's inequality [l]. (For a general background and many other applications, see [3] and [4].)

Theorem 2. Let @,'')(n) be endowed with an arbitrary probability measure. Furthermore, let X be a random variable on 9d,(")(n)such that if D and D'E $9d,(k)(n)differ only in some arcs leaving vertex i, 1S i S t, then IX(D) X(D')I s ci. Then for every a > 0 we have

Let 0 < w0 < 1 be fixed and set to = La0n/k].For t G to set w(t) = kt/n. We shall at time to, i.e. we shall study the states 0,only for t s to. stop the process D(") Let us say that a vertex x dominates a vertex y if there are vertices z,, z2, . . . , q such that x z l , z1z2, ~ 2 ~ . 3. ., , zl-,z1, z l y are all arcs. We shall show that it is very unlikely that some vertex in D, dominates or is dominated by at

B. Bollobbs, S. Rasmussen

58

least mo= [(log n)(iog log n)'l vertices. Set $2, =

{ D ( k= ) (D,):E @k)(n):in D,, no vertex dominates or is dominated by mo vertices}.

Proof. For x E V = [n]let m, = m,(D,) be the number of vertices dominating x in 0,. Then 0,contains a rooted tree on m, + 1 vertices, with the root at x , such that every arc goes towards the root. Also, there is no arc from the outside of the tree to a vertex of the tree. Therefore, if m, S m S 2m0, then m

=s (et/m)mmm (k/n)me-kmr'n I-a(O)m

s (a(t)e

(1)

m ~ ~ - 3 l o g l o g n

)

Note also that if b(") = (0,): is such that in Of, some vertex is dominated by at least rn, vertices then for some x E V and 1 S t 6 to, we have rn, s m,(D,) s 2m,. By (1) the probability that this happens is, very crudely, at most n-210gLogn. A similar argument shows that the probability of dominating many vertices is also sufficiently small. The only change in the argument is that instead of moG m s 2mo we have to take a larger range: m, s m S km,. (7 Occasionally we shall consider probabilities and expectations conditioned on the event SZ,. We shall denote the probability conditional on $2, by Po and the expectation by Eo. Let us introduce some random variables on ?28&).An r-path in D, is a directed path of order r ending in a vertex of outdegree 0. Denote by Y,(t)(b)the number of r-paths in 0,. Let Z , ( t ) ( b ) be the number of pairs of vertices ( x , y ) for which 0,contains an r-path from x to y and let Ur(r)(b) be the number of pairs of vertices joined by a unique r-path. Finally, let K ( f ) ( B )be the number of pairs of vertices joined by an r-path and by no path of strictly smaller order, and let W , ( t ) ( b ) be the number of pairs of vertices joined by a unique shortest path which has order r. Note that


59

is the number of pairs of vertices ( x , y) such that x Z y , the graph 0,contains a (directed) path from x to y, and y has outdegree 0. We wish to use Theorem 2 to show that these variables (with the expection of x ( t ) in which we are not too interested) are close to their expectations on Qo, with probability exponentially close to 1. First we shall estimate the conditional expectations.

Lemma 4. Let n be', 2 6 r S mo and r2 < t = m / k 6 toS aon/k. Then (n - t

y - 1

+ 1a E,(Z,(t)) 3 E,(Wr(t))

+ k / ( n - k t ) ) }- 1

b (n - t)ar-'{1- r 2 ( l / t

(4)

and IV(t) - (n - t ) a / ( l - a)l co(ao)(n- t ) / t .

(5)

Proof. Clearly,

n (1 - i / t )

r-2

E ( Y , ( t ) )= (n - t ) ( t ) r - l ( k / n r - l= (n - 1 ) d - l

i=l

so (n - r)ar-l(l- r 2 / t )s ~ ( x ( t ) (n ) - t)ar-'.

(6)

Let L be the r-path 12 - - - (r - l ) ( t + 1). Denote by d,(t) the probability that 0, contains a path of order at most r from 1 to t + 1 which is different from L, conditional on the event that 0,contains L. By the definitions of d,(t), and Wr(t) we have

v(t)

(1 - Sr(t))E(Y,(t))6 E(Wr(t)) d E(Zr(t))6 E(Y,(t))*

(7)

Note that if there is a path of order at most r from 1 to f + 1 which is different from L then there is a path or order at most r - 2 which starts and ends on L but shares no edge with L. Therefore d,(t) 6

r-2

r-2

s

s=o

2 r2(t)s(k/ny+1 6 - 2 aS s r2k/(n- kt). =o n

(8)

Since Wr(t)G Zr(t)6 n2, by Lemma 2 we have

.

IE,(w,(~)) - E ( w , ( ~ )6) (n2(1 - p(Q0))6 n2-loglogn-= 1

(9)

and, similarly, IEo(Zr(t))- E(Zr(t))l

1-

(10)

Inequalities (6)-(10) imply relation (4). Finally, inequality (5) follows without any difficulty by recalling that if D E SZ, then mo

V(t)(d) =

2 vl,(t)(d).0 i=2

B. BollobL, S. Rasmussen

60

Proof. As these inequalities can be proved in precisely the same way, we shall prove only (I 1). Let D = (Dj); and D' = (0;);be elements of Qo such that D, and 0:differ only in some arcs leaving vertex i, where 1 S i < t. Then, very crudely, I.Z,(t)(D)- Z,(t)(D')lS k(m, - 1)2< k(1og n)*(log log n)',

since the paths of order r created by the addition of k arcs leaving vertex i go from the vertices dominating i to the vertices dominated by i. Therefore, by Theorem 2, for every (Y > 0 we have

P,(IZ,(t) - Eo(Z,(t))l2 a) S 2 exp{ -a2/2tk2(log n)'(log log n)'}). Applying this inequality with a = $tf(logn)4, noting (n(1og n)'/t)+ = and recalling Lemma 4, we find that

that

tf(log n)'/

P,(Iz,(c) - (n - t)a'-'l a tf(1og n)') < n--(logn)*. Since, by Lemma 3, P ( Q ) a 1 - n-'og'ogn, this implies inequality (11).

0

2. Acyclic digraphs

Denote by A, the probability that 0, is acyclic. Our next aim is to find a good approximation for A,, provided t is not too small.

Lemma 6. (i) If t < n / k then A, 2 1 - k t / ( n - k t ) .

(ii) If n 5 G t s r,, then A, 1---

2kt(log n)4) - 2n n(n-t)

k n - kt

(

k

d A , + l S A A 1-----, n-kt

-1oglogn

n)4) + 2ktt(log n(n-t)


61

Proof. (i) Recall that Xr(t)(D)is the number of cycles of length r in D,, where

D = (Di);, so 1-A, = P(C&l Xr(t)> 0). Rather crudely, 1 1 E(Xr(t))6 - t'(k/n)' = - d, r r where, as always, a = a(t)= kt/n, so

s a / ( l - a ) = kt/(n- kt). be an acyclic digraph on [n]= (1, 2, . . . ,n } in which each vertex i, (ii) Let 16 i 6 t, has outdegree k, and each vertex j , t + 16 j S n, has outdegree 0 and is dominated by d j vertices of outdegree k. Set a, = P ( D = (D$ is such that D,+l is acyclic I D, = D,). Then

2(

1 n -4 k - I)/( n - t j=t+l

i),

with the additional -1 term due to the possibility of creating a loop at the newly selected vertex. Setting d = Cin_,+l(dj + l)/(n - t ) we find that

Suppose now that dj 6 mo- 1 for every j and define the integer 1by

+

(n - t)d = Imo h, 0 S h < mo. Then

1

at 6 n - t {I( i m o ) / (

i)+ (' ')I(nk> +

-X) + k

sn -L t {I(1-:)*+ 6 1-- 1

(1

n

I

n -t - j - 1

{I + 1-(1--+-kmo k2mt)

n-t

I

n -t -I -1

n2

-( 1--+kh n

k2h2)} n2

(16)

kd dmok2 Sl--+-, n n2 Inequalites (15) and (16), together with our information about the likely structure of D,, readily imply the required inequalities. Indeed, the probability that

+

v ( t ) ( D )< (n - t ) a / ( l - a) tQog n)4 and D, is acyclic, is at least A, - n-loglogn . Since in this case the average d defined

B. Bollobh, S. Rasmussen

62

for 13, is at most 1/(1 - a ) + ti(logn)4/(n- t ) , by (15) we have kd 2kd22 1 - - - k a,2 1 - -- n nz n-kt

2ktf(log n)4 n(n-t) '

This gives the required lower bound for A,,,. Similarly, the probability that

v(t)(D) > ( n - t)a/(I- a)- t+(logn)4, no vertex in 0,is dominated by mo vertices and DI is acyclic, is at least A, - 2n-IOglogn . If this event holds then the average d defined for 0,is at least 1/(1 - a ) - ti(logn)4/(n- t ) , so by (15) we have kd n

dmok2 n

k n -kt

a, d 1 - -+ 7 c 1 - -+

2ktl(log n)4 n(n - t ) '

This implies the required upper bound for A,+1. From here it is easy to obtain a fairly precise expression for A,.

Theorem 7. For ni 6 t = a n / k =Z to = [ a o n / k ] ,ao< 1, the probability A, of 0, being acyclic is A, = (1 + O ( n - f ) ) / ( l - a) with the constant implied in O(n-f) depending only on ao.

Proof. Let t , = [nsl. Then, by Lemma 6(ii),

+

= (1 O ( n - f ) ) / ( l - a).

Similarly, using both parts of Lemma 6, we find that I--1 k 2kjf(logn)4) A12(1-2k/nf) (1--n -kj n(n - t )

n

+

nz-loglogn

{-?- k

a (1 - 2k/nf)exp

]=,, n - kj

3. The first cycles Given a process b = (Of)& set t = t(b)= min{t: 0,contains a cycle} and let o = o(B) be the minimal length of a cycle in 0,.In fact, this definition of u is

63


rather pedantic because, as the following lemma shows, almost every process I) is such that D, contains a unique cycle, provided the hitting time t is bounded away from nlk.

Lemma 8. P ( D = (DI)::t = t ( D ) s to and D, contains at feast O( U n ) .

IWO

cycles) =

Proof. It suffices to show that in the graph Dto the expected number of pairs of minimal cycles sharing at least one arc is O(l/n). But this expectation is bounded by

2

Osr-2w

r2(to)r(t0)s(k/n)r+s+1 s r2dg+' = O(l/n). 0 n orr-26s

The results in the previous sections enable us to obtain a rather precise approximation of the joint distribution of t and 0.

Theorem 9. Let n: S t = cm/k 6 to and 1 s r d mo. Then P ( t = t + 1 and

(I

=r)= (1

+O(n-f))d-'(l- a)k/n

Proof. Let S2: c 9(')(n) be the event that 0,is acyclic, r-1

~ ( t-)(n - t )

12

2 mi-'

ri=2 --l

I

< &(logn14,

IK(t) - (n - t)nr-'l < tf(1ogn)4 and no vertex is dominated by mo vertices. Then, by (13) and Theorem 7 ,

P ( s ~ := ) ( 1 + O(n+))(l- m).

(17)

Let us fix a process E = (Ei): E a:, with the vertices 1,2, . . . ,t having outdegree k and the others 0. For t + 1 S j 6 n let dibe the number of vertices i , l s i ~ t for , which there is an i-j path of order r but there is no i-j path of smaller order, and let d,! be the number of vertices i , 1 S i S n, for which there is an i-j path of order at most r - 1. Since E E a:,we have di s mo and d,!G mo for all j . Note that Cin_I+ldi = &(n) and Cin_,+ld,!= CTZ: y(t)+ (n - t). Set d = Cin_t+ld j / ( n- t ) = V,(n)/(n- t ) . Denote by B , , the probability that D,+l contains an r-cycle and it contains no cycle of length less than r, conditional on 0, = E,. Clearly,

where in each summand the first term is the conditional probability that D,+, contains no cycle of length at most r - 1, and the second term is the conditional

B. Bollobb, S. Rasmussen

64

probability that Dr+,contains no cycle of length at most r. The jth summand is

il-(

n -dl - d i ) / ( n i d ; ) } ( " i d ; ) / ( ; ) k

n

k-1

=[1-

(1-

I=(J

n

n-i

- d !I - l

Hence

Relations (17) and (18) imply the assertion of the theorem. The following results about the distribution of Theorem 9.

0

and

t

0

can be read out from

Theorem 10. Let 0 < cyO < 1 be fixed and let 1s r 6 mo. Then

P(o = r and

tG

a o n / k )= (1

+O(n-f))

Proof. The expected number of r-cycles in 0,is 1 E ( X , ( t ) ) = - (t),(k/n)' = a r / r r

so P ( o = r and t s n i ) < (kn-f)'/r. Also, by Theorem 9,

P ( o = r and ni < t s cYon/k) = (1

+ O(n-f))

LaidkJ

( k t / n ) r - ' ( l- k t / n ) k / n

Theorem 11. For every fixed r 3 1 we have P ( a = r ) = l/r(r + 1) + o(1). Proof. Theorem 10 implies that P ( a = r )3 Since

C:zzl

l/r(r

1 r(r

+ 1) + o(1).

+ 1) = 1, the result follows.

0


65

Theorem 3.2. Let r 2 1 b e f i e d . Then

1

~ ( 6 z =r)= ( I

rn + o(1))(r + 2)k *

Proof. By Theorem 10 and 11,

= (r(r

+ 1 ) + o(1))2n

I

1 ar(l- a ) d a

0

Theorem W. Let 0 C a. C 1and n f P(z =t )=(1

GtG

aon/k. Then

+ O(n-f))k/n.

Furthermore, E(z)= ( 1 +o(l))n/2k.

Proof. Both assertions are immediate from Theorem 9. 0 Note that in the results above the parameter k plays a very insignificant role. In fact, if instead of t we use m = kt, the number of edges, to measure the evolution of our random graph then our formulae become independent of k. For example, the expected number of edges when the first cycle has t vertices is ( 1 + o ( l ) ) ( r n ) / ( r 2), and the expected number of edges when the first cycle appears is ( 1 0 (1))n/2. It is clear that the methods above can be used to refine considerably the results above. When defining ao,to, mo and Qo, we were far too cautious for there is no need to guarantee that P(Qo) is that close to 1 . In fact, we may take E 0 -- n-'/(logLogn)*, a0 -- 1 - co and mo = n3/(IogLogn)*, and define Qo as before. Then we still have P ( Qo) 2 1 - O(n-') for every constant c and all our theorems hold in this larger range. Theorem 11 implies that E ( a ) , the expected length of the first cycle, is unbounded as n +CQ. However, the results above are too weak to enable one to determine the asymptotic value of E(a). In particular, it is not even clear whether E ( a ) is closer to a power of log n, say, than to a power of n.

+

+

4. Some other models In this brief section we shall discuss some of the many other models in which the emergence of the first cycle may be of some interest.

B. Bollobh, S. Rasmussen

66

First of all, the model closest to the usual graph process model, the space of standard random directed graph processes, is defined as follows. A (standard) directed graph process is a sequence Do, D1,. . . , Dn2 of directed graphs on [n] such that 0, c D,+l and D, has precisely t arcs. The normalized counting measure turns the set of all (n2)! directed graph processes into a probability space, the space of standard directed graph processes. The emergence of the first cycle in this model is rather similar to the emergence of the first cycle in the standard random graph process. This was studied in detail by Janson [ll] and Flajolet, Knuth and Pittel [9], who proved very precise results about them (see also [3] and [4]). In a variant of the model above we construct DI+l by picking at random a vertex of 0,of outdegree at most n - k and sending out k new arcs from that vertex. (For the sake of simplicity, one should assume that k divides n . ) Note that the vertex we pick may have been picked earlier, so it may have positive outdegree. The random process of directed graphs defined in this way is rather similar to the standard random graph process, with kt playing the role of time. The main model we shall consider in this section is, perhaps, the most relevant to self-organizing systems. This model is a refinement of 9 ( k ) ( n ) ,the refinement being that we keep a record of the times the arcs were born, i.e. of the times the vertices were activated. This is, of course, the case when we look at a process fi = (0,);; however, when considering the state 0, of this process at time t, up to now we have ignored the order in which the vertices have been born. Thus let @ ( n ) consist of all sequences E = (E,); in which each E, is a directed graph on [ n ] with precisely t vertices of outdegree k, labelled 1, 2 , . . . , t, such that Eo c E , c . . . c En and for 1C t S n the vertex t has outdegree 0 in Given E = (E,): and a vertex x E [n],denote by f(x) the label of x. Thus l(x) is the time when x was ‘active’. An r-cycle in E, is a sequence of vertices xl, x 2 , . . . , x, such that I ( x l ) < I(x2)< < I(xr)zs t and E, (or En) contains the arcs x I x 2 , ~ 2 x 3 ., . . ,x,-~x, and x j l . Thus in an r-cycle x l x z - ex, the vertex x 1 influences x2 at some time l(xl), then x 2 influences x 3 at a later time I(x& etc. Define X , ( t ) , t and GI before: X,(t) is the number of r-cycles in E,, t(&)= min{t: E, contains a cycle} is the hitting time of a cycle and (I= min{r: E, contains an r-cycle} is the minimal length of a cycle appearing at the earliest time. Setting, as before, LY = a([)= kt/n, we find that the expected number of r-cycles at time t is

-

-

E ( X , ( t ) )= ( ‘r) ( k / n ) ‘ - a‘/r! if r is fixed and

t-m.

If O < a < 1 then

Analogously to the theorems in the previous sections, one can prove the


67

following results. Theorem 14 is straightforward while the others require some work.

Theorem 14. Let k, j and a > 0 be fixed and let ktln + a. Then X,(t), . . . , Xi(t) are asymptotically independent Poisson random variables with means a, 2 1 2 , d / 3 ! ,. . . , dlj!. Theorem 15. For every fixed r 3 1, we have e-euar-l

P(a = r ) = ( r - l)!

d a + o(1).

Theorem 16. m

E ( z ) = I ( e + o ( l ) ) l ae"e-""da. 0

It may be of interest to remark that if we refine the space of standard directed graph processes by keeping track of the time, and define an r-cycle as above, then Theorems 14-16 hold for this case as well.

References [l] K.Azuma, Weighted sums of certain dependent variables, Tokoku Math. J. 3 (1976)357-367. [2]B. Bollobh, Random Graphs (Academic Press, London, 1985) xvi + 447pp. [3] B. Bollobh, Sharp concentration of measure phenomena in the theory of random graphs, to appear. [4] B. Bollobh, Martingales, isoperimetric inequalities and random graphs, in Combinatorics, Eger (Hungary) 1987,Coll. Math. SOC. J. Bolyai, 52,Akad. Kiad6, Budapest (1988)113-139. [5]M. Eigen, Selforganization of matter and evolution of biological macromolecules, Naturwissenschaften 58 (1971)465. [6]M. Eigen, The origin of biological information, in: The Physicists Conception of Nature, ed. J. Mehra (Reidel, Dordrecht, 1983). [7] M. Eigen and P. Schuster, The Hypercycle - A Principle of Natural Selforganization (SpringerVerlag, Heidelberg, 1979). [8] P. Erdos and A. R h y i , On the evolution of random graphs I, Publ. Math. Debrecen 6 (1959) 290-297. [9]P. Flajolet, D.E. Knuth and B. Pittel, The first cycles in an evolving graph, this volume. [lo] P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations (Wiley, New York, 1971). [ll] S. Janson, Poisson convergence and Poisson processes with applications to random graphs, Stochastic Processes and their Applications 26 (1987)1-30. [12]J. Monod, Chance and Necessity (Knopf, New York, 1970). [13]C. Nicolis and I. Prigogine, Selforganization in Nonequilibrium Systems (Wiley, New York, 1977).

68

B. Bollobcir, S. Rasmussen

(141 S. Rasmussen, B. Bollobis, E. Mosekilde, J. Engelbrecht and N. Raagaard, Elements of a

quantitative theory for prebiotic evolution, submitted to the J. of Theoretical Biology. 1151 S. Rasmussen, E. Mosekilde and J. Engelbrecht, Time of emergence and dynamics of cooperative gene networks, Proceedings of MIDIT Workshop on Structure, Coherence and Chaos, the Technical University of Denmark, to appear in a special issue of Nonlinear Science, Theory and Applications Manchester University Press). I161 P. Whittle, The equilibrium statistics of a clustering process in the uncondensed phase, Proc. Royal Society Ser. A. 285 (1965) 501-519.


69

TRIGRAPHS J.A. BONDY Dept. of Combinatorics and Optimization, University of Waterloo, Ont. N2L 3G1, Canada

1. Cycle double covers and trigaphs Definition 1.1. A cycle double cover (CDC) of a graph G is a collection V of cycles of G such that each edge of G belongs to exactly two cycles of V. A small cycle, double cover (SCDC) of a graph G on n vertices is a CDC V of G such that lCel sn - 1. The following conjecture was put forward in [2].

Conjecture SCDC. Every simple 2-edge-connected graph admits an SCDC. Remark 1.1. Conjecture SCDC, the Small Cycle Double Cover Conjecture, is a strengthening of the Cycle Double Cover Conjecture, due to Seymour [lo], which asserts that every 2-edge-connected graph admits a CDC. Conjecture SCDC is studied in [2]. It is easily verified for complete graphs and complete bipartite graphs, and holds, too, for triangulations of surfaces. As is the case with the Cycle Double Cover Conjecture, it can be reduced to 3-connected cyclically 4-edge-connected simple graphs. It cannot, however, be reduced to 3-regular graphs simple for which a stronger assertion seems to be valid [2]. We now define a class of graphs, called trigraphs. They appear to be relevant to the study of SCDCs, and are the central subject of this work.

Definition 1.2. A tritree of a graph G is a spanning tree T of G such that every fundamental cycle of G with respect to T is a triangle. Definition 1.3. A trigraph is a graph which has a tritree. Example 1.1. A spanning star is a tritree. Example 1.2. The square of a tree is a trigraph. We shall see, in Section 4, that the restriction of Conjecture SCDC to trigraphs is a question of independent interest, and that the conjecture holds for those trigraphs which are squares of trees. 0012-365X/89/$3.50 C (J 1989, Elsevier Science Publishers B.V. (North-Holland)

70

J.A. Bondy

2. Decomposition of trigraphs

We show, here, how a trigraph decomposes naturally into trigraphs each of whose tritrees is a star. A closely related decomposition, into trigraphs at least one of whose tritrees is a star, was found, independently, by Seyffarth [9].

Definition 2.1. Let G1 and G, be two trigraphs such that V ( G l )n V(G,) = { x } . The 1-sum of G1 and G2 is the graph G1UG,; it is nontrivial if V ( C , ) # { x } , i = 1. 2.

Proposition 2.1. Let G be a 1-sum of trigraphs G1and G2,and let I;: be a tritree of G,, i = 1, 2. Then U & is a tritree of G. Corollary 2.1. The 1-sum of fwo trigraphs is a trigraph.

Proposition 2.2. Let G = GI U G,, where V(G,)n V(G,) = {x} and V(G,)# { x } , i = 1, 2, be a separable trigraph, let T be a tritree of G , and let I;. = T n G,, i = 1, 2. Then 'I; is a tritree of GI, i = 1, 2. Cordary 2.2. Every separable trigraph is a nontrivial 1-sum of trigraphs. Defiaition 2.2. Let GI and G2 be two trigraphs such that V(Gl) n V(G,) = {x, y} and E(Cl) ilE(G,) = {xy}. Suppose, moreover, that Gi has a tritree T, through the edge xy, i = 1, 2. The 2-sum of G1and G, is the graph G1U G,; it is nontrivial if V(GJ # {x, y } , i = 1 , 2. Propition 2.3. Let G be a 2-sum of trigraphs G1 and G,, and let T, be a tritree of G, through the edge x y , i = 1,2. Then T, U & is a tritree of G. Carollary 2.3. The 2-sum of two trigraphs is a trigraph.

Proposition 2.4. Let G be a nomeparable trigraph with a 2-vertex cut { x , y}, and let T be a tritree of G. Then xy E E( T ) . Proof. Let G = G, U G,, where V ( C l )n V(G,) = {x, y} and V ( G , )# {x, y } , i = 1,2, and let P, be an ( x , y)-path of length at least two in GI, i = 1, 2. Suppose that xy 4 E ( T ) . For each edge uu E E(&)\E(T), there is a (u, v)-path length two in T f l G,, i = 1, 2. Thus there is an ( x , y)-path Q, in T n Gi, i = 1, 2. But now Q , U Q, is a cycle in T , which is impossible. Therefore xy E E ( T ) , as claimed. El CoroUary 2.4. Let G be a nomeparable trigraph with a 2-vertex cut {x, y}. Then G = G, U G2, where V ( C l )n V(G2)= {x, y}, E(Gl) n E(G,) = {xy}, and

Trigraphs

V ( G j )# { x , y } , i = 1,2. Moreover, if T is a tritree of G and a tritree of Gi, i = 1, 2.

71

= T fl Gi, then

is

Corollary 2.5. Every nonseparable trigraph with a 2-vertex cut is a non-trivial 2-sum of trigraphs.

Definition 2.3. An internal edge of a tree T is an edge of T neither of whose ends is an endvertex of T. Remark 2.1. A tree has an internal edge if and only if it is not a star. Proposition 2.5. Let G be a trigraph, T a tritree of G, and xy an internal edge of T. Then { x , y } is a 2-vertex cut of G.

Proof. Denote the component of T - xy containing x by T, and that containing y by Ty . Since T is a tritree of G, no edge of G can have one end in T, - x and the other in Ty - y . It follows that { x , y } is a 2-vertex cut of G. 0 Definition 2.4. A star trigraph is a trigraph every tritree of which is a star. Corollary 2.6. Let G be a trigraph with no 2-vertex cut. Then G is a star trigraph.

Theorem 2.1. Let G be a trigraph. Then G is: 1. a star trigraph, 2. a nontrivial 1-sum of trigraphs, or 3. a nontrivial 2-sum of trigraphs. Proof. Suppose that G is not a star trigraph. If G is separable, then, by Corollary 2.2, G is a nontrivial 1-sum of trigraphs. If G is nonseparable, then, by Corollary 2.6, G has a 2-vertex cut. Therefore, by Corollary 2.5, G is a nontrivial 2-sum of trigraphs. 0 The following algorithm accepts, as input, an arbitrary graph H, and yields, as output, either a tritree T of H or the message that H is not a trigraph. Algorithm 2.1. Set G = H, S = 0, and T = 0. Step 1. If G has a cut vertex x , let G = G, U G2, where V ( G , )n V(G2)= { x } and V(GJ # { x } , i = 1,2. For i = 1,2, replace G by Gj and return to Step 1. If G has no cut vertex, proceed to Step 2. Step 2. If G has a 2-vertex cut { x , y } and xy @ E ( G ) , stop: H is a not a trigraph. If G has a 2-vertex cut { x , y } and xy E E ( G ) , let G = GI U G2, where V(G,)n V(G,) = { x , y } , E(G,) n E(G2)= { x y } , and V(Gi)# { x , y } , i = 1,2. For i = 1,2, replace G by Gjand S by (S r l E(Gj))U { x y } , and return to Step 2. If G has no 2-vertex cut, proceed to Step 3.

72

J.A. Bondy

Step 3. If S is contained in a spanning star S* of G, replace T by T US*. Otherwise, stop: H is not a trigraph.

Definition 2.5. For a trigraph G , the number of tritrees in G is denoted by t(G). Proposition 2.6.

1. Let G be a separable trigraph with blocks Bi, 1S i =sm. Then

n m

t(G)=

t(Bj).

i=l

2. Let G be a nomeparable trigraph on n vertices. (a) If G has no 2-vertex cut, then

with equality if and only if G = K,, n # 2. (b) I f C has a 2-vertex cut, then

t ( G )s 2*-2, with equality if and only if G = K 2 v KCn-2.

Proof. 1. This follows directly from Propositions 2.1 and 2.2. 2. (a) If G has no 2-vertex cut, then G is a star trigraph, by Corollary 2.6. Thus t(G)an, with equality if and only if G = K,, n # 2. (b) By induction on n . Let { x , y } be a 2-vertex cut of G. By Corollary 2.5, G is a nontrivial 2-sum of trigraphs GI and G2. Let n, = IV(C,)l, and let r'(G,) denote the number of tritrees of G, which contain the common edge ny, i = 1, 2 . By (a) and the induction hypothesis,

r'(Gr)s 2 " 1 - ~ if n j # 3, i = 1, 2. Also,

t'(Gj)= 2 = 2"1-' if nj = 3, i = 1, 2. Therefore, t ( G ) = t'(G,)t'(G2)s 2"'-22"2-2 = 2"-'

. o

Trigraphs

73

3. Cycles in trigraphs Definition 3.1. Let G be a trigraph and T a tritree of G. A cycle C of G is a T-star cycle of G if there is a vertex u E V(G)\V(C) such that uv E E(T) for all v E V(C).

Proposition 3.1. Let G be a trigraph, C a cycle of G and T a tritree of G. Then 0 if C is a T-star cycle of G, 2 otherwise.

Proof. If C is a T-star cycle of G, then IE(C) n E(T)J= 0 because T is acyclic. Thus we may assume that C is not a T-star cycle of G. We prove, by induction on IV(G)l, that IE(C) n E(T)I = 2. The result is trivial if T is a star of G, so, by Theorem 2.1, we may assume that G is either a nontrivial 1-sum or a nontrivial 2-sum of trigraphs. If G is a nontrivial 1-sum of trigraphs G1 and G2, then C E Gi, i = 1 or 2. By Proposition 2.2, T = T n Gi is a tritree of Gi. Moreover, since C is not a T-star cycle of G, C is not a 'I-star cycle of Gi. By the induction hypothesis, applied to Gi I E ( C )n E(T)I = I E ( C )n q z ) =~2. 9

Suppose, then, that G is a nontrivial 2-sum of trigraphs GI and G,, where V(G,)nV(G,)={x,y}, and that T is a tritree of Gi through the edge xy, i = 1,2. If C G Gi, i = 1 or 2, then C is not a q-star cycle of Gi because C is not a T-star cycle of G. By the induction hypothesis, applied to Gi,

I E ( C )n E(T)I = p(c)n E ( Q I = 2. If C $ Gi, i = 1, 2, then the edge xy is a chord of C, and partitions C into cycles C , and C,, where Ci E Gi, i = 1, 2. Since xy E E(Ci)n E ( T ) , Ciis not a T-star cycle of Gi. Therefore, by the induction hypothesis, applied to Gi,

IE(C,)n E(T)I = /E(c,) n E ( T ) ~= 2, i = I, 2. It follows that I E ( C )n E(T)J = 2.

Definition 3.2. The set of tritrees of a graph G is denoted by 9=Y(G). Corollary 3.1. Let G be a trigraph and C a longest cycle of G. 'Then

( E ( C )(I E(T)I = 2 for all T E 9

Proof. A T-star cycle can be extended to a longer cycle, by including the centre of the star. 0

74

J.A. Bondy

Corollary 3.2. Let G be a trigraph on n vertices and let % be a CDC of G . Then

I%’}

3n

- 1,

with equality if and only if

IE(C) flE(T)I = 2 for all C E % and all T E Y. It follows from Corollary 3.2 that if %’ is an SCDC of a trigraph on n vertices, then l % l = n - 1. We conjecture that trigraphs are the only simple graphs with this property. Conjecture 3.1. Let G be a simple 2-edge-connected graph on n vertices. If

\%‘l’.n - 1 for every CDC

(e

of G,

then G is a trigraph. While it has not been proved that every 2-edge-connected trigraph admits an SCDC, Seyf€arth [9] has proved that every such graph admits a special type of CDC .

Debxition 3.3. A k-cycle double cover (k-CDC) of a graph G is a set 3 = {&, &, . . . , Z,} of k even subgraphs of G such that each edge of G lies in exactly two members of 3. Remark 3.1. Celmins [4] and Preissmann [8] have conjectured that every 2-edge-connected graph admits a 5-CDC. Tarsi [ll] has proved that every Z-edge-connected graph with a Hamilton path admits a 6-CDC; a simpler proof of this fact can be found in Goddyn [5].

Proposition 3.2 (K. Seyffarth). Let G be a 2-edge-connected trigraph. Then G admits a 3-CDC. Proof. The proof is by induction on IV(G)l, the proposition being evident for IV(C)l= 3. If G is a nontrivial 1-sum of trigraphs GI and G2, then, by the induction hypothesis, Giadmits a 3-CDC %‘ = { Z ; , Zi,, zJ}, i = 1, 2.

Let

zj = zi’u z;, Then = {ZI, z

is a 3-CDC of G.

2 9

j = 1, 2, 3.

Trigraphs

15

If G is a nontrivial 2-sum of trigraphs G1 and G,, then, by the induction hypothesis, Giadmits a 3-CDC

9' = {Zi, Z,: Z : } , i = 1,2. If E(Gl) f E(G,) l = { x y } , we may suppose that XY

E ~ ( 2 :n ) ~(2:)n ~ ( 2 % n E(z$). )

Let Zl = 2: U Z t ,

& =2: A Zf, Z,= Z: U Z$.

Then

3 = { 2 1 9 &, 5 ) is a 3-CDC of G. By Theorem 2.1, it remains to consider the case where G is a star trigraph on at least four vertices with no 2-vertex cut. Let T be a spanning star of G, with centre w. Then Go= G - w is a 2-connected subgraph of G. Let T,, be a spanning tree of Go and uv E E(Go)\E(T,,). Then

T,=T,+vw and T , = T + u v - v w are edge-disjoint spanning trees of G. By a result of Jaeger [6], G admits a 3-CDC. 0

4. The T-graph of a trigraph

Definition 4.1. Let G be a trigraph and T a tritree of G. The T-graph G Tof G is defined by V ( G T )= E ( T ) and E ( G T )= E(G)\E(T), vertices u and v of G T being incident with edge e of G T if and only if { u , e, v} induces a triangle in G.

Proposition 4.1. 1. Let G be a star trigraph and T a tritree of G. Then G~=G-v,

where v is the centre of T. 2. Let G be a nontrivial 1-sum of trigraphs Gl and G,, let T be a tritree of G, and let = T n Gi, i = 1, 2. Then G Tis the 0-sum (that is, disjoint union) of G,'l and GP. 3. Let G be a nontnvial2-sum of trigraphs G1 and G,, let T be a tritree of G, and let = T n Gi, i = 1, 2. Then G T is the 1-sum of G,'l and G?.

z

76

J.A. Bondy

Proposition 4.2. Let G be a trigraph and T a tritree of G. Denote by %;(G,T) the set of cycles C of G such that IE(C) n E(T)I = i, i = 0, 2, by % ( G r )the set of cycles of G', and by P(GT) the set of paths of G'. Then there exist bijections

40:I;e,(G,

v-+% ( G T )

and Cp2: %*(G,TI+ 9 ( G T ) ,

where, for C E 'Go(G,T ) , $ J ~ ( Cis) the cycle of G T with edge set E ( C ) and, for C E Z2(G,T), &(C) is the path of G T with endvertices E ( C ) r l E ( T ) and edge set E(C)\E( T ) .

Proof. We proceed by induction on n = IV(G)(.If G is a star trigraph, then C r = C - v , by Proposition 4.1. Each T-star cycle of G is a cycle of G - v , and conversely. Each other cycle C of G gives rise to a path C - v of G - v , and conversely. Thus the bijections Go and G2 are evident in this case. Otherwise, by Theorem 2.1, G is a nontrivial 1-sum or 2-sum of trigraphs GI and G2. By the induction hypothesis, there are bijections

$h: To(G,, T)+ T(G?) and

44: %(G,, IT;)-+

S(G?).

In the case that G is a nontrivial 1-sum, we may set I$,

=

& u $;,

j = 0, 2.

If G is a nontrivial 2-sum, the union @; U & must be extended to include those cycles in S ( G , T) which are contained in neither G1 nor G2. Each such cycle C is the symmetric difference Cl A C2 of a cycle C1E Y2(G1, Tl) and a cycle C2E %2(G2,T2), and &(C) is defined to be the concatenation of the paths @:(C,) and &(C2). 0

5. Perfect path double covers and trigraphs

Definition 5.1. A path double cover (PDC) of a graph G is a collection P of paths of C such that each edge of G belongs to exactly two paths of 9.A small path double cover (SPDC) of a graph G is a PDC 9 of G such that 1.91=sn. A perfect path double cover (PPDC) of a graph G is a PDC 9 of G such that each vertex of G is an end of exactly two paths of 9;a path of length zero is considered to have two (identical) ends. Conjecture PPDC. The following conjecture is studied in [l]. Every simple graph admits a PPDC.

Trigraphs

I1

Remark 5.1. Conjecture PPDC, the Perfect Path Double Cover Conjecture, is easily verified for complete graphs, complete bipartite graphs, hypercubes, and trees-more generally, a graph admits a PPDC if each of its blocks admits a PPDC. It holds, also, for line graphs of simple graphs, by a theorem of Seyffarth [9], and for simple graphs in which every vertex is of odd degree, by a theorem of LovAsz [7]. Proposition 5.1. Let G be a 2-edge-connected trigraph, and let T be a tritree of G. Then G admits an SCDC if and only if its T-graph G T admits a PPDC. Proof. Suppose, first, that G admits an SCDC V. By Corollary 3.2, IE(C) f l E(T)I = 2 for all C E %. Let

9= 92(Ve), where G2 is as defined in Proposition 4.2. Then 9 is a collection of paths of G T . Moreover, each edge of GT belongs to exactly two paths of 9 because each edge of E(G)\E(T) belongs to exactly two cycles of V,and each vertex of G T is an end of exactly two paths of 9 because each edge of E( T ) belongs to exactly two cycles of V. Therefore 9 is a PPDC of G T . The converse holds because @2 is a bijection. 0

Proposition 5.2. Let H be a simple graph with no isolated vertices, and let G be the graph obtained from H by adjoining a new vertex v and joining v to every vertex of H . Then H admits a PPDC if and only if the trigraph G admits an SCDC. Proof. The spanning star of G centred at v is a tritree T of G. By Proposition 4.1, GT = G - = H . The conclusion now follows from Proposition 5.1. 0

Theorem 5.1. Conjecture PPDC, and Conjecture SCDC for trigraphs, are equivalent. Proof. This follows directly from Propositions 5.1 and 5.2. 0

Corollary 5.1. Conjecture SCDC holds for squares of trees. Proof. Let T be a tree, and let G = T2. Then G is a trigraph and T is a tritree of G. Moreover, G T is precisely the line graph of T. Thus each block of GT is complete. Since complete graphs admit PPDCs, GT itself admits a PPDC, by Remark 5.1. Proposition 5.1 now implies that G admits an SCDC. 0

J.A. B o d y

I8

6. Weighted graphs and trigrapbs

In this section, we show how trigraphs arise naturally in the study of cycles in weighted graphs.

Definition 6.1. A weighted graph is one in which each edge e is assigned a nonnegative number w ( e ) , called the weight of e . Let G be a weighted graph. The weight of a subgraph H of G is defined by

w(H)=

{ w ( e ) : eE E ( H ) } .

An optimal cycle of G is a cycle of maximum weight. Suppose that Conjecture SCDC holds. Let G be a simple 2-edge-connected weighted graph on n vertices, and let % be an SCDC of G. Then

2 { w ( C ) : C E%} = 2 w ( G ) and so the average weight of the cycles in % is

L' { w ( C ) : CE %} -->2w(G) I %I

[%I

2w(G) /

n-1

'

Conjecture SCDC thus implies that an optimal cycle of G has weight at least 2w(G)/(n - 1). This implication of Conjecture SCDC has, indeed, been verified

[31. Theorem 6.1. Let G be a simple 2-edge-connected weighted graph on n vertices. Then G contains a cycle of weight at least 2 w ( G ) / ( n - 1). The extremal graphs for Theorem 6.1 have been completely determined [3], and are of interest because they lend support to Conjecture 3.1. In order to describe them, we need two more definitions.

Definition 6.2. A simple 2-edge-connected weighted graph G on n vertices is cycle-extremal if its optimal cycles are of weight precisely 2w(G)/(n - I). Definition 6.3. Let G be a weighted graph, and let X be a collection of subgraphs of G. If there is an assignment of positive real numbers aH to the members H of X such that, for every e E E ( G ) ,

we say that G is a weighted union of the members of X .

Theorem 6.2. Let G be a simple 2-edge-connected cycle-extremal weighted graph. Then either w ( G ) = 0 or G is a weighted union of tritrees.

Trigraphs

79

References [l] J.A. Bondy, Perfect path double covers of graphs, preprint. [2] J.A. Bondy, Small cycle double covers of graphs, in G. Hahn and G. Sabidussi (Eds) Cycles and Rays (Reidel, 1989, to appear). [3] J.A. Bondy and G. Fan, Cycles in weighted graphs, preprint. [4] U. Celmins, On Cubic Graphs That Do Not Have An Edge 3-Colouring, Ph.D. Thesis, University of Waterloo (1984). [5] L.A. Goddyn, Cycle double covers of graphs with Hamilton paths, J. Combinat. Theory, Ser B., to appear. [6] F. Jaeger, Flows and generalized coloring theorems in graphs, J. Combinat. Theory, Ser B. 26 (1979) 205-216. [7] L. Lovbz, On covering of graphs, in Theory of Graphs, P. Erdos and G.O.H. Katona, eds. (Academic Press, New York, 1968) 231-236. [8] M. Preissmann, Sur les Colorations des Ar8tes des Graphes Cubiques, Thkse de Doctorat de 3bme cycle. Universite de Grenoble (1981). [9] K. SeyEarth, personal communication. [lo] P.D. Seymour, Sums of circuits, in Graph Theory and Related Topics, J.A. Bondy and U.S.R. Murty, eds. (Academic Press, New York, 1979) 341-355. [ll] M. Tarsi, Semi-duality and the cycle double cover conjecture, J. Combinat. Theory, Ser B. 41 (1986) 332-340.



81

ON CLUSTERING PROBLEMS WITH CONNECTED OPTIMA IN EUCLIDEAN SPACES* Endre BOROSt and Peter L. HAMMER RUTCOR -Rutgers Centerfor Operations Research, Rutgers Universiv, New Brunswick, N .J., U.S.A. Let X be a finite subset of a Euclidean space, and p be a real function defined on the pairs of points of X , expressing the “unsimilarity” of points. The problem is to find a partition P,, . . . ,Pp of X into p groups which maximizes the sum of unsimilarities of all those pairs of points which do not belong to the same group. It is shown here that for some typical unsimilarities p, there exists an optimal partition such that the intersection of 5 with the convex hull of pi is empty for all i <j . In particular, it is shown that if X is on a sphere then the convex hulls of the groups of an optimal partition are pairwise disjoint.

1. Introduction A typical problem in cluster analysis is the following: let X be a finite set of points in some Euclidean space, X c Rd,and let p : X x X + R be a function defined on pairs of points, expressing their unsimilarity. (Sometimes p is defined only on a subset of X X X.) The problem is to find a partition X = S, U S, U - - u S, of the base set into p groups (p is fixed), such that some objective function h(S,, . . . , S,) is minimized. The solution of such a problem is usually very difficult, however is some cases the optimal partition has special properties which make the optimization problem easier. One such case, studied in [l],consisted in the minimization of

where g was a quasi-concave function, implying that conv Sj n conv Sj = 0 for i # j at the optimum. (In this case the unsimilarity p has no influence on the problem.) This paper deals with another special case. Let n, d, p be fixed non-negative integers, X c Rdbe a given n-elements set of points, p(x, y) be a given non-negative real function on pairs of points of X. For * The authors gratefully acknowledge the partial support of NSF (Grant ECS 85-03212) and AFOSR (Grant 0271). t On leave from the Computer and Automation Institute of the Hungarian Academy of Sciences, 1502 Budapest, P.O.B. 63, Hungary. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

E. Boros, P.L. Hammer

82

any partition P = (Pl, . . . , Pp) of X into p groups let Isi=sp x.ycP,

and let us consider the problem of minimizing h(P) over all p-partitions of X. Introducing

we get an equivalent problem, consisting of the maximization of f(P)over all p-partitions of X. Let 6 denote the collection of optimal partitions, i.e. P E 0 iff f(P)is maximal. We shall say that a partition P = ( P I ,. . . , Pp) is nested if Vj#ie

n (conv P , ) = 0 or 8 n (conv 4 ) = B

(1.3)

or equivalently, if the groups can be relabelled such that Vi <j : p/

n (conv P,) = 0.

(1.4)

Such a labelled partition we shall call connected. In this paper we show that for some unsimilarity functions the optimum is always nested. We start with a more difficult case, when p is defined as the Euclidean distance, p ( x , y ) = d m . In this case in Section 2 we prove the following.

Theorem 1.1. If X c R and p ( x , y ) = Ix - y ( , then every optimal partition P E 0 is nested.

We shall also show that this theorem does not hold in higher dimensions. We present an example in the plane for p = 2, where the unique optimal partition is not nested. The case when p is the square of the Euclidean distance will be shown to be much easier. In Section 3 we shall show that using this distance the optimal partitions are nested for any dimension.

Theorem 1.2. If X c Rd and the unsimilarity of points x , y p ( x , y ) = ( x - y)', then any P E 6 is nested.

EX

is defined by

More precisely we show that if P E 6 is an optimal partition, P = ( P , , . . . , f,), then each group can be separated from another one by a sphere. An immediate consequence of this is the following

Theorem 1.3. Let X c Sd-' be a finite set of points on the sphere in Rd,and let p ( x , y ) = (x - y)'. Then for any optimal partition P = ( P I ,. . . ,P,) the convex hulls conv P I , . . . , conv P, are painvise disjoint sets.

Clustering problems with connected optima

83

Before presenting the proofs we mention that if P = (Pl, . . . , P,) is an optimal partition of a given set X,then obviously P' = (s,4) is also an optimal partition (into two groups) of X ' = S U q . This and the definition of nestedness implies that it is enough to consider the case p = 2 in the above theorems. 2. Partition of points on the line

To prove Theorem 1.1 we show it in case of p = 2, since the general case follows from this immediately. We need some additional notations and lemmas. A multiset X of real numbers means a finite set X = {xl, x 2 , . . . ,x,} of reals, with non-negative real multiplicities, m&) assigned to the elements x E X. We say that x belongs to X, or x E X if m&) > 0. Another multiset S is called a subset of X,or S E X, if from x E S it follows that x E X and ms(x) s mx(x). We say that X is the union of S and T, X = S U T,if mx(x) = ms(x) + mT(x)for any x E S or x E T. If S c X, then its complement with respect to X, = X\S, is defined on the same points as X, and m&) = mx(x) - m&) for x E X ,where ms(x) = 0 for x @ S, by definition. The cardinality of a multiset S, card S, is the number of points in S having positive multiplicities. The size of a multiset S, 181, is the sum of the multiplicities ms(x) of the elements of it, i.e. IS( = C x s S m S ( x ) . Let convS denote the interior of the convex hull of points of S with positive multiplicities, let c, be the weight center of S, i.e. c, = (l/lSl) Cxesms(x)x,and for given multisets E and F let

s

c

f (E, F ) = x s E . y e F m E ( x ) m F ( y ) Ix

-Yl*

(2.1)

Finally let f (T) =f (T, F). First, we mention that if the sets 0 and 0 form an optimal partition, then they are not necessarily convex separable, as the following example shows.

Example. Let x1 = -10, x 2 = 0 and x 3 = 10, X = {xl, x 2 , x 3 } with mx(xi)= 1 for i = 1, 3, and mx(x2)= 3. Then clearly 6 = {{q, x 3 } , { x 2 } } , but conv{x,, x 3 } n {xz} f 8.

The first lemma will help us to reduce the problem.

Lemma 2.1. Let E, F be disjoint subsets of reals. Then = IEI IF1 JcE- cFJif conv E n mnv F = 0, 3 IEI IF1 IcE - cFI if conv E n mnv F 2 0 .

- - -

Proof. If card F = 1, e.g. F = { x i } , then the lemma is obvious, since cF = x i . It is also clear by the definitions that if E n E' = 8, then f(E,F) +f(E',F ) = f ( E U E', F). Thus, applying the lemma at first for the elements of F one by one, and then for the pair { c E } ,F, we get the lemma. 0


84

Next we show a very special case of Theorem 1.1. Lemma 2.2. Let X = { x l , x2, x3, x4, x5, x g } be a given multket of reals, x1 <x2 < x, < x4 <x5, x4 < x6, and let A , B , C , D , E , F be the partition of X , given by A = {xl}, B = {x2}, C = {x3}, D = {x4}, E = { x 5 } and F = {xg}, where lAl, IBI, ICI and ID1 are positive, IEI 2 0 and IF1 2 0. Then T = A U C U E cannot

maximize f.

Proof. We shall show that f(A U C U E) is always less than the value off for one of the following seven sets; A U E, C U E, A U B U E, A U D U E , B U C U E, A U B U C U E and A U C U D U E. Introducing 5, = x 2 - xl, E2 = x 3 - x2, E3 = x4 - x3, E4 = x5 - x4, E5 = x6 - x4, and a = IAI, b = IBI, c = lCl, d = 1 0 1 ,e = IEI and f = IFI, we have the following table for the values off on pairs of sets from { A , B, C, D,E , F } . J

I

A

B

C

D

E


85

To prove the lemma, let us suppose indirectly that f(A U C U E) is the largest of the above values off. Using the positivity of El, E2, E3, E4, E5 and a, b, c, d we show that the inequalities 0s f ( A UCUE ) - f ( A U E ) 0 s f ( A U C U E) -f(C U E ) 0 s f ( A U C U E) - f ( A U B U E )

O S f ( A U C U E) - f ( A U D UE)

O c f ( A U C U E ) -f(B U CUE) O s f ( A U C U E ) -f(A U B U C U E ) O S f ( AU C U E ) - f ( A U C U D UE) cannot hold at the same time. Using (2.2) and substituting the values from the table above, after simplifications, we get the following system of inequalities: (i) 0 s --&a - &(a - b ) + E3(d - e +f) - E4e + E5f, (ii) 0 s El@ - c + d - e +f) + E2(-c + d - e + f ) + E3(d - e + f ) - E4e + g5f, (iii) 0 s - c) - E2[ac+ b(d - e +f)]- E3(b- c)(d - e + f ) + E4@ - c)e - E5(b - elf, (iv) 0 s -t1a(c - d ) - &(a - b)(c - d ) + &[(a - b)d - c(e -f)] - E4(c + E5(c - dlf, (2.3) (v) 0 s -E1a(c - d + e -f)) - E2(a - b)(c - d + e -f) + E3(a - b)(d - e +f) - E4(a - b)e + - blf, (vi) 0 s L a + EAc - d + e -f) - E3(d - e +f) + E4e - Esf, (vii) 0 6 E1a + E2(a - b ) + E3(a - b + c) + E4e - f5f. For the sake of simplicity, a linear combination with the non-negative real coefficients LY and /3 of the above inequalities, say of (i) and (ii) will simply be denoted by a(i)+ /3(ii). Then, (i) + (vi) implies &(-a + b + c - d e - f ) 2 0, and therefore

+

a

+ d +f s b + c + e.

(2.4)

We distinguish between two cases, corresponding to the equality and strict inequality in (2.4), and in both cases we shall reach a contradiction. Cuse 1 a

+ d +f < b + c + e.

(2.5)

+

Supposing a 3 b, by (a - b)(vi) + (v), we get E1a(a- b - c d - e + f ) 3 0, in contradiction with (2.5); therefore b > a . But then, from (b - a)(ii) + (v) it follows that ijlb(-a b - c + d - e +f ) 2 0, and therefore

+

a

+ c + e s b + d +f.

(2-6)


86

+

Supposing d 3 c, by (d - c)(i) + (iv), it follows that E,d(a - b - c d - e + f ) 5 0, i.e. a + d + f a b c + e in contradiction with (2.5). Therefore c > d is implied, and thus by (iv) + (c - d)(vii), it follows that E3c(a - b + c - d e + f ) 3 0, i.e.

+

a

+ c +f 2 b + d + e.

(2-7)

Now, supposing b 3 c, (b - c)(i) + (iii) implies that E26(-a + b - c - d + e -f) S O , i.e. b + e 3 a + d +f + c in contradiction to (2.7) and the positivity of d. Supposing c > 6, (c - b)(vi) + (iii) implies that g2c(-a - b + c - d + e - f ) 2 0, i.e., c + e 3 a + b + d +f, in contradiction to (2.6) and the positivity of a. Case 2.

+ d +f = b + c + e. (2.8) Supposing b c, (b - c)(i) + (iii) implies that E,b(-a + b - c - d + e - f ) 3 0, 1.e. 6 + e > a + d +f + c in contradiction to (2.8) and the positivity of c. Supposing c > 6, (c - b)(vi) + (iii) implies that lj2c(-a - b + c - d + e -f) 2 0, i.e. c + e 2 a + b + d +f, in contradiction to (2.8) and the positivity of b. a

We reached a contradiction in all cases, thus (2.3) cannot be consistent, proving the lemma. 0

P d of Theorem 1.1. Suppose that there is a partition (T, T ) E 0 which is not nested, i.e.for which convTnT#0

and c o n v T f l T T 0 .

(2.9)

We may suppose that min T S min F. Now searching the points of X from left to right, T and T can clearly be decomposed into painvise disjoint subsets S,, . . . , S,, such that k

us;=x,

r=l

conv S, nX\Si = 0, and

(2.10)

T if i is odd, Si c T i f i iseven. Clearly, (2.9) forces that k 3 4. Lemma 2.1 can be applied, by (2.10), and a set S; can be substituted by a point cs, with multiplicity ISJ, i = 1, . . . ,k. Then defining xi =csi for i = 1, 2, 3, 4, x5 = c ~ \ ( ~and , ~x6~=~c *)\ ( % ~ ~ ,Lemma ), 2.2 applies for the multiset {xl, . . . , x 6 } , and implies that at least one of the sets T\S3, T\S1, (TU&)\S,, (TUS,)\S,, (TU&)\S,, T U&, or T US., is better than T, in contradiction with the optimality of T. This contradiction shows that any partition {TI T } E 0 is nested. 0


87

Next we show that the analogous results does not hold in higher dimensions. Consider the rnultiset of points on the plane given by their affine coordinates Point

Coordinates

Multiplicity 100 100 1 1 1 100 100 1 1 1

Now if X = {xl, . . . ,xl0}, then an easy calculation shows that the unique partition into two groups of X is P = ({xl,. . . ,x 5 } , {x6, . . . ,x l 0 } ) . However, here x8 E conv{x3, x4, n5} and x5 E conv(x8, x g , xl0} showing that this partition is not nested.

3. Partition of points of R~ To prove Theorem 1.2 we shall examine only the case p = 2, since this implies the general one. Let us suppose that X c Rd and p(x, y) = ( x -y)’ for any x , y E Rd.In this case for any (A, B) E 0 we have

2

f(A)=

(a -b)’= IAI

a EA,b E B

c b’+ 1B1 2 a’-

beB

aEA

2

(a, b),

(3.1)

aeA , 6 6 8

where (a, b) denotes the scalar product of the vectors a and b. For any x E A we have f(A\{x}) s f ( A )

(3.2)

because of the optimality of A. But by easy calculations we have

(3.3)

Thus, by (3.2) for every x E A we have 2(x,

(2 a - c b ) ) 3 ( 2 a’- 2 b2) + (IAI - IBl)x2. aeA

be8

aoA

boB

(3.4)


88

Similarly for any y E B we get

aeA

brB

oeA

beB

(3.5)

Hence we have proved somewhat more than Theorem 1.2.

Theorem 3.1. Zf ( A , B ) E 0 and IAI # IBI then A and B are separated by the sphere

If JAl= /BI then they are separated by the hyperplane 2(x.

(2 a - 2 b ) ) = (a'-z oeA

bcB

aeA

c b').

beB

An immediate consequence of this theorem is the following

Corollary 3.2. If X is on a sphere, then for any (0, 0)E 0 we have conv o n conv 0 = 0, i.e. the sets of an optimal partition of a sphere are convex separable.

Corollary 3.2 clearly implies Theorem 1.3.

References [ 11 A. Hoffman, Communication of the XII. Internation: Symposium on lathematical Programming (Boston, August 5-9, 1985). 121 U. Rothblum, Clustering is optimal or optimal partitions having disjoint conic or convex hulls, colloquium talk (RUTCOR, Rutgers University, February 1988). 131 A.C. Williams, The separation problem: Linearization and the equivalence of all degree 2 binary optimization problems, RUTCOR Research Report, RRR #21-88 (April 1988).


89

SOME SEQUENCES OF INTEGERS Peter J. CAMERON School of Mathematical Sciences, Queen Mary College, Mile End Road, London E l 4NS, U . K . Combinatorialists are interested in sequences of integers which count things. We often find that the same sequence counts two families of things with no obvious connection, or that a simple translation connects the answers to two counting problems. In this way, unexpected connections have come to light.

1. The Handbook What I want to describe is a kind of experimental mathematics, ideal for doing at times when honest thinking is not going well. The requirements are a small computer (pencil and paper suffice, though the calculations are tedious), and Neil Sloane’s “A Handbook of Integer Sequences” [15]. This book, a kind of hitch-hikers’ guide to the universe NN,consists mainly of a list of 2372 sequences of nonnegative integers, arranged lexicographically, with an index, references, and notes for users. The main criterion for inclusion of a sequence is that somebody must have found it sufficiently interesting to record it in the literature. The Handbook can be used, then, like a book of tables, using the index to locate a sequence. A more exciting possibility is this. Suppose you find youself in possession of an “unknown” sequence. (This is not an uncommon event; a glance through the Handbook confirms that sequences occur in all provinces of mathematics, and well beyond its frontiers.) If you can locate your sequence in the Handbook, you have both a problem (of showing that your sequence really is the one listed) and a source of information (the references to the sequence). I know of several cases where new results have been discovered this way. I propose a third way of using the Handbook. There are some naturallyoccurring transformations of sequences, two of which I will consider in detail. Finding instances where a known sequence is transformed into another can give rise to new mathematical insights in the way described above. Also any sequence which is transfirmed into a closely-related one gains significance independent of the objects it counts. Sloane adops the convection that all sequences commence 1, n, when n > 1. To ensure this, he deletes “superfluous” leading ones and zeros, and inserts a 1 if necessary. Some valuable information is lost in this way, namely the “natural” starting point of the sequence. But, on the positive side, the weakness of the 0012-365X/89/$3.500 1989, Elsevier Science Publishers B.V. (North-Holland)

P.1. Cameron

90

convention draws our attention to the operation of “shifting” a sequence, which will prove fruitful. I will refer to known sequences by their number in the Handbook.

2. The background

I will now describe the area from which the sequences of most interest to me arise. There are three main sources. (i) In combinatorics, sequences come from counting problems: if C is a class of objects of some kind, let x, be the number of objects of cardinality n in C , up to some well-defined notion of isomorphism. I will refer to ( x , , x 2 , . . .) as the sequence enumerating C. be a graded algebra over (ii) Graded algebras. Let A = F 1 63 V, 63 2@ F - that is, each V; is a vector space over F, and there is a multiplication on A satisfying V, . V, 5 V;+,. If each V; is finite-dimensional, we have a sequence ( x l , x 2 , . . .), where x, = dim V,. Its generating function 1 + Cnalxnrn is the PoincarC series of A. Another example: it may happen that A is a polynomial ring generated by a set of homogeneous elements (i.e. each lying in V, for some n ) ; then there is a sequence enumerating generators in V,. (iii) Permutation groups. Let G be a permutation group on an infinite set X. Then G acts on the set X, of n-element subsets of X in an obvious way. Let fn(G) be the number of G-orbits in X,. An interesting class consists of those permutation groups C for whichf,(G) is finite for all n. (This class has been of interest to model theorists, in view of a celebrated theorem of Svenonius [ 161: the countable first-order structure M is w-categorical (i.e. determined up to isomorphism by first-order axioms and the assumption of countability) if and only if the permutation group Aut(M) has finitely many orbits on n-subsets of M for all n.) For such groups, we have a sequence (fi, f2, . . .); we say it is realised by G. It is an interesting problem to characterise sequences realised by groups. Any such sequence must be non-decreasing, and there are gaps in the spectrum of growth rates (Macpherson [13], [14]). Many examples appear in this paper. In fact, (iii) is a special case of both (i) and (ii). For (i), take a permutation group C on X ;it is easy to construct a relational structure M on X such that G s Aut(M), and any isomorphism between finite substructures of M is induced by an automorphism of M - such structures are called homogeneous - and even by an element of G. Thus fn(G)is the number of isomorphism types of finite substructures of M. Furthermore, FraissC [9] gave a necessary and sufficient condition for a class of finite structures to be the finite substructures of a homogeneous structure. For (ii), let V,(G) be the space of G-invariant functions from X, to Q, as

-

Sequences of integers

91

Q-vector space. Set

A ( G ) = Q * 1 @ Vi(G) @

*,

and define a multiplication as follows: for f E V;:(G), g E F(G), let fg E y + j ( G ) map the (i +j)-set K to

It is easily checked that A ( G ) is commutative and associative and that dim K(G) =fn(G) (if this number is finite).

3. The operators I will be considering the operators S and A defined on the set of sequences of non-negative integers as follows: for x = (x,), set Sx = (y,) and Ax = (zn),where 1+

2 y,tn = n (1 - t ?I31

y

n

nal

and 1+

c z,t”= (1 -

,==l

flP1

xntn

1-l

.

These definitions will look either obvious or artificial, depending on your background. Here is the motivation. (i) Suppose that x enumerates a class C. Then Sx and Ax each enumerate the class of disjoint unions of members of C where, for Sx, the order of the “component” members of C is unimportant, while for A it is significant. (Think of the process of building a structure, e.g. a graph, from its connected components.) (ii) Suppose that the graded algebra A is a polynomial ring generated by homogeneous elements, and let x enumerate its generators. Then Sx gives the dimensions of the homogeneous components. The operator A plays a similar role for free associative (non-commutative) algebras. (iii) We need the concept of wreath product of permutation groups. Let G and H be permuation groups on X and Y respectively. The set on which the wreath product acts is X X Y,regarded as a covering of Y with fibres isomorphic to X (that is, a disjoint union of copies of X indexed by Y). Now the wreath product G Wr H is generated by two kinds of permutations: those which permute the fibres according to the action of a member of H on Y, and those which fix every fibre and induce independently-chosen elements of G on the fibres. Let S denote the symmetric group on an infinite set, and A the group of order-preserving permutations of the rational (or real) numbers. Now, if x is realised by G, then Sx and Ax are realised by C Wr S and G Wr A respectively.

92

P.J. Cameron

Y Fig. 1.

The last remark can be translated into either of the other interpretations. Let G Wr H act on X X Y as above. An n-subset of X X Y is partitioned by the fibres it meets; the “connected” subsets are those contained in a single fibre, and they are enumerated by (fn(G)). The structure of the set of connected components is governed by H. Also, regardless of the algebraic structure of A ( G ) , A(G Wr S ) is always a polynomial ring, and ( f n ( G ) )enumerates its generators. (However, there is no known analogue for G Wr A . ) I remark in passing that some interesting combinatorics, involving Stirling numbers, comes from considering S Wr G instead of G Wr S; see Cameron and Taylor [4]. The remainder of this paper is a sequence of commented examples. 4. The random graph

Erdos and R6nyi [7] showed that there is a graph R on a countably infinite set of vertices, which has the following remarkable property: If a countable graph is chosen at random (by considering all pairs of vertices in turn, and for each pair, tossing a coin-fair or biased- to decide whether to join those vertices with an edge), the resulting graph is, with probability 1, isomorphic to R. This property clearly characterises R up to isomorphism. Naturally, R is called “the random graph”. R has many further interesting properties (see Cameron [2]). Two of the most basic are these: Every finite (or countable) graph is embeddable in R, and any isomorphism between finite subgraphs of R extends to an automorphism of R. (In our earlier terminology, R is homogeneous.) Thus the orbits of the group G = Aut(R) on X,,correspond to n-vertex graphs up to isomorphism, and the sequence realised by G enumerates graphs by number of vertices. This is Sloane #479: 1, 2, 4, 11, 34, 156, 1044, 12346,. . . (Here, as in most other cases, Sloane gives many more terms, enough to fill two lines of text.) The considerations of the last section show that, if X is the sequence enumerating connected graphs by vertices-i.e. Sloane #649: (I), 1, 2, 6, 21, 112, 853, 11117,. . . -then Sx enumerates all graphs, i.e. Sx =f(G). This might


93

lead us to guess that the algebra A(G) is a polynomial ring whose generators are enumerated by x. This is true; indeed, the generators are identified with connected graphs under the natural identification of basis vectors with all graphs. This phenomenon holds much more generally. What is needed is a “good” decomposition of arbitrary objects into connected ones; sufficient conditions can be given. Examples include the pairs f ( G ) and f(G Wr S) described earlier, and directed graphs, partially ordered sets, finite topologies, etc. (Sloane ##1133 and 648, 545 and 985, 588 and 1152).

5. The all-1 sequence

Let 1 denote the all-1 sequence. (This is not in the Handbook- Sloane’s convention excludes sequences with no term greater than 1- but it is obviously important. It enumerates sets and totally ordered sets; it gives the dimensions of homogeneous components of a polynomial ring in one variable; and it is realised by both S and A, among others.) What is S l ? Clearly it enumerates partitions of a set with no structure and with no distinguished order of the parts; in effect, partitions of an integer. Thus, S1 is the partition function p (Sloane #244). The definition of S gives the familiar generating function 1+ C,,,p(n)t“ = (1 - W . If we apply S again and refere to the Handbook, we find Sloane #1019, with the name “functional determinants” and a reference to Cayley [5]. It turns out that Cayley was counting the types of projective transformation over an algebraically closed field, in other words, the Jordan forms of n X n matrices. There are infinitely many Jordan forms; but, if we neglect the actual eigenvalues and note only whether the eigenvalues in two blocks are equal or not, the number is finite. (For example, all generic matrices, i.e. diagonal matrices with distinct eigenvalues, are identified.) We have a partition of n corresponding to the eigenvalues, and a partition of each part corresponding to the Jordan blocks. The identification with S21 is clear. We turn now to A l . As before, this sequence enumerates partitions of n, but

n,,,

I

1

Fig. 2.

J

P.J. Cameron

94

+

now the parts are ordered. For example, (Al), = 4, since 3 = 2 1 = 1 + 2 = 1 1 1. Experimentally we find that A 1 = (1, 2, 4, 8, 16, . . .) (Sloane #432: powers of 2), proving this is a pleasant exercise. More generally, A'1 is the sequence of powers of r + 1. (Sloane lists several of these: ##1129, 1428, 1630, 1765, 1874, 1937, 1992, 2054, 2084, 2107, 2120, 2164, 2182, 2192, 2198). Further experimentation reveals that S - ' A l is Sloane #287: (1,) 1, 2, 3, 6, 9, 18, 30, . . . , called "irreducible polynomials, or necklaces". The first description suggests the proof. There are 2"-' polynomials of degree n over GF(2) with nonzero constant term; each has a unique factorisation into irreducibles, with order unimportant; and all irreducibles f ( t ) except t occur. We notice, incidentally, that is x enumerates all irreducibles, i.e. x = (2, 1, 2, 3, 6, 9, . . .) (Sloane #46), then Sx enumerates all polynomials, i.e. (Sx), = 2", which can be regarded as either a shift or a double of #432. There is a problem here. Sloane #432 is realised by (at least) two quite different groups G. First, we can take C = H WrA, where H realises the sequence 1. (Such groups H are called highly homogeneous. Both S and A are examples.) Second, we can take a partition of CD into two dense subsets, and let g be the subgroup of A which preserve the partition (i.e. fix or interchange the subsets). For either group G, if it holds that A ( G ) is a polynomial ring, then Sloane #287 enumerates the generators. But is A(C) a polynomial ring? Similarly, the subgroup G' of the second G fixing both subsets realises the double of #432, and the same problem arises with #46.

+ +

5. "he natural numbers Let x be the sequence (1, 2, 3, . . .) of natural numbers (Sloane #173). Empirically, Sx is Sloane #1016: planar partitions, and Ax is Sloane #1101: bisection of (i.e. alternate terms of) the Fibonacci sequence. I leave the proofs as exercises. I do not know whether either #lo16 or #1101 is realised by a group. (The Fibonacci numbers themselves, Sloane #256, are realised; for they form the sequence A(1, 1, 0, 0, . . .), and so are realised by Z 2 Wr A. The combinatorial interpretation is that F, is the number of ways of writing n as an ordered sum of ones and twos.) #173, as it stands, is not realisable; but if we shift this sequence, obtaining y = (2, 3, 4, 5, . . .), we obtain a realisable sequence. The group in question is G = S X S, acting on the disjoint union of two sets which are orbits of the factors. (The orbit of an n-set is determined by the cardinality r of its intersection with the first G-orbit, and r can be any integer in the range [0, n]). One would expect that Sy and Ay would count objects similar to those for Sx and Ax; but neither sequence is in the Handbook. (For the record, the first few terms of Sy and Ay are 2, 6, 14, 33, 70, 149, 298,. . .


95

and 2, 7,24, 82, 280, 956, 3264,. . .

respectively. 7. Self-generating sequences

I turn now to sequences which are only slightly modified by S or A. No interesting sequence can be wholly unaltered: the only fixed point of A is the zero sequence, and no sequence x with x1 > 0 is fixed by S. Consider the problems: Which sequences x with x1= 1 are (a) shifted one place to the left, or (b) doubled (apart from the first term, which is unaltered) by the action of (i) S, (ii) A ? There are four distinct problems here. We note that each problem has a unique solution. For (Sx),

=f(x1,

x,)

* * * 9

for some function fn; so the solution to (i) (a) is given by the recurrence x1=

1 * xn+l = f n ( x l ,

. . . ,x,).

Also, fn has the form &(XI,

* * * >

+f&,

x,) = x ,

--

*

9

x,-1);

so the solution to (ii) (a) satisfies the recurrence

x1 = 1, x, =fA(xl,

. . . ,x n V l )

for n > 1.

The argument for (b) is the same. Moreover, the solutions are easy to calculate. For example, for (i), use the procedure for evaluating S or A with each output as next input. But can we anticipate the results? The solution x to (i) (a) should count a class G of structures for which the connected structures on n 1 points correspond to all structures on n points. A little thought shows that rooted trees (Sloane #454) fill the bill. For (i) (b), we want a class with a bijective correspondence between connected and disconnected objects on n points for n > 1. The obvious correspondence to use is complementation of graphs - the complement of a disconnected graph is,

+

Fig. 3.

P.1. Cameron

Fig. 4.

after all, connected -and there is a class with the required property. It can be described either as the smallest class containing the l-vertex graph and closed under complementation and disjoint union, or as the class of N-free graphs (graphs containing no path of length 3 as an induced subgraph). The sequence enumerating connected N-free graphs is Sloane #558, and its “double” enumerating all N-free graphs is Sloane ##M. These are described as “Series-reduced planted trees” and “Series-parallel networks” respectively. At first glance , it is not clear what these descriptions have to do with one another or with N-free graphs. Answering these questions and realising the sequences by groups has led to some fruitful research on treelike objects (Cameron [3], Covington [ 6 ] ) . Now consider problem (ii). We would expect the solutions to be ordered analogues of those for (i). Thus, for (ii) (a), we count rooted trees in which the set of branches above the root is ordered, and each branch has the same property. This is just the recursive specification for depth-first search on the tree. In other words, our objects are rooted plane trees. The enumerating sequence is the Catalan numbers (l), 1, 2, 5, 14, 42, 132,. . . (Sloane #577), one of the most celebrated sequences in combinatorial mathematics. (The nth term is (:I:)/n.) The fact that the Catalan numbers form the unique solution to (ii) (a) helps identify them in some of their many guises. For example, consider the number x, of paths in the plane from (0, 0) to (2n, 0) such that (i) each step is from (x, y ) to (x + 1, y 1) or (x 1, y - 1); (ii) apart from the end-points, the path lies strictly above the x-axis. The sequence An enumerates ordered unions of such paths; these are just paths satisfying (i) and (ii) with “strictly above” replaced by “above or on”. But any such path from 0 to 2n yields, by extending its ends down one step and then translating by (1, l), a solution to the strict problem, with n replaced by n 1. So (Ax), = x,+, . Since x 1 = 1, x is the Catalan sequence. The sequence A2x also appears in the Handbook, as #1144 (“Central binomial coefficients”); the nth term is [(?I;)=+(%)]. W h y is this? Consider ordered unions of paths satisfying the “weak” specification above, where alternate components are reflected in the horizontal axis. These account for half of all the possible paths (since the first step is upwards). But there are (%) paths

+

+

+


97

altogether, since we merely have to choose the set of positions where the n upward steps are taken. This can be viewed another way. Consider the set of paths defined by the “strict” specification and their reflections. These are enumerated by the Catalan numbers doubled (Sloane #128). Ordered unions of these give all possible paths, enumerated by the sequence with nth term (2)(Sloane #643). I will return to this oddity in the next section. Note, by the way, that the inverse image under A of the Catalan sequence is Sloane #635. Why? The unique solution to (ii) (b) enumerates complementary pairs of N-free posets. (A poset is N-free if it has no four elements a, b, c, d with a > c, a > d , b > d, and other pairs incomparable - see Fig. 4. Two posets on the same set are complementary if each pair is comparable in precisely one of the posets.) This sequence is Sloane #1163 (“Dissections of a polygon, or parenthisizing a product”). In fact it is also listed (with a small misprint) as #1170 (“Schroder’s second problem”). Once again, explanation proves fruitful. A question of some interest to statisticians (Bailey [l]) is that of enumerating N-free posets. Let x1 =yl = zl= 1 and, for n > 1, let x,, y,, z, be the numbers of connected, disconnected, and arbitrary N-free posets on n points. Then clearly z,=x,+y,

forn>1

and Sx=Ay=z. From this, the sequences can easily be calculated. We have x = ( l , 1, 3, 9, 30, 103, 375,. . .)

y = (1, 1, 2, 6, 18, 64,227,. . .) z = ( l , 2, 5, 15, 48, 167, 602,. . .) Note that the solutions to (i) (b) or (ii) (b), doubled, are lower and upper bounds respectively for z. But I do not have a good asymptotic estimate for z. The unique sequence x with x1 = 2 which is shifted by the application of A turns out to be twice the solution to (ii) (b)! I shall explain why in the next section.

8. Generating functions and functional equations Many readers will know, or will have spotted for themselves, that the operator A lends itself readily to analysis by means of generating functions. Given a sequence x , let

x(t)= 2 x,tn rial

P.J .

98

Cameron

be its generating function. A induces a map on formal power series, which I will denote by a:thus, by definition, = (1 - X)-*, (1 + ffx)

whence aX = X/(l- X).Now an easy induction shows that d X = X/(l- rX). It follows immediately that r d X = a ( r X ) , or, for sequences, r A‘x = A(rx). As Patrick McCarthy pointed out to me, for r = 2 this is an instance of the Feigenbaum-Cvitanovie Eq. [8], albeit for functions on NNrather than Iw . He also remarked that A (in its action on generating functions) is formally identical with the solution f(x) = x / ( 1- x ) of the F-C equation

f(f(x)) = f f ( W , discovered by Hirsch, Nauenberg and Scalapino [lo]. For a survey of this area of mathematics, I refer to McCarthy [12]. The case r = 2 also generalises our observation about Catalan numbers and central binomial coefficients in the last section; it shows that the Catalan numbers were really irrelevant there. But the argument suggests a combinatorial proof of the identity in this case. Let C be a class of objects enumerated by a sequence x. Then 2x enumerates C-objects with an additional distinction into “red” and “blue” objects. Hence A(2r) enumerates ordered unions of C-objects where the points are coloured red and blue so that points in the same component have the same colour. On the other hand, Ax enumerates ordered unions of C-objects, and A2x ordered unions of these, which we may regard as being coloured alternately red and blue; these account for half of all the general coloured ojects, namely, all those starting with a red component. So 2 A 2 x =A(&). I do not know a similar proof of the general identity. Now let M,denote multiplication by r ; let M : denote multiplication of all terms except the first by r; and let T denote the shift one place left. Now, among sequences x with x , = 1, each of the following conditions defines a unique one, and in fact they all define the same sequence (for fixed r 3 1): (i) A‘x = Tx (ii) AM,x = TM,x (iii) AM:x = Tx (The equivalence of (i) and (ii) is immediate from AM, = M A ‘ and TM, = M , T ; that of (ii) and (iii) is proved by a generating function argument.) The unique sequence defined by these conditions for r = 1 is the Catalan numbers (Sloane #577), of course. For r = 2 , conditions (i) and (iii) imply that Ax = M:x - this was the problem (ii) (b) which characterised Sloane St1163. Now condition (ii) for this sequence, i.e. A(&) = T ( 2 x ) , explains the observations right at the end of the last section.


99

9. Exponentiation and convolution McCarthy [ l l , 121 has emphasized the analogy between the FeigenbaumCvitanovie function, whose self-composition is just a re-scaled version of itself, and the exponential function. We have seen that A is a F-C function; by a delightful coincidence, S is an exponential function! Let x $ y denote the pointwise sum of sequences x and y, and x y their convolution, given by 0

n

(x

y)n =

0

XkYn-k, k=O

with the convention no = y o = 1. (Thus x y corresponds to the product of the generating functions 1+ X and 1+ Y of x and y.) Then it is immediate from the definition of S that S(x @ y) = Sx 0 Sy. The operation of convolution ties in naturally with our examples (ii) and (iii). If A and B are graded algebras, the PoincarC series of A 69 B is the product of those of A and B. (And, if A and B are polynomial rings, then so is A 69 B, generated by the disjoint union of the generating sets for A and B, according to the exponential equation for S.) Also, if G and H act on X and Y, respectively, then G x H acts on the disjoint union of X and Y,and 0

(fn(G))o ( f ( H ) >= (fn(G X HI).

A number of sequences in Sloane arise as convolutions of smaller sequences. For example, the kth convolution of 1 has nth term ("t!;;') (Sloane ##173, 1002, 1363, 1578, 1719, 1847, 1911, 1976, 2013, 2046, 2073). As a special case, x 1 is the sequence of partial sums of x (again with the convention xo = 1); this accounts for Sloane ##374, 392, 394, 395, 396, 397, 1007, 1050, 1382, 1398. Other examples include Sloane ##128,525,533, 535, 536,537,1124, 1413, 1600, 1738, 1865. Three more samples must suffice here. (i) S-'l= (1, 0, 0, . . .); and, if x is the sequence of powers of 2 (#432), then 1 o x = 2 y . Thus 0

S-%

= S-lx

a3 (1, 0, 0, . . .),

as we observed in Section 5 . (ii) Let T be the left shift. Then any sequence satisfies x oAx=2Ax;

and, if x1 = 1, then also Tx

0

Ax = TAX.

(These are proved by generating function arguments.)

P.J. Cameron

100

(iii) Let x be the Catalan sequence (#577), starting 1, 2 , 5 , . . . . The familiar convolution property of Catalan numbers shows that x

0

x = Tx.

From this, it follows that T x 0 A x = x 0 2 A x = T A X . (For T x 0 A x = TAXby (ii); the rest follows from the associativity of the convolution x x 0 Ax.) These yield such consequences as 2S-’x = S-’Tx. 0

10. Final remarks

Apart from the obvious comment that many connections remain to be explored, I want to draw attention to other operators in the class including S and A. If G is any permutation group satisfying our condition that fn(G)is finite for all n, then there is an operator G on sequences, specified by the rule that, if H realises x , then H Wr G realises G x . Note that G maps the sequence (1, 0, 0, . . .) to the sequence realised by the group G. An interesting candidate is the group C preserving the cyclic order on the unit circle (or on the set of complex roots of unity). As with any highly homogeneous group C realises the sequence 1, so

C(l,O, 0, . . .) = (1, 1, 1, . . .). A short calculation shows that this is equivalent to the identity etl(l-t)

=

n

(1 - t n ) - O ( n ) / n

fl>l

Jaap Seidel encouraged me to consider the operator S* defined by

Among its properties are: (i) If x lists the dimensions of homogeneous components of a graded vector space V, then S*x lists those of the exterior algebra of V. (Compare S, which does the same job for the symmetric algebra). (ii) Like S, S* is an “exponential”: S*(x @ y ) = s * x

0

S*y.

(iii) If x enumerates a class C of objects, then S*x enumerates the class of disjoint unions of objects in C satisfying the “exclusion principle”, that is, no two the same. This has some amusing consequences: (a) S*l enumerates partitions with all parts distinct (Sloane #loo). (b) If x enumerates asymmetric connected graphs, then S * x enumerates asymmetric graphs. (c) The unique sequence x with x 1 = 1 which is shifted by S* enumerates asymmetric rooted trees.


101

(iv) The identity (1+ t ) (1 - t2)-' = (1 - t)-' has the following consequence. For any sequence x , s*x

0

(SX)"

= sx,

where y" denotes the sequence obtained by alternating zeros with terms of y .

Note added in proof As I hoped, my lecture at the conference elicited some further connections from members of the audience. (i) Several people gave a combinatorial proof of the identity A(rx) = rA'x. (ii) Concerning Ax, where x is (1, 2, 3, 4, . . .): Ron Graham and Donald Knuth informed me that that identifying this sequence with the alternate Fibonacci numbers is an exercise in their forthcoming book. (To recapitulate: the nth term is found by expressing n in all possible ways as an ordered sum of positive integers, multiplying the terms in each sum, and adding these products. The problem is to prove the identification without using generating functions). Knuth also pointed out that this sequence counts spanning trees in a fan. (iii) Some explorers discovered that the other bisection of the Fibonacci sequence, viz. (1, 2, 5, 13, 34, . . .) is A y , where y = (1, 1, 2, 4, . . .) (powers of 2, shifted right and preceded by 1). (iv) Knuth also remarked that the double of the sequence satisfying (ii) (b) of Section 7 counts permutations which can be produced by a deque (double-ended queue).

References [l] R.A. Bailey, Discussion of T. Tjur, Analysis of variance models in orthogonal designs, Internat. Statist. Review 52 (1984) 33-81. [2] P.J. Cameron, Aspects of the random graph, in: Graph Theory and Combinatorin, ed. B. BollobAs (Academic Press, London, 1984) 65-79. [3] P.J. Cameron, Some treelike objects, Quart. J. Math. Oxford 2, 38 (1987) 155-183. [4] P.J. Cameron and D.E. Taylor, Stirling numbers and affine equivalence, Ars Combinatoria 20B (1985) 3-14. [5] A. Cayley, Recherches sur les matrices dont les termes sont des fonctions lineaires d'une seule indeterminee, J. Reine Angew. Math., 50 (1855), 313-317. [6] J. Covington, A relational structure for N-free graphs, J. London Math. SOC., to appear. [7] P. Erdos and A. Rknyi, Asymmetric graphs, Acta Math.,Acad. Sci. Hungar. 14 (1963) 295-315. [8] M. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Stat. Phys., 19 (1978) 25-52. [9] R. Frdisse, Sur certains relations qui genbralisent l'ordre des nombres rationnels, C. R. Acad. Sci. Paris 237 (1953) 540-542. [lo] J.E. Hirsch, M. Naunenberg and D.J. Scalapino, Intermittency in the presence of noise: a renormalisation group formulation, Phys. Lett., 87A (1982) 391-393.

I02

P.J. Cameron

[ 111 P.J. McCarthy, Ultrafunctions, projective function geometry and polynomial functional equations, Proc. London Math. Soc.(3) 51 (1986) 321-339. [I21 P.J. McCarthy, Non-linear projective geometry, ultrafunctions and applications, Proc. Roy. SOC., submitted. 113) H.D. Macpherson, Orbits of infinite permutation groups, Proc. London Math. Soc. (3) 51 (1985) 246-284. [14] H.D. Macpherson, Infinite permutation groups of rapid growth, J. London Math. SOC.(2) 35 (1987) 276-286. 1151 N.J.A. Sioane, A Handbook of Integer Sequences (Academic Press, New York, 1973). [ 161 L. Svenonius, &,-categoricity in first-order predicate calculus, Theoria (Lund) 25 (1959) 173-178.


103

1-FACTORIZING REGULAR GRAPHS OF HIGH DEGREE - AN IMPROVED BOUND A.G. CHETWYND Department of Mathematics, University of Lancaster, Bailrigg, Lancaster LA1 4YL, U.K.

A.J.W. HILTON Department of Mathematics, University of Reading, P. 0. Box 220, Whiteknights, Reading RG6 2AX, U.K .

We showed eariler that a regular simple graph of even order satisfying d ( G ) 5 9 IV(G)l was the union of edge-disjoint 1-factors. Here we improve this to regular simple graphs of even 1) ( V ( G ) ( . order satisfying d ( G )5 $(fi-

1. Introduction The graphs we shall consider will be simple, that is they will have no multiple edge or loops. An edge-colouring of a graph is a map @: E ( G ) - %, where % is a set of colours and E ( G ) is the set of edges of G , such that no two incident edges receive the same colour. The chromatic index f ( G ) of G is the least value of I%( for which an edge-colounng of G exists. A well-known theorem of Vizing [7] states that

A(G) s f ( G ) S A(G)

+ 1,

where A(G) is the maximum degree of G. Graphs for which A(G) = f ( G ) are said to be Class 1, and otherwise they are Class 2. A regular Class 1 graph is often called 1-factorizable,as it is the union of edge-disjoint 1-factors. For a regular graph G, let us denote the common degree of the vertices by d ( G ) . A well-known conjecture which may be due to G.A. Dirac (he told one of us that it was ‘going around’ in the early 1950s) is as follows.

Conjecture 1. A regular graph of even order satisfying

d ( G )3 t IV(G)l is 1-factorizable. The present authors took the first significant step towards solving this conjecture by proving it with the more restrictive bound d ( G ) 2 4 IV(G)l in [l]; they actually proved it with d(G)30.849 IV(G)l. Here we improve this to 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

A.G. Chetwynd, A.J.W. Hilton

104

d(G)3 IV(G)l; in fact again we prove a slightly stronger bound, namely d ( G )3 0.823 IV(G)l. In a regular graph G of even order on vertices v l , . . . , u2n,let pij = p,(G) be the number of paths in G , the complement of G, of length 2 which join vi and v,, and let p = p ( G )= maxi,jpjj. Clearly p s d ( G ) = IV(G)l- d ( G ) - 1. First we prove the following result.

Theorem 1. Let G be a regular graph of euen order satisfying 1 d ( G )3 2 IV(G)(- $ - 6.

Then G is l-factorizable. By proving an easy bound on p 2 _= 0.833 and $(fi - 1 ) = 0.823.)

we obtain the following corollary. (Note that

Theorem 2. Let G be a regular graph of even order satisfying - 1 ) IV(G)l. d ( G )3 $(fi Then G is l-factorizable. For the case when p = ( V ( G ) (- d ( G ) - 1, Theorem 1 reduces to Theorem 3.

Theorem 3. Let C be a regular graph of even order containing two vertices which, in G , are joined by IV(G)(- d ( G ) - 1 paths of length 2. Furthermore, let d ( G )3 2 IV(G)l+ 1. Then G is l-factorizable.

Let Gd be the subgraph of a graph G induced by the vertices of degree A = A(G). We call G,, the core of G. A very useful result, due to Fournier [5], is that if G Ais a forest, then G is Class 1. As a preliminary to our proof of Theorem 1 , we extend Fournier’s theorem. A general discussion of the possibilities for extending Fournier’s theorem was provided by Hoffman and Rodger in [ 6 ] ;see also [2] and [3].

2. Preliminary results For a vertex u in a graph G, let d*(u) denote the number of vertices of C of maximum degree to which u is adjacent. The following lemma was proved in [l].

1 -Factorizing regular graphs

105

Lemma 1. For a graph G, let e E E ( G ) be incident with w E V ( G ) .Let d*(w)G 1. Then A(G - e ) = A(G) Jf(G

- e )=f ( G )

and A(G - W ) = A(G) Jz‘(G - W ) = x’(G). The next lemma is a well-known result of Dirac [4].

Lemma 2. Let G be a graph whose minimum degree 6(G) satisfies

Then G possesses a Hamiltonian circuit.

3. Extensions of Fournier’s theorem We first prove the following theorem. Define a proper tree to be a tree with at least one edge.

Theorem 4. Let the connected components of G A be GA(l), . . . , GA(r).For each i E (1, . . . ,r } assume that GA(i) consists of disjoint proper trees K1, . . . , Th(i) which are rooted on a graph Hi, where, for each j E ( 1 , . . . , s(i)}, Hi n is a single vertex vii (the root vertex), and such that GA(i)\V(Tl U - - - U T,,,,) contains no edges. Then G is Class 1.

rj

The type of graph permitted for a GA(i)is illustrated in Fig. 1. In the particular special case when each is a single edge, Theorem 4 was used (without being explicitly stated) in [l].

xj

Proof of Theorem 4. We first colour all the edges of G\E(G,) with A ( G ) colours. Since the only vertices of degee A(G) in this graph are non-adjacent, it follows from Fournier’s theorem that this is possible. For each i we colour all the edges of Hi using Vizing’s fan argument; we first colour the edges of the subgraph using vertices in ( v i l ,. . . , vbCi)}as pivots, and then of Hi induced by vil,. . . , we colour the remaining edges of Hi, using the vertices of V(Hi)\{vll, . . . , vh(,)} as pivots. Finally we colour the edges of each Ti as follows. We may order the edges e l , . . . , e, of a tree Tj so that el is incident with vij, and, for 1s k s t, the edges e l , . . . , el, induce a subtree. We then colour e l , . . . , e, in that order using Viing’s fan argument, always choosing as pivot the vertex of ek which is non-adjacent to any of the vertices of e l , . . . , ek-l. 0

106

A.G. Chetwynd, A.J. W.Hilton

Fig. I . A connected component of the graph Gd of Theorem 4.

Next we show that Theorem 4 can be extended.

Theorem 5. Let GA be the edge-disjoint union of two graphs B and R , having the following properties. ( i ) l f R A ( l ) , . . . , RA(r) are the connected components of R d , then for each i E { l , , . . . , r } , R A ( i ) consists of disjoint proper trees TI, . . . , Tut1)which are rooted on a graph H I , where, for each j E { 1 , . . . ,s ( i ) } , HI n T, is a single vertex I),, (the root vertex), and such that RA(i)\ V ( T , U . * U T,,,)) contains no edges. (ii) The graph B i s bipartite and has the property that the set of all proper trees can be written in an order T,, . . . , Tp such that the edges of B join vertices of V ( &) \ { v k } to vertices v, with k < 1, where v k and vIdenote the root vertices of Tk and TI respectively. Then G is Class 1.

-

r,

In the theorem above, the hypotheses on RA are the same as the ones on GA in Theorem 4. The graph B and the trees T,, . . . , Tp are illustrated in Fig. 2 in the

1 -Factorizing regular graphs

107

Fig. 2. The graphs B together with trees TI, . . . , Tp in the case when T,,. . . , 5 each consists of one edge, and p = 5.

case where each Tk consists of a single edge (this is, incidentally, the special case we shall use in the proof of Theorem 1). Proof of Theorem 5. Adapting the proof of Theorem 4, first we colour the edges of G\E(G,). Then we colour the edges of H,,. . . ,H,.Then we colour the edges of B, using Vizing's fan argument with the vertices on the trees TI,. . , Tp as pivots. Finally we colour the edges of the trees as before, but colour them strictly in the order T,, . . . , Tp. 0

4. The proof of Theorem 1 Let G be a regular graph of order 2n and degree

d = d ( G ) 3 2 IV(G)l- ip -

a.

Let w, u* E V ( G ) be such that the number of paths of length two between w and u* in G is p. Let W be the set of vertices of maximum degree in G. Then G - w has 2n - 1 vertices, IWI = 2n - d - 1 of them having degree d, and the remaining d of them having degree d - 1. The vertex v * is non-adjacent to p of the vertices of W . Thus d*(v*)= IWI - p or IWI - p - 1. Let X be a set of IWl - p - 1 vertices of V ( G - w ) which are non-adjacent to v * ; as there are in G - w at least IWl - 1 vertices non-adjacent to Y * , such a set X does exist. Let s=IXI=IWI-p-l. Let q=((XUW)\{v*}ls (IWl - p - 1)+ IWl=21WJ - p - 1 =4n - 2 d - p -3. Now consider the subgraph H of G - w induced by (XUW ) \ {v*}. Let Mo be a set of edges of H forming a maximal matching, and let m = IMol. Let

(XUM)\{ u * } = L u R, where L f l R = 0, JLI= m, and where each edge of Mo joins a vertex of L to a

A.G. Chetwynd, A.J.W. Hilton

108

vertex of R. Let L" = L U {v*} and let H* be the subgraph of G - { w } induced by L" U R. Let the elements of L* be denoted by f l , . . . , where = v*, and let the elements of R be denoted by r,, r,,,, . . . , r,, where t = 2m - q + 1 (so t may be negative). Suppose that

M,= { l l r l , . . . , f,,,r,}. Let E' consist of all edges of H*, except those of the form fir,, where i " j , and those with both endvertices in R. We now observe that E' is contained in the union of q edge-disjoint matchings of the complete graph on V ( H * ) , M:, . . . , M:, where M : is defined as follows:

M: = { l , r , - , , f2r2-,,. . . , l,r,-,, fm+lrm+l--r}rif 1S i a 1- t , M : = { f j f i + r - 1 , 121r+r-2, . . . f l ~ 1 + r - i ~ / 2 J f r ~ i + r - * ~ / z l + ~ } t

U { f i + ~ t ,f r + i + l r r + l ,

if 2 - I ai S m

M:

. . . , lmrm-i,

+1

-

= { ~ i - m + t - ~ ~ r n +l i -~- m, + r l r n ,

if m

+2 -t S i Sq.

lrn+lrm+l-i}r

t,

..

*

~l~i+r-,~~llr~I+r-i~~l+i},

Notice that UQzI M: contains all the edges of the complete graph on the vertices of V ( H * ) except for the edges of M,, the edges with both endvertices in R, and the edges which join 1, E L to r, E R with i <j. Finally we notice that

IM:I i ( q + 2 - i) (1 9 i =Z 4). The matching M: (1=G i 6 q ) are illustrated in Fig. 3. Or

0 =t 0

.1'

v'

l(i(1-t

2 - t ( i ( m +

1

Fig. 3. The matchings MT

-

t

m + 2 - t ( i i q

(1 s i S 4).

t

1-Factorizing regular graphs

109

Let W, be a set of s elements of W which are adjacent to v*. (Recall that there are either s or s 1 such elements.) Let the vertices of X be xl,. . . , x, and the vertices of W, be w l , . . . , w,. If an edge xw is in Mo with x E X and w E W,, we may suppose that x E R and w E L. We may moreover suppose that Zl, . . . , l,, r,, . . . , r,, xl, . . . , x,, w l , . . . , w, are labelled so that, for 1sj s s - 1, xi comes before wi in the list (r,, . . . ,r,, I ] , . . . ,l,). (In fact, except in the case when each edge of Mo joins either two vertices of X or two vertices of W,, we could suppose that x, comes before w, also.) We now construct matchings MT (1 s i S q + 1) by slightly modifying the M + . If x, comes before w,, define M f = Mi+ (1 S i s q ) and M,*+l= #. If x, comes after w,, then we may suppose that v*w, E M i for some io. If xo is not incident with any edge in M c , then define M ) = M i + ( l s i s q ) and M,*+,= #. If there is an edge in MC incident with xo, say ei,, then define M ) = M t ifiE(1, . . . , q } \{io}, Mi*,=Mi,\(eia} andM,*,,=(ei,}. Note that

+

IM: I s 4(q

+ 3 - k)

(1 s k s q + 1).

A near l-factor F of G - w is a set of i(IV(G - w)l - 1) independent edges of G - w . We say that the vertex which is not incident with any edge of F is “missed” by F. We choose q 1 edge-disjoint near l-factors F,, . . . , F,,, of G - w such that

+

4u M0n(4u . - .u ~ , + ,=)O E + n (Mf u

’

UM:)

*

UFk

‘

(1 sk S q + 1))

and furthermore, if v*wi E M: for some (1, and

. . . ,s},

then v*wi E Fk and Fk missesxi,

if v*wi $ Mk for all i E { 1, . . . ,s}, then Fk misses v*.

To choose Fk (1 s k s q already. Let Mk

Then

= ( E + n M:)\(Fl

+ l),

suppose that F,, . . . ,Fk-1 have been chosen

u .u Fk-1 u &)’

a

IM:I s 4(q + 3 - k) (1 s k S q + 1).

Consider Gk-1=

(G - W ) \ ( &

u

* *

.u 4 - 1

u Mo).

We choose Fk to be a near l-factor of Gk-l containing Mk and missing xi if v*wiEMk for some irz (1,. . . , s}, or missing v * if v*wi$M k for all i E (1, . . . ,s}. Let V(Mk)denote the set of vertices of G which are incident with the edges of M k , and define G l - , by G:-l = Gk--r \ V(Mk).To see that we can choose 4 in the way described, we apply Lemma 2 (Dirac’s theorem) to show that G:-] has a

A.G. Chetwynd, A.J. W.Hilton

110

Hamiltonian circuit. We have

d(G;-1)

3

( d - 1) - {(k - 1) + 1)

- IV(Mk)I = d - k - 1 - lV(Mk)l.

Also

f lV(Gi- J l =

${IV(G~-I)I- I V ( K ) I }

= 4{2n - 1 - lV(Mk)l}.

- n - I -2 L -

2

IV(Mk)l.

Therefore 6(G:-,) - 4 lV(Cl-l)l 2 d - k - 1 - IV(Mk)I - n + $ + $ IV(Mk)I = d - n - k - I - 2 tlV(Mk)I

a d - n - k - f - $(q + 3 - k ) = d - n - 4q - $k - 2 ad-n-q-2 a d - n - (4n - 2d - p - 3) - 2 = 3d - 5n + p + 4 3 0,

since d a g ( 2 n ) - f p -$. Therefore by Lemma 2 , Gl-, does have a Hamiltonian circuit. It follows that Gk-l contains a near 1-factor Fk which contains Mk and misses x i if v*wi E Mi for some i E (1, . . . , s}, or misses u* if u*wi$ Mi for all i E { 1, . . . , s}. It is easy now to check that Fk has all the various properties required of it. Let J = {i: misses v*}. The graph ((G - w)\(F, U * . U Fq+*))\{u*} has core of the form of Theorem 5, with each tree qj just consisting of a single edge. Therefore ((G - w)\(F, U U &+l))\{v*} is Class 1. The graph

-

-

-

{e:

( ( ( G - w)\(Fl U . U Fq+J)\{u*}) U i EJ} = ( ( G - w)\{&: i E ( 1 , . . . , + ~ } \ J } ) \ { u * }

is therefore Class 1. By Lemma 1, it now follows that

(G - w)\(F,: i E (1, . . . , q + 1}\J) is Class 1. It therefore follows that C - w is Class 1. In any edge-colouring of C - w with d ( G ) colours, it is easy to see by counting that each colour is missing from exctly one vertex. Therefore an edge-colouring of C - w can be extended to an edge-colouring of C. Thus G is Class 1. This completes the proof of Theorem 1. 0

5. The proof of Theorem 2

The argument in the last section improved our result from 4 IV(G)l to an improvement of $ - 2 ~ 0 . 0 2 4 In . this section we use a counting argument to improve our bound further by about 0.01.

2 IV(G)l,

1-Factorizing regular graphs

111

First we give a bound on p .

Lemma 3.

p a (2n - d - 1)(2n - d - 2 ) 2n - 1

Proof. Each vertex in G is the centre of G of (” -2” - ’) paths of length 2. There are therefore altogether 2n(2”-2”-1)paths of length two in G. Therefore the average number of paths of length two joining an arbitrary pair of vertices is 2n-d-1 2n(

2

(3

1

-(2n-d-1)(2n-d-2) 2n - 1

Clearly p is greater than or equal to this average number. This proves Lemma 3. 0

Proof of Theorem 2. From Theorem 1 and Lemma 3 , it follows that if

then G is Class 1. After multiplying out and simplifying, the inequality becomes

+

d2 2nd - (6n2- $)3 0,

so that d 3 -n

+ d{7nZ- 3).

suffices. Therefore G is 1-factorizable if

d 3 4(fl-

1 ) IV(G)l.

This proves Theorem 2. 0

6. The proof of Theorem 3 If we substitute p = IV(G)l- d ( G ) - 1 into the inequality d ( G ) 3 2 IV(G)l4p - f , we obtain the inequality d ( G )3 a IV(G)(+ i.

Acknowledgement The authors would like to thank the referee for suggesting various improvements to the text, and, in particular, for spotting a rather serious blemish.

112

A.G. Chehvynd, A.J.W. Hilton

References [I] A.G. Chetwynd and A.J.W. Hilton, RegL..,r graphs c high degree are 1-factorizable, Proc. London Math. Soc. (3) 50 (1985) 193-206. 121 A.G. Chetwynd and A.J.W. Hilton, A A-subgraph condition for a graph to be Class 1, J. Combinat. Theory (B), 46 (1989) 37-45. (31 A.G. Chetwynd, A.J.W. Hilton and D.G. Hoffman, On the A-subgraph of graphs which are critical with respect to the chromatic index, J. Combinat. Theory (B), to appear. [4] G.A. Dirac, Some theorems on abstract graphs, Proc. London Math. SOC.(3) 2 (1952) 69-81. [ 5 ] J.-C. Fournier, Colorations des ar2tes d'un graphe, Cabers de CERO, 15 (1973) 311-314. [6] D.G. Hoffman and C.A. Rodger, Class one graphs, J. Combinat. Theory (B) 44 (1988) 372-376. [7] V.G. Vizing, On an estimate of the chromatic class of a p-graph (in Russian), Diskret. Analiz. 3 (1964) 25-30.


113

GRAPHS WITH SMALL BANDWIDTH AND CUTWIDTH F.R.K. CHUNG, P.D. SEYMOUR Bell Communications Research, Morristown, NJ 07960,U.S.A.

We give counter-examples to the following conjecture which arose in the study of small bandwidth graphs. “For a graph G, suppose that IV(G‘)l< 1+ c1 . diameter (G‘) for any connected subgraph G’ of G, and that G does not contain any refinement of the complete binary tree of c, levels. Is it true that the bandwidth of G can be bounded above by a constant c depending only on c , and c,?” On the other hand, we show that if the maximum degree of G is bounded and G does not contain any refinement of a complete binary tree of a specified sue, then the cutwidth and the topological bandwidth of G are also bounded.

1. Introduction For a graph G with vertex set V ( G ) and edge set E(G), a numbering of G is a one-to-one mapping JG from V ( G )to the integers. The bandwidth of a numbering n is max{ln(u) - n(v)l: {u,v} E E ( G ) } . The bandwidth b ( G ) of G is the minimum bandwidth of all numberings. The cutwidth of a numbering n is max I{ {u, v } E E ( G ):n ( u ) i < x(v)}I. i

The cutwidth c ( G ) of G is the minimum cutwidth of all numberings. The bandwidth problem and the cutwidth problem are associated with many optimization problems in circuit layout. In a circuit design or a network system, the maximum length of the wire is often proportional to the delay for transmitting messages, and so bandwidth is a graph-invariant of importance in circuit design. On the other hand, the cutwidth problem is of particular interest in designing microchip circuits and is often associated with the area for the layout (see [7]). One of the interesting problems about bandwidth is to understand what substructures force up the bandwidth of a graph. There are two known factors which may make bandwidth large. The first is the density lower bound (see [I, 21):

where D ( G ) is the diameter of G, that is, the maximum distance among all pairs 0012-365X/89/$3.50 01989, Elsevier Science Publishers B.V. (North-Holland)

114

F.R.K . Chung, P.D.Seymour

of vertices in G. A somewhat stronger lower bound, the so-called “local density” bound, can also be easily obtained:

where G’ ranges over all connected subgraphs of G with 3 2 vertices. One natural problem arises: “If local density is small, is it true that the bandwidth is small?’’ This question was answered in the negative by ChvAtalovA [4] by examining refinements of the complete binary tree Bk of k levels. A graph G’ is said to be a refinement of G if G’ can be formed by replacing some edges in C by paths. For each integer k, every refinement of Bzk has bandwidth ak, and there is a refinement of BU, with local density at most 3. Now if a graph contains a refinement of BZk,its bandwidth is at least k. Again, containing a large complete binary tree is sufficient but not necessary for the graph to have large bandwidth (as we see from the star K l , n . )That suggests the following question. Suppose that the local density of a graph G is no more than c , , and that G does not contain any refinement of Bc2. Is it true that the bandwidth of G is bounded above by a constant depending only on c1 and c,? We mention that ChvAtalovA and Opatrinq [ 5 ] proved a somewhat similar result for infinite trees. They showed that if an infinite countable tree T satisfies that (i) the maximum degree is at most c , , (ii) the number of edge-disjoint semi-infinite paths is at most c Z rand (iii) T does not contain a refinement of B,, as a subgraph, then some refinement of T has finite bandwidth bounded above by a function depending only on c,, c2 and c3. in this paper, we will prove two results. One of them answers the above question in the negative and identifies a third structure which drives up the bandwidth. The other result answers positively an analogous question for cutwidth (or “topological bandwidth”, defined below).

Theorem 1. For each integer k , there exists a tree with the following properties (i) its local density is at most 9 (ii) it does not contain any refinement of B4 (iii) its bandwidth is at least k.

Theorem 2. Suppose that G has maximum degree c l , and does not contain any refinement of Bc2. Then the cutwidth of G is bounded above by a constant depending only on c1 and c2. The topological bandwidth b * ( G ) of a graph G is the minimum bandwidth h(G’) over all refinements G‘ of G. The topological bandwidth problem can be viewed as the optimization problems of circuit layout when vertices of degree two (interpreted as “drivers” or “repeaters”) can be inserted to help minimize the length of the edges. Cutwidth and topological bandwidth are known to be closely

Graphs with small band- and cutwidths

115

related, and it has been shown [3, 61 that b * ( G ) S c ( G )for any graph G. In particular, for trees [3]

b*(T ) S C( T ) S b*(T ) + log2 b*(T ) + 2. But it is not hard to see that for some graphs, such as G = K,,, the cutwidth c ( G ) can be much larger than b*(G). Nevertheless, Theorem 2 implies the following relation between c(G) and b*(G).

Theorem 3. There is a function f such that for any graph G , c ( G )S f ( b * ( G ) ) . (One interpretation of Theorem 3 is that if the topological bandwidth is bounded above by a constant c,, then the cutwidth is bounded above by another constant c2 which depends only on ct.) The paper is organized as follows. In Section 2, we will construct some special trees, the so-called Cantor combs, which imply Theorem 1. In Section 3 we will give the proof of Theorems 2 and 3.

2. Cantor combs

In this section, we will show that the two conditions, local density S C , and containing no binary tree of c2 levels, do not imply small bandwidth. A comb is a tree T with two special vertices, called its roots, such that every vertex of T with degree 3 3 has degree 3 and lies on the path of T between the roots. For k 3 1, we define the Cantor comb c k as follows. C , is the 2-vertex tree, where both vertices are roots. Inductively, having defined C k - 1 , we define c k as follows. Take two disjoint copies T,, T2 of c k - 1 with roots s,, tl and s2, t2. Let P and Q be paths with 4 l v ( c k - 1 ) l and 6(k - 1) I v ( c k - 1 ) I edges respectively, such that P, Q,T, and T2 are mutually vertex-disjoint except that p has ends t1 and t2, and one end of Q is the middle vertex of P. We define c k to be & U U P U Q, with roots s,, s2. This completes the inductive definition of c k . We observe that there is an automorphism of c k exchanging the roots. Let I v ( c k ) l = Nk ( k 2 1). We shall show that c k satisfies Theorem 1, by means of the following assertions.

(2.1) For k 3 1, the bandwidth of

c k

is at least k.

Proof. If possible, choose k 3 1 minimum such that c k has bandwidth O an ordinal (possibly finite), and let BA be an induced subgraph of G for every A < a. The family F = (BA)A KO for some regular cardinal a. Suppose that G $ K a , and that pG(x, y ) < a whenever x and y are non-adjacent vertices of G . Then G admits a simplicial decomposition F = (BA)a KO. Then G =I TK,,,. Corottary 4.8.

If an uncountable graph G is n-connected, n E N, then G 3 TK,,.

5. Related decompositions Among the motivations suggested in the introduction of this paper for decomposing graphs into simplicia1 factors was the prospect of being able to ‘lift assertions about the factors to similar assertions about the whole graph’; k-colourability was given as an example to illustrate the idea. The value of a particular kind of decomposition for this purpose clearly depends on two features of the graph property under investigation. Firstly, the property must be wholly or at least to a controllable extent preserved in the pasting operation, and secondly, it must be easier to investigate the factors of the graph than the graph itself. However, these two objectives obviously work against each other; the more specificly we define our attachment rule, the fewer graphs will be decomposable, and the larger the primes we get -and vice versa. Finding the right kind of decomposition for a given problem is therefore a task of striking a balance. As was illustrated (and to some degree explained) in Section 1, simplicial

Simplicia1 decompositions of graphs

139

decompositions seem to be just the right kind of decomposition for investigating minor-closed properties, and properties defined in terms of forbidden subdivisions. In general, however, the requirement (S2) that all attachment graphs be simplices seems to be rather on the strict side. If one focusses on monotone increasing classes of attachment graphs (and there seem to be reasons for doing so), simplicial decompositions are even an extreme case, based on the smallest possible class of attachment graphs. And indeed, while quite a few graph properties are compatible with attachment along a simplex (i.e. can be lifted from simplicial factors to the whole graph), simplicial decompositions do tend to leave rather large primes, which are often not fewer in number or simpler in structure than arbitrary graphs with that property. The k-colourable graphs are again a case in point. Halin 1401 suggested to take account of this problem by relaxing condition (S2) in the definition of simplicial decompositions if appropriate, while keeping the tree structure of the decomposition by imposing (S4). This, together with a few additional constraints, would ensure that the structural properties of the decompositions obrained would be similar to simplicial ones, enabling us to transfer some of the existing theory. However, if we place the emphasis firmly on decomposability to a high degree, keeping the tree structure seems unnecessarily restrictive: it may prevent us from decomposing a factor further, even if it has a separator that would be admissible as an attachment graph (an example will be given below). The simplest way to ensure maximum decomposability (with respect to a fixed class 9 of admissible attachment graphs) is to use as an attachment rule the direct reversal of the process of successive separation. For properties 9 and 9 of finite graphs let us define the 9-sumof graphs in 9 recursively: (1) Every G E $3 is a 9-sum of graphs in 9$ (2) If G, G‘ are 9-sums of graphs in 9 and G f l G’ E 9,then G U G’ is a 9-sum of graphs in 9. For p k = {G :/GI= k} and PSk= {G :/CISk}, we abbreviate ‘Pk-sum’ to ‘k-sum’, ‘ 9 S k sum’ to ‘(dc)-sum’, ‘(9n 9k)-sum’ to ‘9-k-sum’ and so on. Moreover, we shail loosely speak of simplicial sums, connected sums etc. if 9 is the property of being complete, connected etc. Using well-known facts about simplicial decompositions it is not difficult to show that any simplicial sum of certain graphs admits a simplicial decomposition into precisely these graphs as factors (provided only that none of them is contained in another), and with precisely those simplices as simplices of attachment that were used as attachment graphs for building the sum. For a simplicial sum we may therefore usually assume without loss of generality that it was obtained by adding only one factor at a time. For other sums however this is not the. case. As an example, consider the graphs shown in Fig. 2: it is a connected 3-sum of four K i s , obtained by first pasting the K4’s together in pairs along triangles and then joining the two arising

140

R. Diestel

Fig. 2. A connected 3-sum of K,’s.

K;’s along a common path of order 3. This graph cannot be obtained as a 3-sum of K,‘s in any other way. Connected sums are used in a recent paper of Duchet, Las Vergnas and Meyniel [26] to describe two interesting graph properties: ‘well-connectedness’ and ‘null-homotopy’. A graph G is well-connected if every minimal relative separator is (non-empty and) connected, and G is null-homotopic if every algebraic cycle of G is the sum (mod2) of triangles. Both these properties are compatible with connected summing: if G = G‘UG” where G’ and G” are well-connected (null-homotopic) and G’ n G“ is connected, then G is wellconnected (null-homotopic). Using Wagner’s characterization of the finite graphs without a K5 minor (see Table l ) , Duchet, Las Vergnas and Meyniel obtain the following result:

Theorem 5.1 [26]. For a finite graph G E %(HK,) the following statements are equivalent: (i) G is null-homotopic; (ii) G is well-connected; (iii) G is a connected ( ~ 3 ) - s u m of disc-triangulations. (A disc-triangulation is a plane graph in which at most one face is not a triangle.) It is interesting to note that if G is required to be planar, the connected 3-sums in Theorem 5.1 can be replaced with simplicia1 3-sums [26]. Using other homomorphism bases from Table 1 , Theorem 5.1 can be extended as follows:

Theorem 5.2. If G i s a finite connected graph from any of the classes %(HK3,3), 48fHW,,,), %(HL) or %(H(C, x K 2 ) ) , the following statements are equivalent: (i) G is null-homotopic; (ii) G is well-connected; (iii) G IS a connected (S3)-sum of disc-triangulations and copies of K s . If G E %(H(C3x K 2 ) ) , the disc-triangulations in (iii) can be chosen as wheels, triangles, K;’s or K2’s. An infinite analogue to k-sums can be obtained from the definition of simplicial decompositions by replacing (S2) with a condition on the order of the attachment

Simplicia1 decompositions of graphs

141

graphs. Given an infinite cardinal a and a graph G, call a family (B,),,, of induced subgraphs of G an a-decomposition of G if it satisfies (Sl), (S3) and (S,2) Each subgraph (UL.=p B,) fl B,, =:S,,has order 0 and infinitely many n, we would have say I{x EAk,:d(x,A;) *cn}I * c n

and I{x ~ A k , : d ( A;) x , 2 cn}lZ cn.

Then, by the assumption, for infinitely many n,

I{xEAk,:d(x,A;)>$ IAbl}l2cn and

1 { x E A; :d(x, A;) > $lA;1}I 2 cn hence for some y E A;, f,(y) > f n for infinitely many n, a contradiction.

0

Lemma 7. For every k there is an 1 such that whenever y E V, A c T(y), B c f(y), c >0, IAI, IBI b c n and i(G) S O(n'') then there are W, c V and a Go for which IW,l= o(n), Go is 1-canonical on ( A U B)\W, and A(G[A\W,, B\W,]AGo)~l. Proof. We use the notation f,,g, introduced in the proof of Lemma 6 for the graph G' = G[A ,B] with V' = A U B. For an x E ( V ) r and i S r we denote the restriction of x to i by x I i. For every fixed I and for every n 1 we define a sequence ( x i : i< I ) by recursion on i, using a greedy algorithm: we let xi be an element of V'\ {xi : j < i} satisfying

&(xi) = gxli(n)* We now claim that gx(n)= o ( n ) for an x E (V')'l with 1, 0 for infinitely many n, then for all these n we have

V, 1

I
0 infinitely often, provided 1 is large enough. By Lemma 7, there exists an 1, and 1,-canonical graphs G, :< 1 such that ;,jc/ A;]AGi)=Sl,. A ; ,j # U

Using a Ramsey type argument we can select a subsequence {xI,:j < l , } , c2> 0 and At c A, such that by putting y, = x,,, A;= A:, we have IA;I

(1)

[A;, Ail] c E, for j < t < I 2

(2)

[A;, Ail

3 c2n and

either

or

n E = 0, for j < t < 12,

provided 1 is large enough compared to k, l , , and 12. If case (2) holds, by Lemma O(a) we have

for some c3 > 0. If case (1) holds, then either for some c4 > 0 and for more than

Distinct Induced Subgraphs of a Graph

151

12/2 values of j , G[Aij has a component of size at least c4n4 and in this case Lemma O(a) implies that i ( G ) 3 ~ ~ nfor ' ~some ' ~ c5 > 0, or else we may assume that for more than 12/2j, the components of G[AiI have sizes at most k. This follows from Lemma O(b). Then for some c6>0 we can choose A , c A ; , lA,l 3 c6n for more than 12/2 values of j < l2 in such a way that [A,]' c E. By Lemma 1, we have i(G)P(z). 0 We are now in a position to prove Lemma 2. Proof of Lemma 2. Just as in the proof of Lemma 3, if the lemma fails with 1 = 2 k +2, then we may assume that omitting o ( n ) vertices W, arbitrarily, A(G[V\W,])>nnf holds and that for every A c V , A # 0 and for every m, I { x E V :d(x, A ) 3 l / m lAl}l= o ( n ) . Using these, for every m and sufficiently large n, we can choose disjoint sets { x i : i < r n } , A i : i < m in such a way that [{xi:i < m}]' r l E = 0 and for A = U i r m A i , T(xi,A ) = Ai and lAil 3 llmn; hold for i < m. Now applying Lemma 8 for the graphs G [ { x i : i< m } UA ] we get a contradiction. 0 Now we can prove our main lemma. Lemma 9. Assume i ( G ) = o(nk+l),k 3 1. Then there are W, c V , 1 and a Gosuch that IW,l = o(n), Go is 1-canonical on V\ W, and

A(G[V\W,]AGo)sl.

Proof. We use the notation fx, g, introduced in Lemma 6 and we repeat the greedy algorithm described in the proof of Lemma 7, i.e. for every fixed 1 and for every n 3 1 we define a sequence {xi:i < l } by recursion on i < 1 as follows: xi is an element of V \ { x j : j < i } satisfying f,li(xi) =g+(n). If for some 1 we have g x ( n ) = o ( n ) , then by Lemma 6 there are W A c V , lI and Go such that IWAl = o(n), G is 2'1-canonical on V\ WA and A(G[V\ W,!JAGo)= ~ ( n ) . Then, by Lemmas 2, 3, and 4, we can omit W,, IW,l=o ( n ) vertices so that for some 1 A(G[V\W,]AGo[V\W,]) sl. Hence we may assume that the following holds infinitely many n: (*) There is a sequence {xi :i < 1 } of distinct elements such that Vi < 1 3 E~(2)' d(xi, ~ ( I i,x q )2 cn) A &xi, for some c > 0.

r(xi,cp) 3 cn)

P. Erd&, A . Hajnal

152

We may as well assume that (*) holds for all n and prove that if (*) holds for large enough 1, then i ( G )B conk+'for some co> 0 infinitely often. First remark that (*) holds for any subsequence of ( x i :i < 1 ) . Now, by Lemma 5, we may assume that (V& < 2 ) I {O< i < ! : x i E T ( ( x o ) , ( E ) ) =1-E

A

d(xi,T(x I i, rp))

3 cn A

A

3rp(rp(o)

d(x,, T(x I i, 9))) 2 cn} I 6 k + 1,

as otherwise we are done. It follows that for either the graph or its complement the following statement is true. There is a set

such that {x, :i E T } c T(xo), and we can omit W, vertices, IW,l = o(n), of @,) in such a way that for all i , j E T and for all z E r(xo)\W,, { z , x i } E E a { z ,x i } E E. Now by a repeated application of this argument we obtain that if 1> 4.5'1(~+') then for either the graph or its complement the following holds: (1) There is a set Y = { y, : i < 11}, [ Y l ' c E, a c1> 0 and a sequence of pairwise disjoint subsets of V such that lAil 2 cln, Ai c r ( y i ) for i c 2 fn o r i + l < l , o r A , c T ( y j ) f o r i < j < l , , forc,>O. We will assume that (1) holds for G. If in the last statement the first alternative ~ get that holds, then applying Lemma 5 with y = Y ~ , - we i(G) 2 c2n'1-3 with some c3> 0. Thus we may assume that A , c T(yi) for i <j < ZI. However, in this case Lemma 8 yields i ( G ) 2 conk+'provided 1, is large enough. 0 To conclude the proof of Theorem 1, it remains only to prove the following.

Lemma 10. Assume G has n vertices, i ( G ) = o ( n k " ) for some k > 1. Assume further that 1 is minimal with respect to the following property: (*) There are c > 0 and s and an 1-canonical graph Go= ( V , E,) with canonical classes ( A , : i < l ) , lAiJ>cnf o r i < l and A(GAG,,)Ss. Then 1 S k and we can find W, c V , IW,l = o ( n ) such that setting G , = GAG,, all components of GI[V \ W,] have size at most m = k + 1 - 1. Proof. Set m = k + 1 - 1 if 1 S k and m = 0 otherwise. Assume for a contradiction that the claim is not true. Then for some c l , c2 > O , c,< bc, we can find pairwise disjoint sets {A,!:i < I} and a set B such that (1) IA,!(=cln, A f c A i f o r i < l .

Dbtinct Induced Subgraphs of a Graph

153

(2) For A = U i 0. In the case m > 0 and all the components G1[Xn B ] have size at least two, then

n ( X n B ) = Y n B and n ( A ; n X ) = A ; n Y for i 0, we are done. 0

2. One more result and some problems One may conjecture that if G is a strong Ramsey example, then G is close to a random graph, hence i(G) is very large, say exponential. As is shown by the

P. E r a , A. HajmI

154

attempt described in 111, this will be difficult to prove. We only have one result pointing in this direction.

TBeorem 2. Assume G is a graph with n-vertices c > 0, k > 2c log 2 and Kc log n.c log n Q Gi G-

Then, for every sufficiently large n, i ( G )3 hoof. We may assume that there is an x E V with d ( x ) 3 (n/lo$ n ) , d ( x ) k 4 n.

Let A c T(x), B c p(x) with JAI= [(n/log2 n ) ] JBj= 111, Let 9= (T(x)n A : x e B ' } , I B ' l > $ , B ' c B . Assume first ! % I > % . Let C c B ' , ICI= [ ( n / 3 k ) ] be such that T ( y )n A # T ( z )n A for y # z E C. Consider the graphs G [ { x }U A U Y ] for Y c C. If n - \ A / !+ 1 of them are pairwise isomorphic, then there are two, say G [ { x }U A U Yo] and

G [ { x }U A U Y,]

which are isomorphic by an isomorphism n keeping x and the elements of A fixed. Clearly such a 3d must keep the elements of Yo fixed, hence Yo= Yl. It follows that in this case 2Ln13kJ . ( n . nnfiogZn)-l > 2n/4k i ( ~ ) holds for sufficientiy large n. Hence we may assume that there is a sequence Bi :i s 1 of pairwise disjoint subsets of B such that IB,I = k and T ( y )f l A = T ( z )n A whenever y, z E B, for i 2 c logn, i.e. for an I = [c,(log n/log2)] with c1 < 1. L e t D = U, R(n, 2) > R(n, 3) > R ( n , 4)

(6)

and for a =2b, 2 s b S 9 , R(n, 26

+ 2) < R(n, 26) < R(n, 26 + 1)< R(n, 26 - 1).

At the end of the paper, a table of R ( n , a ) is given. It has been calculated by J. Dixmier, H. Epstein and O.E. Lanford, using the induction formula.

f ( n ,p , a)=

C f ( n - i, i, d u SB - i ) . i =sp

Here, f (n, p , d)denotes the number of partitions of n in parts s p such that no subsum belongs to d,and d - i = { a - i ; a E d,a - i > O } . It has been independently calculated by F. Morain and J.P. Massias. They have used computer algebra systems MAPLE and MACSYMA to compute polynomials mentioned by Diximier (cf. [4], 4.3 and 4.10). Unfortunately these polynomials are of degree ( ( a + l)(a + 2)/2) - 2, and it is not easy to deal with them for large values of a. As observed in [4], R(n, 2) < R(n, 3) for 1 0 s n S 106, which contradicts (6). But (6) is true only for n large enough. The aim of this paper is to study R(n, a ) for a depending on n, and smaller where ;11, is a small positive constant. The tools for that are an than estimation for r(n, d )(cf. Lemma 2 below), and inequalities involving R(n, a), extending (1) and (2). We shall prove the following result.

at/;;,

Theorem 1. There exists &,> 0, such that uniformly for 1 == a s LOG, we have, when n goes to infinity,

where ya = $ if a is odd, and, i f a is even, ya = 4

a + log3 - 2 log2 + c -log = a

+0.79.

-

log a +cwhere c is a f i e d constant. a

Partitions of n without a given subsum

157

Let us observe that, when a goes to infinity, (4) or (5) gives

( - l + F+3 o(1))a

-

< 1og u(a) - ;a

log a s (log 2 - ;+ o(1))a

(7)

while (3), (i) and (ii) yield that

- y,a

+ o(a) s log u(a) -

$2

log a s o(a)

which is better than (7) except for the lower bound when a is even. 6 6 a in an other paper, by a different method, We intend to treat the case &, which will give also an estimation for Q(n, a). For this quantity, we here give only a lower bound.

Theorem 2. There exists Al > 0, such that, uniformly for 1s a s A,fi,

we have:

We thank very much J. Dixmier for several interesting remarks.

2. Preliminary Results Let us first recall the definition of the mth Bessell polynomial y,(x); (cf. [9]): Y&) = 1 yrn(x) = (1+ m)Ym-l(x) -I-x2~:,-l(x)-

(9)

From that definition, it is easy to see that, if we set ~ ( x =) (exp (fi))/fi,

then we have (cf. [5], Lemma 1)

Furthermore, it follows from (9) that ym(0) = 1.

Lemma 1. For m odd, the function x+ym(x) is increasing for x its zero a,,,satisfies

~ ] - m , +m[,

and

Y s exp(

m(m2+

"x).

(17)

Now, let us write the polynomial P,, defined in Lemma 2 , in the form rn

Then we have

Thus, the absolute value of the general term of yrn(x)yrn(-x), that is (dkxzk(is decreasing, and, as it alternates in sign, for m s l / a , we have 1-

m(m + 1) x2 Gyrn(x)yrn(-x) = Prn(x2) 1, 2

which, with (17) completes the proof of Lemma 3. 0 More accurate estimations have been obtained by M. Chellali, using Agarwal's integral representation yrn(X)

+

=n . [ t n ( 1

dt

and the saddle point method (cf. [3]).

Proposition. There exists k2> 0 such that, if d = {al, . . ,ak} satisfies s = a l + a, + ak 6 A2n, then, when n ten& to infinity, we have

- +

Proof. It is very similar to the proof of Theorem 1 of [5].First we introduce the operator D'"). Let f : Iw +R, m 2 1, u l , . . . , urnbe positive. We set D(l)(u1;f,x ) =f(x) -f(x - u1)

D(rn)(U1,. . . , u,;f, x ) = D ( m - l ) ( U ] , . . . , u,-,;f, x )

-p

- 1 )

( ~ 1., . * urn-,;f, x - urn). 9

From the generating functions, we observe

r(n, {al, . . . ,ak}) = ~ ( ~ ) ( a. .l ., ,ak;p, n).

P. Erd& et al.

160

Now, with F ( x ) = (exp(fi))/fi, the classical result of Hardy and Ramanujan can be written (cf. [5]) as follows:

c3 ht/Z

p ( n ) = -F'(CZ(n - 1/24)) + f i ( n )

(19)

with C = n m , and

Furthermore, using Lemma 3 of [5], (18) and (19) give

with n -s

=S(5 s

n.

(21)

- 1/24)). The function x--, exx( kp+ 2(1V/23 is We now use (10) to estimate F(k+')(C2(E

-

increasing for x z= (k + 2),. As s = a , + . * + a k a k(k plies k s for A, small enough, from (21) we have

6,

+ 1 ) / 2 3 k2/2, which

Now let us turn to the proof of (i). By (22), the main term of (20), is at most

For A2 small enough, we have

~

-1

%

Y

-

and thus Lemma 1 gives

-I Y k + l ( Z i 7 = g T E )

1.

Using the estimation

the main term of (20) is at most

z

im-


161

To complete the proof of (i), it remains to check that the error term of (20) is included in the above error term. First, (24) is

>> k ! by Stirling’s formula. So it is enough to show that ( 4efi xrsexp(gfi). But the left hand side of the above inequality is an increasing function on k , for 4 k s - fi,and we know that k C fiC To conclude, we observe that for C A2 small enough, we have:

a.

In order to prove (ii), first we apply Lemma 3 , to obtain

which with (23) yields

Then we observe that e x p ( C V K G - 7 ) = exp(Cfi

+~(s/fi))

and ( n - s - l ) ( k + z ) n= exp(

(log n + O ( s / n ) ) )

- n(“’)’’ exp(O(s/fi)) since k = O(fi). Furthermore, by ( l o ) , (22) and (25), the main term of (20) is at least

The end of the proof of (ii) goes in the same way as for (i). 0

Remark. A similar proposition is given in [7] in the case of restricted partitions. A more general estimation is given by J. Herzog (cf. [lo] and [ l l ] ) ,using a Tauberian theorem.

P. E&"s et al.

162

3. The upper bound in Theorem 1

---

First let us say that a partition n = n1+ + n, of n represents a if there is a < ij 6 t which is equal to a. Thus R(n, a) counts subsum nil,. . . , ni,, 1S il < the number of partitions of n which do not represent a. Clearly if b < a , b and a - b cannot be together parts of a partition which does not represent a. Let us suppose first that a is odd. From the above remark, we deduce that for all integers, i, with 16 i 6 [ a / 2 ] , at most one of i and a - i can be a part, and thus

- -

where in the summation E~ E (0, l}. Now we apply our proposition, with k = W ( a ) , and obtain that

S S E ~ and ~ we ~ ~ ~ ~

But this summation is exactly

n (i +

Lon1

a

( a - i)) = a'(a)

i=l

which proves (i) for a odd. When a is even, the part a / 2 can occur but only once. Thus we have 1G' {iE,@

- i)l-E,

i=l

where the first summation counts partitions without any part equal to a/2, and the second counts partitions with one part equal to a / 2 . For the second sum we obtain the upper bound

But, as already observed, the function x-,expfi/xk is increasing for x and for &, small enough, the expression between brackets is smaller than

3 k2,


163

So, the second sum in (27) is not bigger than the first one, which was already estimated when a is odd. This completes the proof of (i). 0

4. The lower bound in Theorem 1 Let us suppose first that a is odd. Then R(n, a)>rr(n, {1,3,5,. . . , a } ) and, observing that @ ( a )= (a + 1)/2, by the Proposition we have

By Stirling's formula, (2u)!/2"u! 2 ~ " 2 ~ e -and " , since @ ( a )3 a/2, we obtain (ii). Let us suppose now that a is even. In fact, the following reasoning works also for a odd, but it gives a worse estimation than the preceding one. For real numbers x and y, let us denote the set of integers belonging to the real interval ]x, y [ by ]x - . y [ . We set d = [1* ~ / 3U ] [ ~ / *2* * h / 3 ] U { a } . +

--

Then, it is not difficult to see that

R(n, a ) 3 r(n, d) (which is slightly bettter than (2)), and considering the three possible cases a = 0,2,4 mod 6, that card d = @(a).By the proposition, we get

Using Stirling's formula in the form [u O(l)]! = uUe-"exp(O(1og u ) )

+

we obtain (ii) with an effectively computable constant c. 0

5. Proof of Theorem 2 We consider now only partitions without any repetition, and we look at a subset of [l - - (a - l)], say d with the following property:

-

[ - - - i]belongs to d for each j €1; - - ;[, there are 3 possibilities: no element j E 1

j E d and a -j @ d,j 6 d and a - j E d,j 6 d and a - j 6 d

for each j E

[f - - a - 11, there are 2 possibilities, j *

E d or j 6 d.

P. Era% et al.

164

For any such d,we have: Card d s c,a2. How many such d ' s are there? As

[

C a r d ( ] ~ . . ~ ~ [ ) 2 ~ and - 1 Card( -. 2a 3

- - (a - l)]) 3 , 12

there are more than 3(i1/6)-1 013

2

such sets d.Further, to build a partition of n, we choose such a set d ,and we complete by a partition of n-Card d,without any part smaller than a 1. Thus, since p(n, rn) is non-decreasing in n (cf. [S]),

+

Q(n, a ) 3 3a'6-12013p(n - c ~ u ' ,u + 1).

Using Theorem 1 of [8], which gives p(n, r n ) 2 q ( n ) / 2 " - ' , estimation

and the classical

Table of R(n,-a) 2

3

4

6

5

1

101

135 176

385 490 627 792 1002 1255 1575 1958

I

I

I I I

I

I I

I

5 7 9 11 15 19 20 23 34 30 30 41 32 38 46 51 -70 Be 58 46

~

-72 10: 70 88 137 105 109 165 119 133 156 21( 161 25: 177 198 32( 228 240 38: 262 ~

..

7

8

9

1

0

1

1

1

1

2


165

Table of R(n, a) (continued).

which implies

we obtain Theorem 2. Cl

References M.Chellali and A. Salinier, Sur les zeros des polynbmes de Bessel, CRAS, t. 305, Sene 1 (1987) 765-767.

M. Chellali, Sur les zeros des polynbmes de Bessel II, CRAS, t. 307, Stne 1 (1988) 547-550. M.Chellali, Sur les ZRros des polynbmes de Bessel 111, CRAS, t. 307,S&ie 1 (1988) 651-654. J. Dixmier, Sur les sous sommes d‘une partition, prepnnt de I’IHES (1987) and to appear in Memoires SOC.Math. France. J. Dixmier and J.L.Nicolas, Partitions without small parts, to be published in the Proceedings of Colloquium in Number Theory, Budapest (July 1987).

P. Era%

166

et al.

[6j P. Erdiis and M. Szalay, On some problems of J. DBnes and P. Turan, Studies in pure Mathematics to the memory of P. Turan, Editor P. Erdiis, Budapest (1983) 187-212 [7] P. Erd& and M. Szalay, On some problems of the statistical theory of partitions, to be published in the Proceedings of Colloquium in Number Theory, Budapest (July 1987). 181 P. Erdtjs, J.L. Nicolas and M. Szalay, Partitions into parts which are unequal and large, to be

published in the Proceedings of Journ6es ArithmCtiques, Ulm, Germany (Springer Verlag Lecture Notes). [9] E. Grosswald, Bessel polynomials, Lecture Notes in Mathematics No. 698 (Springer Verlag, 1978).

[lo] J. Herzog, GleichmWige asymptotische Formeln fiir parameterabhangige Partition funktionen, Thesis of Univ. J.F. Goethe, Frankfurt am Main (1987). [ll] J. Herzog, O n partitions into distinct parts a y , preprint.


167

THE FIRST CYCLES IN AN EVOLVING GRAPH Philippe FXAJOLET,* Donald E. KNUTH,** and Boris PITTEL? Computer Science Department, Stanford University, Stanford CA 94305,U.S.A.

* Permanent address: INRIA, Rmquencourt, 78150L.e Chesnay (France). ** Permanent address: Computer Science Department, Stanford University, Stanford, CA 94305 (U.S.A.). t Permanent address: Mathematics Department, Ohio State University, Columbus, OH 43210 (U.S.A.).

Revised November 1988

If successive connections are added at random to an initially disconnected set of n points, the expected length of the first cycle that appears will be proportional to n t , with a standard deviation proportional to nk The size of the component containing this cycle will be of order nf, on the average, with standard deviation of order nh. The average length of the kth cycle is O(n-f) that the graph has proportional to nf(logn)’-*. Furthermore, the probability is no components with more than one cycle at the moment when the number of edges passes an. These results can be proved with analytical methods based on combinatorial enumeration with multivariate generating functions, followed by contour integration to derive asymptotic formulas for the quantities of interest.

e+

A classic paper by ErdBs and Rknyi [6] inaugurated the study of the random graph process, in which we begin with a totally disconnected graph and enrich it by successively adding edges. Algorithms that deal with graphs often mimic such a process, inputting a sequence of edges until some stopping criterion occurs, based on the configuration of edges seen so far. To analyze such algorithms, we wish to estimate relevant characteristics of the resulting graph. For example, we might stop when the graph first contains a particular kind of subgraph, and we might ask how large that subgraph is. The purpose of this paper is to introduce analytical methods by which such questions can be answered systematically. In particular, we will apply the ideas to an interesting question posed by Paul ErdBs and communicated by Edgar Palmer to the 1985 Seminar on Random Graphs in Posnad: “What is the expected length of the first cycle in an evolving graph?” The answer turns out to be rather surprising: The first cycle has length KnA + O(nA) on the average, where

for a certain contour

r. The

form of this result suggests that the expected

This research was supported in part by the National Science Foundation under grant CCR-8610181, and by Office of Naval Research contract N00014-87-K-0502. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V.(North-Holland)

168

P. Fib@et ei al.

behavior may be quite difficult to derive using techniques that do not use contour integration. The methods to be described start with comparatively easy techniques of combinatorial analysis based on generating functions, and finish with more difficult (yet standard) techniques of complex analysis. The main novelty in this approach is the use of contour integration to give parametric estimates of a function that appears within an ordinary integral. Such methods may well find application in other studies of random graphs, so they are presented here in an expository fashion and in somewhat greater generality than is needed to solve the special problems used as examples. Section 1 introduces two basic models of evolving graphs that will be studied in the sequel, corresponding roughly to sampling with and without replacement. Section 2 discusses bivariate generating functions suitable for studying these graphs. Such generating functions can be used to derive probabilities in both of the models, as shown in Section 3. Asymptotic calculations in Section 4, based on the saddle point method, lead to results in Section 5 about the limiting distribution of first cycle lengths. Section 6 proves the main theorem about expected cycle length, and Section 7 derives auxiliary results about the expected waiting time and expected component sizes. The joint distribution of cycle lengths and edges is studied in Section 8, which also demonstrates a connection between waiting times and the parametric functions of Section 3. Section 9 extends the ideas to another problem in which we consider the first “bicyclic” component instead of the first cycle. An alternative approach to waiting times is considered in Section 10, where we also give an affirmative answer to a long-standing conjecture of Erdiis and RCnyi about the probability that a graph is planar. Finally we consider the first k cycles, in Section 11.

1. Models of graph evolution We shall consider two related ways to enrich an initially empty graph on the vertices (1, 2 , . . . , n } . The first procedure, called the uniform model, is the simplest: At each step we generate an ordered pair (x, y), where x and y are uniformly distributed between 1 and n , and all n 2 pairs are equally likely. The (undirected) edge x - y is then added to the graph. In this way we obtain a multigraph, which may have duplicate edges or self-loops x - x. Interesting variants of this model can be obtained by imposing other distributions on the pairs (x, y ), but we shall not pursue such generalizations in the present paper. Another way to generate a sequence of random edges may be called the permutation model; this model corresponds directly to random graphs as studied in the classic papers by Erdos and RCnyi [6, 71. Here we consider all (;)! permutations of the pairs (x, y ) with 1=zx < y 6 n to be equally likely, and we generate new edges x - y by considering the pairs as they occur in such a

First cycles in an evolving graph

169

permutation. The resulting graph contains no self-loops and no multiple edges; we are essentially sampling without replacement. The permutation model can be derived from the uniform model if we generate ( x , y ) uniformly but disregard any pairs with x = y or pairs that duplicate a previous edge. Our goal is to study the generation of random edges in such models until a cycle first appears in the resulting graph. (This would be the first time that a sequence of random “union” operations specifies a redundant union; see [111.) In the uniform model, the process might stop with a self-loop ( x , x ) , which is a cycle of length 1. Or it might stop with a duplicate edge (a pair ( x , y ) such that either ( x , y ) or ( y , x ) has occurred before); this is a cycle of length 2. In the permutation model all cycles have length 3 or more. For example, Fig. 1 illustrates a “random” graph on n = 100 vertices, based on the representation of n = 3.1415926 . . . in decimal notation. (Here the vertices have been labeled 00 to 99 instead of 1 to n.) A cycle first appears when the 45th random pair, ( 0 5 , 5 5 ) , is added. In this case the uniform and permutation models produce identical graphs, because the first cycle has length >2; in other words, no duplicate edges or self-loops are generated before there is a cycle. (We will see in Theorem 2 below that both models give the same graph with probability approaching & .) @ @ @ @ ) @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @,@@@@@ @-@

@@@)a@ @ @ @ @

@-@ 0-41-97

~

@-ll-@

!+ z 02

, - , - !@ -

8

g

@-@

@

@-@

zx

@-@

0-0 @-@-@-@-@

rg3

Fig. 1. The final state of a graph on 100 vertices that has evolved until a cycle first appears. The successive ordered pairs (31,41) (59,26) (53,58) (97.93) (23,84) (62,64) (33,53) (27,95) (02,88) (41,97) (16,93) (99,37) (51,05) (82,09) (74,94) (45,92) ( 3 0 7 8 ) (16,40) (62,86) (20,89) (98,62) (80,34) (82,53) (42,11) (70,67) (98,21) (48,08) (65,13) (28,23) (06,64) (70,93) (84,46)( 0 9 , 5 5 ) (05,82) (23,17) ( 2 5 3 5 ) (94,08) (12,84) (81,11) (74,SO) (28,41) (02,70) (19,38) ( 5 2 , l l ) produce nothing but free trees in the initially empty graph, but then ( 0 5 , 5 5 ) yields a cycle of length 4. (This cycle appears in the lower right comer.) At this point there still are 40 isolated vertices (shown at the upper left) that have not yet been mentioned.

170

P. Flajokt et al.

The permutation model of graph evolution is often called the “random graph process”. In these terms we can call the uniform graph model the “random multigraph process”. Let us recall briefly some of the main results of Erdos and RCnyi from [7], to establish a context for the facts proved below. (A detailed discussion of the theory appears in [2] and [12].) The following propeties hold “almost surely” (i.e. with probability tending to 1 as n+m) at the time when rn random edges have been added to an initially disconnected set of n vertices: Only isolated vertices and edges will be present when rn 1, equation (4.18) implies that sl,,(A) decreases exponentially to zero as n +CQ, because A - l / A > 2 In A. (If A = e‘, we always have e‘ - e-‘ > 2t.) On the other hand, the difference between A - l / A and 2 In A is of order (A - 1)3, so formula (4.18) says that e-~316

s,,,(l+ En-:) =

~

1 n6

2fi

(l+O(&)

+o($)+o($)),

when n-i(ln n)2 s E < inf. Thus q n ( l+ n-f) is of order na, but the nearby value sI,,(l + n-4 In n) is already exponentially small. In other words s ~ , ~ ( Ais) unbounded as n + m, but it decreases very rapidly when A passes 1. Lemma 1 tells us that sl,,(l - ni) is of order n i and that sl,,(l - n-f Inn) bni(lnn)-i. But the error estimates in both Lemmas 1 and 2 blow up when A is near 1, because the two saddle points at A and 1 come together; indeed, we have h‘(1) = 0 and h”(1)= 0 but h”’(1) # 0 when A = 1, so the magnitude of e”(=)near z = 1 has a graph that looks like a three-legged saddle-as used perhaps by Martian horsemen. A third lemma closes the gap in our knowledge by focussing

-

P. Flajolet et al.

182

on the region near A = 1:

Lemma 3. if lpl S n h and 1 S n f , the function s,,,(A) defined in (4.1) satisfies (4.19) uniformly in p and 1 as n +m, where

(4.20) the parameter

ir

is any positive number, and the contour

p a )is defined by

for t =s-2a;

It1

for -2a- - = t S 2 a ;

(4.21)

fort22a.

t

Proof. The integral over p a ) does not depend on a, since eP(lr*')has no singularities. We will find it convenient to let a be the positive solution to p = a - CY-'.

(4.22)

Let v = n-i and I = e-'". -4e-lr"

+ h(e-"?

A straightforward calculation proves that, for any s, =1

+ v3P(p, s) + R',

(4.23)

where the remainder term is

uniformly in any region where (1p1+ 1sl)v is bounded. The terms in v and v2 have cancelled out beautifully from the right-hand side of (4.23), thereby making the asymptotic behavior simple when we multiply by n = v-'. Formula (4.1), with z = e-'", becomes st,,(e-P") = n!n-"envfexp(-sfv+P(p, 4nie-"" s)+O((l~l+Is1)~v))ds, (4.24) where s traverses a path from 8 - inn; to + innf for some @. We will choose f? = 2nh and let s = 8 + i8 for -nnf S 8 s - ~ / ? this ; brings s to the point /3 - ~ ; S=is(-2/3) on the contour f i a ) defined by (4.21) and (4.22). (Notice that > a,since p s n k ) Then we shall continue with s = s(t) on I f a )for -28 =st s 26. Finally we take s = + ie again, for *#3 S 8 d m i . This contour keeps z = e-'" inside the unit circle.


183

c)

Let be the portion of fi" that we are traversing, namely the portion for -28 d t d 28, and let C,,be the other part of the contour just described. We will show that the integral over C,, is negligibly small. Indeed, the integrand has the nice monotonicity property that we described earlier in connection with (4.11), because C, corresponds to a circle in the z plane of radius r = e-Bv< (1 + e-Pv)/2 = (1 A)/2. Therefore the integrand is largest at the point where C,, meets fi;), and we will see that it is exponentially small there. On f i m )we have s = 0 ( n h ) , hence the 0 term in (4.24) is bounded and so is the term -sZv. We can therefore write this part of the integral as

+

(4.25) Now we have P ( p , s) = 3p3 - ips2 + 4s3,so when s = eni'3tthe real part of P(p, s) is 2n 3

+

3p3 - fpt' cos - 4t3 cos

3Jd = tp3 + fpt3 - 4t3; 3

its first derivative, j p t - t2, is negative when t 3 2a. When t = 28 at the end of E),we have p d f t , hence the real part is at most &t3

+ A t 3 - 4t3,

t =4th;

the integrand is indeed exponentially small when this point is reached. Furthermore when s = a + j l b t the real part of P ( p , s) turns out to be ia-1-l

6(Y- 3 - 3

*(a+ a - ' ) t 2 S

4,

(4.26)

Therefore lep(P.s)lis uniformly bounded on fina), and we have

ds = 0(1)

Ispep(p*s)I

fGU)

+

~(lalp-*)

for any fixed nonnegative power p. (When a is large, the integrand is 0(&) when It1 = O(a-t), and approximately zero for larger Itl.) The 0 terms can now be removed from the integral (4.27). Finally we can extend the domain of to the full contour obtaining (4.19). 0 integration from F) The integral (4.19) is investigated further in the appendix below, where the following result is derived as a special case of a general series expansion: (4.27)

5. Distribution of cycle lengths We can now combine the three lemmas with Theorem 1 and obtain the limiting probability distribution of cycle lengths:

P. Flajolet et al.

184

Theorem 2. For Jixed 1 as n-m, the graph evolution procedure of Section 1 generates a first cycle of length 1 with probability

in the uniform model, and with probability 1

em+A2'4 dA + O(n-i), 1 2 3,

A'-'in the permutation model.

(5.3) so it suffices to determine Jts,.,(A) dA. The integral for A 1- n-f is O(n-i), by Lemmas 4.2 and 4.3. Therefore we may restrict consideration to the interval O G A s 1 - n - f , when we find that the total error in (4.15) is n-l/l-n-'O((l

- A)-$)

dh = n-'O(nf) = o(n-a).

The integral from 1 - n-f to 1 of (1 - A)-$ is also O ( n d ) ;hence

And this is a Beta integral,

is, similarly, 4B(l, $), and we obtain (5.1). The difference Pal,nEquation (5.2) follows from (3.12) and (3.14). 0 Thus the cycle lengths approach a stationary distribution, without any normalization. Formula (5.2) was first obtained (without the error bound) by Svante Janson [lo], using a general theory of Poisson processes, and independently by BCla BolIobAs [3], using the theory of martingales. Since the extra factor eU2+A2'4 lies between 1 and ei = 2.117oooO2 for 0 6 A s 1, both probabilities and & have the same order of growth as I increases. Indeed, let

e,,

First cycles in an euoluing graph

185

Then we can write A/2 - A2/4 = 9 + O(A - l), obtaining

fi

PI = -+ O(1-t); 411

fiet 41; + o(1-n). p / =-

(5.7)

In both cases the average value Crpl is infinite; therefore the expected cycle length must be unbounded as n -+ a. The limiting probabilities pI for the uniform model obey simple recurrence relations:

21 Pl+1==PI;

21 - 21pp 21+Pr-

pa+1=-

Hence it is natural to wonder if the corresponding numbers BI and Bar for the permutation model satisfy similar recurrences, and in fact they do. First we note that

A similar integration shows that @21+3

= 2&l,

(5.10)

and it follows that we have the recurrence p1+2

(5.11)

= 2(1- 1)pI-l- 2&.

Is there a “simple” graph-theoretic explanation of (5. lo)? Setting 1 = 2 in (5.10) yields $5

= 1 - 7p3 - 8 4 ,

(5.12)

+

hence the values of @I can all be expressed in the form al + b4, clp4where al, bl, and cl are integers. Recurrence (5.11) is numerically unstable, but we can obtain accurate values 8 3

= 0.12160 82217 14483 58918,

(5.13)

p4

= 0.08491 50995 26335 99860,

(5.14)

by calculating PI and $I+1 accurately for some large 1 and then solving backwards. Do the fundamental quantities s,,(A) defined in (4.1) obey a recurrence relation? Yes, but it is a bit more complicated: We have 1

s1+2,n(A)

= (1 + A)s1+1,n(A)- A( 1 - -)s1,n(A)* n

This relation follows since s ~ + ~ , ~ has ( A ) the form

(5.15)

P. Flajolet et a(.

186

where f ( n , A) does not depend on 1. Differentiating the integrand with respect to z yields a function with nothing but zero residues, hence

The recurrence (5.15) can be used to calculate s&) the values s,+,,,(~)

= 0,

“backwards”, starting with

sn,,(A) = i n ! n-nA“-’e-nm,

(5.16)

and working down to slJA). This does not appear to lead to any simple consequences about the asumptotic behavior of qn(A). We can, however, use (5.15) to prove by induction that the coefficients of the polynomials (s,,,(A) sI+l(A))enmare nonnegative. Furthermore, (5.15) implies the remarkable identity

C h,,n(A) = n(sl,n(A) - A-’sZ,n(A)),

(5.17)

131

which can be used to study the variance of the cycle lengths.

6. The average cycle length We have seen in Section 3 how to set up a bivariate generating function F ( w , z ) for a set of stopping configurations, thereby allowing us to compute the probability GnFthat such confiurations occur in a graph of n vertices. But we can, of course, also use GnF to compute expected values, if F(w, z ) is a bgf in which each stopping configuration has been multiplied by a weight representing the random variable in question. For example, T(wz)leu(wzf’w is the bgf for stopping configurations with cycles of length I, hence

+- -

+

A ( w , z) = (T(wz) 2T(wz)’+ 3 T ( w ~ ) ~ -)e‘(wz)’w

is a bgf such that @,A is the expected cycle length in the uniform model. According to Theorem 1, this expected cycle length is

where h ( z ) is the familiar function of (4.2). Notice that we have n


187

c

0

x

Fig. 3. The weighted distribution functions a,(A) for n = 20 and n = 40.

Since sl,,(A) is exponentially small for A 3 1+ n-f In n, we need not consider large values of A. However, the presence of 1- z in the denominator of (6.2) means that values of a,(A.) near A = 1 will be crucial. A slight modification to the proof of Lemma 1 shows that the asymptotic formula

holds uniformly for 0 S A C 1- n-i(ln n)' as n +. 03. If we integrate this quantity as A varies from o to 1- n-f In n, say, we get

hence we-may conclude that the value of a,(A) is negligible except when IA - 11s n-4 Inn, if we can show that the integral of a,(A) over that range is of order nf. Fig. 3 shows the behavior of a&) for n = 20 and n = 40. As n increases, the function has sharper and sharper peaks, apparently reaching a maximum when A is very slightly greater than 1. The contour integral that arises when A is near 1 is just like the integral we studied in Lemma 3, except that there is an additional factor (1 - z)-'. If we set z = e-sv as in that lemma, we have

P. Flajolet et al.

188

uniformly in s, provided only that Is1 S 6 n i , since the series converges for I$(< 2nn3. Therefore the calculations of Lemma 3 can be applied almost without change, and we obtain

uniformly over all p such that lpl S n k Once again (Y can be any positive constant. Finally we can compute the asymptotic path length, proving the formula claimed in the introduction:

Theorem 3. The expected length of the first cycle in an evolving graph is Kn8 + O ( n 4 ) in the uniform model, and e$Kna + O ( n s ) in the permutation model, where

and

r = r")is the contour defined in (4.21).

Proof. Setting A = e-"'-',

A,

= e-"

-+I

, and Az = e+,-',

we have

where p , = nh and pz = -nK. (These magic constants will be explained below.) When p is between p2 and pl, the integrand factor exp(-pn-i) is 1+ O(n-A), so we can ignore it. Thus we obtain an integral whose integrand matches (6.8). This integrand is exponentially small as p+ -00, and we will prove in the appendix that it is O ( p - t ) as p-* m. Extending the integral from --oo to -oo, instead of from pz to p , , therefore introduces an error of nbO(p;i) = O(nA). To obtain the total expected length J$a,(A) dA, we must add (12 JT2)an(A)dA; this give a further error of O(n&),by (6.4), so we have established the result claimed for the uniform model. (If we had chosen p1 = n', these error estimates would have been O(n*-"/'),while the error in (6.9) would have been O(n5'-A));the value x = 6 gives the best bounds.) The permutation model requires an additional factor

+

exp(tA + $ A ~ = ) exp(2 + o ( p n - f ) ) , which is treated similarly. There also is a (negligible) factor e-2rv in the inner integral, because the numerator of the bgf in (6.1) must be changed from T ( w z ) to T ( w z ) in ~ order to get the expected value of I - 2. 0


189

7. Additional statistics

To find the variance of the cycle length, we can compute

which is the expected value of bl(1- 1). We get b,(A) from a,(A) by essentially changing (6.6) to

The net effect is to multiply the formula for JFan(A)dA by nil and to change the constant of proportionality by replacing s by s2 in the denominator of (6.8). Thus the expected value of fl(1- 1) is of order nf; the standard deviation is therefore asymptotically proportional to ni, somewhat greater than the mean. In general, if we have a bgf of the form (7.3)

where 1 is fixed as n 4~ 0 , the resulting value of j; cl,JA) dA will be of order n(2k-3)/6, by the same argument. Therefore we can grind out more facts by setting up appropriate bgfs. Let us introduce (temporarily) a trivariate generating function

in which the coefficient of f;iwmzn/n!is the number of stopping configurations with cycles of length 3 1 having m edges and n vertices, with j vertices in the cyclic component. If we take the partial derivative with respect to f; and then set f; = 1, we get a bgf for the expected value of j , namely

Di(1, w,2 ) =

U(wz)/w

(7.5)

(This follows from the well-known relation

T ' ( z )=

T(z) ~ ( 1 -T ( z ) )'

a consequence of (2.4).) Another derivative gives the expected value of j ( j - 1) and introduces another (1- T ( w z ) ) in ~ the denominator. Therefore (7.3) applies and we can state:

Coroilary 1. The expected size of the first cyclic component in an evolving graph on n vertices is asymptotically proportional to n4, and the standard deviation is of order n;.

P. Flajolet et al.

190

Proof. Take 1 = 1 in (7.4) for the uniform model and 1 = 3 for the permutation model; use k = 3 in (7.3) for the mean and k = 5 for the variance. 0 A similar derivation, with T ( ~wr)T(wz)’eU(wZ)’w/(l - T(wz)) in place of (7.4), shows that the expected number of vertices in the tree that leads into the first vertex x, of the first cycle is the same as the expected length of that cycle. The same holds for any individual tree in the cyclic component. Thus the cyclic component consists of @(na) trees, on the average, each of which has O(n4) vertices, on the average; a dependency between these two statistics causes the overall expected size to be e ( n i ) . We can find the limiting distribution of the number of vertices in the first cyclic component by considering the coefficient of in (7.4). Indeed, we have j’ L I ~ (w, ~ ,t)= f”7(wz)Jeu(wz)‘w; la1 I! and we can write this as a function of w and T(wz) by using identity (2.4), which says that wz = T(wz)e-*(’”’). Our general method now tells us to evaluate the integral

c’

C

asymptotically as n -P 0. We find as before that the only relevant contributions occur when A < 1, and an argument like that of Theorem 2 shows that a proper probability distribution appears in the limit: cordlary 2. For fixed j as n 403, the random graph evolution procedure generates a first cyclic component of size j with probability 9,,”=

1

5lj’5

2 - 1 e - J A mdr3. + O(n-%), j

3 1,

(7.7)

in the uniform model, and with probability

-

- 2)

q,II= 2-1jJ-2(j-j !

+ O(n-b),

j

dA

3 3,

(7.8)

in the permutation model. These limiting probabilities qJ and

4, sum to 1. We have, for example,

AJ-’e+m c i ~ =4 =4

1

A-’(Z

I,

(jAe-A)l -)j!

T(Ae-A) m 1 - T(Ae-A)

dA

I, 1

d

A =$

(1 - A)-j dA= 1.


191

Both q . and dj are of order j-3 as j grows; indeed, the substitution A = 1shows that qj = cj-' + O(j - 4 ) and qj = ejcj-3 O(j - : ) , where c =2%-*r(g). We have seen in Corollary 1 that the expected value of the component size is unbounded. Here is a table of approximate probabilities when j is small:

h

+

41 = 0.23096

6 3 = 0.01804

q2= 0.09501

a4= 0.02181

= 0.05649

6s = 0.02153

q3

q 4 = 0.03909 410 =

0.01214

qu, = 0.00504

46

= 0.02015

d l o = 0.01436 420

= 0.00754

The value of q1 is fi - te-% i erf(i), according to MACSYMA. To get the expected value of m, the number of edges, we can use the fact that n - m is the number of acyclic components. The relevant trivariate generating function is

and we have

7

Ei(1, w,2 ) = u(wz)E/(l, w,2 ) (7.10)

The factor w-l contributes a factor of n/A to the corresponding function e,,"(A), according to (3.13), hence we have (7.11) The integral ~ ~ s ~ , ~ ( A )is~ofA order / A n-t, by the results of Section 4, and we have in fact (7.12) Therefore the waiting time has a simple relation to cycle length probabilities: Corollary 3. The expected number of edges when an evolving graph obtains its first cycle ik i n + O(ni), in the uniform model. It k $(l -B3)n + O(n2) in the permutation model, where is the constant in (5.13).

a3

P. Flajolet el al.

192

Proof. Take I = 1 and 1 = 3 in (7.11), getting n(P21,n-4Pa2,n)or n(p23,n+ O(n2) as the expected values of n - m . 0 The variance can be shown, similarly, to have the respective values (7.13) We will examine another way to compute the expected waiting time in Section 10 below. Finally, let us investigate the number of vertices that remain isolated when the first cycle appears. The relevant trivariate generating function is (7.14) since we put the 5; marker on the unordered components of size 1. In this case we find

F ; ( I , w , z) = ZS'(W, z) = w-lT(wz)e-T(wz)Sl(w, z).

(7.15)

Corollary 4. The expected number of isolated vertices when the first cycle appears in an evolving graph is (7.16) in the uniform model, and

(7.17)

in the permutation model. MACSYMA finds the integral in (7.16) to be -e-'fi i erf(i); the coefficient of n is therefore -0.53808. The corresponding coefficient in (7.17) is ~ 0 . 4 2 0 4 6 .

8. Cycle lengths versus edges Let us now try to study the joint distribution of 1 and rn, the cycle length and the number of edges when the evolution procedure of Section 1 is applied to n initially disconnected vertices. The corresponding probabilities will be called P,,,,, in the uniform model and PI,,,, in the permutation model. We can express these probabilities directly from univariate generating functions, instead of using the more elaborate machinery of Theorem 1. Let C,,m,n be the number of stopping configurations in which the process can stop with a cycle


193

of length 1 and with m edges on n vertices. Then there are n - m components in the acyclic part, and we have

- n! Cl,m,n - (n - m ) ![z"]T(z)'U(z)"--". These numbers, incidentally, satisfy the recurrences

The corresponding probabilities, as we have seen in Section 3, are

Let us set A = 2m/n. Erd6s and RCnyi [7]observed that an evolving graph on n vertices changes its character dramatically when m grows so that A passes the critical value A = 1. It turn out that, for sub-critical graphs (A < l ) , the quantity Pal,,,, behaves very much like the function s1,,(A) in Lemma 1, except for a factor 2/n (which corresponds to dA):

Theorem 4. Zf 2mln = A < 1 as n+W, where 6 s A s 1 - 6, we have PI,m,n

= "-'-(l+o(-$)+O(-&)), n

uniformly in 6 > 0 and 1 3 1. Proof. We will apply the saddle point method to estimate the coefficient of zn in T ( Z ) ~ U ( Z ) " -~ /T((~z ) ) , thereby obtaining an asymptotic value of Csl,m,n. Again we replace z by ze-' in order to obtain a simpler integral:

where in this case we have

A h(z) = z - - In z + (1 - $A)ln(1- tz), 2 A 22

2-A 4-22'

h ' ( z )= 1 - -- -

(8.9) (8.10)

There are two saddle points, at z = A and z = 1, just as we observed for a different function h ( z ) in Section 4. (Is there an "obvious" reason why this should be true?) Again we have h"(A)> 0 and h"(1) C 0, so we want to integrate on the circular path 111 = A. The real part of h(Aeie) is now Acos 6 - $ A h A + +(I - 4A)ln L, L = 1 + $A2- A cos 6,

(8.11)

and its second derivative is

;

(2 A - cos 6 -A4

(

4) - A(2 - A)sin26)

L2

This is negative when cos 6 2 0, because

2-I. -L

4 s

2-12

-4=

- 4 ( 3 .

(1 - A/2)2

Furthermore (8.11) is less than %h(A)- A + In(1 + t A ) < %h(A) - ;A when cos 6 C 0. Therefore we can restrict attention once again to the neighborhood of 6 = 0, and the result is

(8.12)


195

Now we can use (8.4) to conclude that A’- 1 P*r,m,n-

(8.13)

9

as desired. 0 Theorem 4 gives us the promised “physical” interpretation of the parameter I in the machinery of Theorem 1: The running time m of the random process is represented by ;An, at least when A 1, integrating on the circle Izl= 1, to deduce that (8.15)

(Compare with (4.18)) The probability Pl,m,n is obtained if we insert the factor (1 - z ) into the contour integrand; this introduces the factor fez at the saddle point 0 = 0, and the result is P:,m,n

- 2 n ( A1- 1)

P*l,m,n

=

1 4m - 2n P*,m,n,

1+6dAS2-6.

(8.16)

The method of Theorem 1 seems preferable to working directly with the actual probabilities &,m,n for m 5 in, because S ~ , ~ ( A is )a “smooth” function of A by which we can use uniform methods like Lemma 3 to span the critical region near A = 1.

P. Flajolet et al.

1%

9. Bicyclic components

Let's turn now to a related problem that can be handled with similar techniques. Instead of stopping the random graph or multigraph process when a cycle appears, let's keep it running until the first time there is a bicyclic componenr-a component with more than one cycle. If the first such component contains j vertices it will have j + 1 edges. The solution to this problem sheds more light on the generating-function-based techniques we have been discussing. As before, we begin by defining and enumerating all of the stopping configurations in which our random process might terminate. The first bicyclic component can arise in one of two ways: Either (1) the final edge lies entirely within a component that was already unicyclic (a component that already contained a cycle), or (2) the final edge joints two different unicyclic components. Our experiences so far suggest that we ought to look first at the uniform model, in which each step selects from n2 ordered pairs ( x , y ) at random, since the uniform model tends to give formulas that are simpler than the ones arising in the permutation model. The generating function for unicyclic components on n labeled vertices turns out to be

V ( z )= j In

T(2)' 1 --T ( z ) +-+-+. l-T(z)- 2 4

T(z)3

6

..

Here's why: Every cycle of length 1 3 3 corresponds to 21 sequences of I rooted trees, because we can list the trees of the cycle by starting at I different places and we can traverse the cycle in two directions. Cycles of length 1 < 3 have the form ( x , x ) or ( x , y ) ( x , y); we will want to divide by 21 in these cases also, because of the weighting function 2"-'(m - l)! that will be applied later. (This weighting function assumes that a given multiset of m edges containing no bicyclic components can arise in 2"-'(m -I)! ways as a sequence { x , , y l ) . { x m - , , y m - , ) of ordered pairs; but the actual number of ways is 2"-'-"(m- l)!, where k is the number of 1-cycles and 2-cycles, so we want to introduce a factor of 5 for every such cycle.)

-

In case (1) the stopping configuration consists of a unicyclic component together with two special vertices ( x , y ) in that component, plus a set of any number of additional acyclic or unicyclic components. In case (2) the stopping configuration consists of an ordered pair of unicyclic components together with a vertex x in the first and a vertex y in the second, plus a set of additional acyclic or unicyclic components as before. Let 6 = t(d/dz) be the operator that multiplies the coefficient of Z" by n. Then the egf for stopping configurations in case (1) is (6'V(z))exp(U(z) V(z)), and in case (2) it is (6V(z))'exp(U(z) + V ( z ) ) .(The operator 6 selects a vertex, and

+


197

U(z) + V(z) enumerates acyclic and unicyclic components.) Using (7.6) we have 6V(z) = 4T(z)/(l- T(z))2; 62V(Z) = 1T(z)(l+ T(z))/(l

- T(z))4.

Therefore the overall egf for stopping configurations comes to

this is only slightly more complex than formula (2.7), the analogous egf for stopping configurations in the first cycle problem. Once again we need to work with bgf's, so that we have access to the number of edges. The appropriate bivariate generating function for stopping configurations in the uniform model is easily deduced from our derivation of (9.4): We have

And as in Section 3, we can state that S(w, t) expends to the sum Em,,S , , , W ~ Z ~ / ~ ! , where 2"-'(m - l)! s,,,/n2m is the probability that the process stops when the mth edge in introduced. As a check, let's look at the coefficients for small n: w2 2

S(w, z) =-z

+ 261w4z 3 + . + w 2+27w3Z2 + 4w2 + 60w3 16

.

When n = 3, the respective probabilities that we stop at time m = 2, 3, 4 are 21*1!.3!*4 _1 34*16 27'

2 2 * 2 ! * 3 ! - 6-0_ 58 -_ - 20 23.3!.3!*26136 * 16 81 ' 3' - 16 81 '

and these sum to 1 as they should. In general, we have

@,,S = 1 for all n 3 1;

(9.6)

the operator Gn of Section 3 applies to the bicyclic problem as well as to the unicyclic problem, and we can use the simplifications of Theorem 1 just as we did before. Now let's turn to the permutation model, in which cycles of lengths 1 and 2 are forbidden. The appropriate egf for cycles is therefore

a formula noted by Wright [15]. The egf for stopping configurations in case (2) is (f@(z))'exp(U(z) + p(z)), because we choose x and y in distinct components as before. But in case (1)the number of ordered pairs ( x , y ) is twice the number of

P. Flajolet et al.

198

edges not already present in the unicyclic component, so the appropriate egf for case (1) is

(e2- 3b)V(z)exp(~(z)+ P(z)). Adding these cases together and introducing w as before gives us the bgf for stopping configurations in the permutation model: S ( W , 2) = w

T ( w z ) ~ ( ~-O6T( w z )+ T(wz)’) 4(1- T(wz))f

-----

2

(9.8) n(n

- 1)/2 ) is the probability

If we write s ( w , z ) = Cm,ni,,,nwmzn/n!, then im,,/2m( that the first bicyclic component arises in the permutation model when the mth edge appears. The coefficients for small n are 5w5

3 ( ~Z), = z4 + 2

5w5

+ 37w6z5+ 5w5 + 79w6+ 367w7 p + .. . . 2

4

thus when n = 6, the process stops at time m = 5 , 6 , 7 with probabilities We have

e.

ynS= 1 for all n 2 4 ,

a,

&,

(9.9)

where Y, is the operator of Section 3. Notice that the coefficient of z3 in s ( w , z ) is zero; a graph on 3 vertices never has more than one cycle, so we should not look for bicyclic components in the permutation model unless n 3 4. But when n 3 4, we obtain a bicyclic component after at most n + 1 edges have been added. What is the size of the first bicyclic component? In the uniform model, the generating function

wT(f;wz)(2+ 3T(@z)) eu(wz)/w 4(1- T ( ~ W Z ) ) ~ (1 - T ( W Z ) ) ;

(9.10)

puts C’ into each stopping configuration whose bicyclic component contains j vertices. After differentiating with respect to g and setting f; = 1, we obtain an expression for the expected bicyclic component size: (9.11) A similar formula, with the same denominator (1 - T ( w z ) ) y , applies to the permutation model. If the factor w were not present, we would have a generating function of the form (7.3), with k = y ; the anoperator would then produce a result of order d Z k - ’ ) l 6 = n;. The factor w changes the integrand by A/n, and ,I-1 in the region where the integral becomes unbounded; hence the w essentially divides by n, and we can state the following result: Corollary 5. The expected size of the first bicyclic component in an evolving graph is of order n f . The standard deviation & also of order n f.


199

Corollary 6. The expected number of cycles in unicyclic components, at the moment when the first bicyclic component appears, Ij. 2 In n O(1). The expected total length of these cycles is proportional to n f.

+

Proof. For the first result, replace V(wz) in the exponent of S(w, z) by fV(wz); for the second, replace (1- T(wz))i in the denominator by (1 - cT(wz))f.Then in both cases, differentiate with respect to f, and set f = 1. 0 We can find the expected waiting time m by using the trick (7.9) that led to Corollary 3. In this case n + 1- m is the number of acyclic components, so the expected value of n + 1- m is (9.12)

depending on the model. In both cases the multiplication by U(wz)= T(wz) T ( ~ z ) ~yields / 2 a numerator polynomial in T(wz) whose value mod(1- T(wz)) is half what it was before. Since @,S(w, z) = Yn$(w, z) = 1, and since division by w contributes a factor of n, the waiting time must be asymptotically In.

10. Waiting times revisited When our goal is to find the average value of m, we can use another method based on generating functions for "going configurations" instead of stopping configurations. Namely, if fmn is the number of graphs with n vertices and m edges such that the random process is not stopped, we can use this information to calculate the probability that the process is still going after m steps. The sum of these probabilities, over all m, is the expected waiting time. In the first cycle problem, a going configuration is simply a. forest (a collection of edges with no cycles); hence the bgf for going configurations is simply (10.1)

Each going configuration occurs with probability 2"m !/nh in the uniform model, so the expected waiting time for a graph with n vertices is Cm2"m!f,,,,/n*". The operator a, of Section 3 computes Cm2'"-'(m - l)! fmn/nh, so it's almost what we want. We can obtain the desired operator for going configurations by first multiplying by W , getting the bgf Cnfm,nwm+lzn/n!; then applying @, to get C 2"m! fm,n/n2m+2;then multiplying by n2. In other words, the expected waiting time in the uniform model is n2@,wF(w, z).

(10.2)

Alternatively, we can obtain the desired operator by first differentiating with

P. fijolet

200

respect to w (getting @., In other words,

C,

et

al.

mfm,nwm-lznln!),then multiplying by 2w and applying

gives the same result as (10.2) (We must add in the by differentiation.) indeed, we have the operator identity

which is annihilated

a

n2@,w = 2@,,w-, dW

(10.4)

a d valid when applied to any bgf with F(0, z ) = 0. Since w -= -w - 1, we can dw dw rewrite (10.4) as follows: (10.5)

Comparing (10.3) with our formula (7.10) for the average of n - m yields the interesting identity 1 eu(wz)/w Qn( - 1- T + 3 T Z ) = n - 1, T = T (wz), (10.6)

(-1 - T

7)

which does not obviously follow from (10.5) and any other identities that we know. It may be possible to find a family of formulas such as this, allowing us to deduce nonobvious relations between different statistics on random graphs. In the permutation model, the relevant formula for expected waiting time is

as in (10.3). There is apparently no simple analog of (10.2), although we can derive a formula that is somewhat like (10.5):

)

(

+ 1 Y " = Y n( 1 + w ) - - w - ' aw a

)

,

(10.8)

The identity analogous to (10.6) is 1 yn((i=T-

1-T

+TZ-

eu( w z ) / w

~

3 2-4) +

lw) = n - 1,

(10.9)

T = T(wz), valid for n 2 3. The bgf for going configurations in the problem of bicyclic components is eu[wz)/w+ V ( w ) or eu(wz)/w+v[wz) , (10.10) depending on the model, because the process keeps going if and only if the graph


201

components are acyclic or unicyclic. The formulas in (9.12) now lead, via (10.3) and (10.7), to further identities like (10.6) and (10.9): T2 3 3 - 4 T +6T2 e"(wr)lw T)4 =n;

@n((w-8+8(1-

) G)

(10.11)

(10.12) (Again T stands for T(wz), and the identity for Ynholds only when n 3 3.) Is there is simple combinatorial or algebraic principle that accounts for amazing formulas like this? We have observed in Section 9 that the waiting time for the first bicyclic component is approximately in; thus, the graph tends to become bicyclic when m passes the critical value where random graphs rapidly gain a complex structure. It is interesting to look more closely at this transitional phase, by studying the probability that there is not yet a bicyclic component when m ==in. For this purpose we can combine the ideas used to prove Lemma 3 in Section 4 and Theorem 4 in Section 8.

Theorem 5. Let A=2mln =e-Pv, where v = n - f . Then the probability that a random graph with n vertices and m edges has no bicyclic component is (10.13) uniformly for lpl d nh, where r = r'" is the contour defined in (4.21).

Proof. We have (10.14)

where P(z) is defined in (9.7) and N = (g). Let h ( z ) be the function defined in (8.8); then, as in that derivation, (10.15) Let z = e-'". A tedious but straightforward calculation shows that (10.15) equals (10.16)

P. FIajoIet et al.

202

if we argue as in Lemma 3. Here are the new details: We rewrite (8.8) in the form 1

n-rn h(2) = 2 - n

+

and analyze h(e-"') by using the uniform estimates e-pv= 1 - pv 0(p2v2) and ln(1- (e-"' - 1)2) = -s2v2 s3v3+ 0(s4v4). To show that the integral over C, is negligibly small for this new integrand, we note that for l h p s 8 s n ; n f the derivative of the real part of h(e-pv+ie")is

+

where r = e-Bv.This is negative, because 1 - f r > 4 and 1 - $r a 1 - 4A when l p l s nit. The integral corresponding to (4.25) is

+

s 4 e ~ ( ~ ~ s ) - q 3 / yo 1

\ern)

( ~+~0 ( )~ 2 ~ 2 +~ )0 ( ~ 4 ~ ) ) d ~ .

Instead of (4.26) we need the real part of P(p, s) - p3/6, which is -16(r 3 +

~ ( Y - ~ ( ( Y + ( Y - 1 ) f 2 ~ ~ ;

cm).

hence (eP(p*s)-p3'61 is uniformly bounded on Furthermore the quantity n ! / ( n- m ) ! in (10.14) can be shown to equal Get+p'/6-n 2n -m (1 + 0((1+ p4)v)). (10.17)

(z)

Multiplying (10.16) by (10.17) yields (10.13). 0 (Theorem 5 applies also to multigraphs: If we use

in place of (10.14), we obtain the same asymptotic result (10.13). Multigraphs are assumed to be generated as in the uniform model, with each of the n2 edges ( x , y ) equiprobable. Hence each self-loop ( x , x ) occurs with probability l/n2, while edges x - y with x # y occur with probability 2/n2 since they arise from either ( x , y ) or (y, x ) . ) When p- -00, the value of the integral in (10.13) is exponentially small; in fact it is O(eP3'6-pn), because

On the other hand, when p+ +a we can prove that the integral is 1+ O ( p P 3 ) , by integrating on the path s = p + i y / f i for --oo < y < 00. For we have P ( p , p + i y / G ) = -y2/2 - iy3/(3p*); the integral can be restricted to lyl C In n, in which range the integrand is e-y2aitimes 1+ fiyp-4 - 2'Y c1 - 3 + 0 N Y 2 +Y6)cL-3).


203

Therefore the random graph process almost always keeps going without bicyclic components until the number of edges is on the order of !ne-pv= in - 4pn:. If we take M large enough, the probability is 21- E that the first bicyclic component occurs when in - Mn; =sm S in Mn3. Informally we can say that the graph almost certainly becomes bicyclic when the number of edges is an + o(n3). When A is strictly less than 1, say A 6 1- S, we can show that xm,n= 1- O(n-'S-') - O(n-$6-3)by integrating on the contour z = heie as in the proof of Theorem 4. (See [7, Theorem 5el.) We can now sharpen the result of Erd6s and RCnyi statedin (8.14):

+

CoroUary 7 . Let L be a set of positive integers, and say that an L-cycle is a cycle whose length is in L. Then Pr(Graph or multigraph with n vertices and m edges has no L-cycle)

(10.18) ItL

if limn+- 2m/n = A < 1.

(This result applies to graphs as well as multigraphs; we assume that 1 4 L and 2 4 L when we are considering graphs. A multigraph can have self-loops (l-cycles) and/or repeated edges (2-cycles), but a graph cannot.) Proof. The multigraph either contains a bicyclic component or it does not. The first case occurs with probability O(n-4). In the second case we want the probability of a "going configuration" that consists entirely of acyclic components and unicyclic components whose cycle lengths are 4 L. The number of such configurations is

IdL

so we are able to complete the estimates by repeating almost verbatim the argument of Theorem 4. 0 If we set L = (1, 2) in (10.18), we get the asymptotic probability that a random multigraph is a graph, namely e-A.n-A2'4 . If we set L = {3,5,7,9, . . .}, we get the asymptotic probability that a random graph is %-colorable,namely

1-Ai exp(bA - ln(1- A2)) = 1 + A

a

(-)

(10.19)

Otherwise [7, $101, such a graph is almost surely 3-colorable when A < 1. Choosing L = {k + 1, k 2, . . .} in (10.18) gives the limiting distribution of the

+

P. Flajolet et al.

204

longest cycle in a random graph: All cycle lengths are 6 k with probability (10.20)

(An analogous result has been derived by Pittel [13, Theorem 13, for random graphs in which each edge occurs independently with probability 1ln.) Erdos and RCnyi [7, 481 stated that, if r is any real number, the probability that a graph with n vertices and i n + rnf edges is non-planar "has a positive lower limit, but we cannot calculate its value. It may even be 1, though this seems unlikely". We can now show that this probability is definitely less than 1. Indeed, a graph with n vertices and i n + rnf edges has p == 2rn-a in the hypothesis of Theorem 5, so the probability that it has no bicyclic component (and is therefore planar) approaches the limiting value stated for p = 0 and a = 1. We can prove, in fact, that this limiting value nnn,,is rather large:

Corollary 8. The probability that a graph with n vertices and an edges has no bicyclic component is O(n-i).

+

Proof. The contour integral in (10.15) is particularly interesting when 1 = 1 because it has a three-legged saddle point. One way to evaluate it is to consider a path of the form z = 1 teh~3n-4for t SO; this accounts for half of (10.15), and the result turns out to be

+

-fen

-1

[fie-'"

dt(1 + O(n-4)).

$6.

We will see in formula (A.8) below that this integral is The auxiliary e?-"2"n(l + O(n-')) when m = in. 0 coefficientn ! / ( n- m ) !(E) is

+

Erdos and RCnyi 171 also remarked that a graph with i n q,fi edges has a cycle with any given number of diagonals, with probability -1 when w,++m and n--t m. However, we have just proved that this is not true when on= nb. Therefore the claim that a graph with exactly f n edges has positive probability of nonplanarity might also be false; an explicit proof of disproof would be desirable.

11. The first k cycles As a final example of the techniques we have been considering, let us study the distribution of the first k cycles that appear in an evolving graph. We have seen in Section 10 that this problem is well-defined, at least asymptotically, because the first cycles in a sufficiently large graph will almost always occur in distinct components.


205

For simplicity let us once again consider the uniform model first. We will run the random multigraph process until there is either a bicyclic component or a set of k unicyclic components, whichever occurs first. In the latter case we let 11, 12, . . . , l k be the lengths of the first k cycles, in order of appearance. A stopping configuration in the non-bicyclic case will consist of a sequence of cycles of rooted trees, having respective lengths (Il, . . . ,l k - l ) , together with a sequence of lk rooted trees, plus a Set of number of unrooted trees. A cycle of 1 rooted trees has the egf T(z)'/21, as discussed in (9.1). Therefore if we form the multivariate generating function

the coefficient n! [Cp - - . . . , g k , w, z ) will be the number of stopping configurations with m edges, n vertices, and cycle lengths (11, . . . , l k ) . In order to convert these coefficients to probabilities, we need to consider how many of the n sequences (xl, yl) - - (xm, y m ) of edges will yield a stopping configuration with parameters m,n, 11, . . . ,l k . For this we need a slight generalization of the argument at the beginning of Section 3; the appropriate factor is now not 2"-'(m - l)!/nZ" but rather (11.2) where

Li = l1+ -

- + lj. *

(11.3)

The reason is that the (rn - l)! permutations of the m - 1 non-final edges are not all admissible. Exactly l k - 1 / L k - 1 of them have the final edge of the (k - 1)st cycle occurring after all the edges of the first k - 2 cycles; and l k - z / L k - 2 of these have the final edge of the (k - 2)nd cycle occurring after all the edges of the first k - 3; and so on. The stopping configurations in the bicyclic case can be ignored, because we know that this case occurs with vanishing probability as n --* m; but we might as well describe the generating function, so that we can see how rapidly the probability approaches zero. We mimic the derivation of (9.4): Either k 3 2 and there is a unicyclic component with two marked vertices, plus an additional set of acyclic and (at most k - 2) unicyclic components; or k 2 3 and there are two unicyclic components with marked vertices plus an additional set of acyclic and (at most k - 3) unicyclic components. The egf is therefore (11.4)

P. Flajolet et al.

,206

Converting to a bgf gives a formula like (9.5) except that it has the form wfk( T (w2 )) eu(W L ) (1 - T ( w z ) ) ~

l ~

(11.5)

where fk is a polynomial. By reasoning as we did after (9.11), we conclude that the @, operator produces a result that is .(n-1n(8-3)16)=0(n-g), for every fixed k. We can now determine the asymptotic probability that a given sequence of cycle lengths will appear: Theorem 6. The probability that the random multigraph process produces the first

k cycles in distinct componenfs with respective lengths (11, 12, . . . , l k ) is 91-k

(11.6)

for a l l f i e d 11, . . . , 1&21, where L, is defined in (11.3) and p I k defined in (5.6). The same formula holds for the random graph process, if p is replaced by p and if we require 11, . . . , l & > 3. Proot, The desired probability, according to (11-1), (11.2), and (3.1), is

91-k

(11.7) plus O ( n - i ) for the probability of failure due to the early occurrence of a bicyclic component. And @,,(T(wz)'eU(wz)'w)is the probability that the first cycle has length I, computed in Theorem 2. This proves (11.6) in the uniform model; the same ideas apply to the permutation model, with minor changes. 0 The probability distribution in Theorem 6 was first derived by Svante Jansen [lo], without the error term, in the case of random graphs. We can show that the sum of probabilities (11.6) over all ( I , , . . . , 1,) equals 1, by using the identities m

OD

(11.8) already mentioned in (5.8) and (5.10). Notice that the asymptotic probability in the uniform model that the first k cycles will all be loops of length 1 is P&/2'-'(k - I)! = 1/(3. 5 ' ' (2k + 1)). On intuitive grounds we expect the second cycle to be larger than the first, and


207

the third should be larger yet, because the trees that yield cycles gradually get bigger. And indeed, this is true:

Theorem 7 . The average length of the kth cycle, for fixed k, is of order nf(1og n)k-l.

Proof. It suffices to give the proof for the uniform model, since the other model is similar. The basic idea is to apply the identity

which is readily verified by induction. The average value of

c

lkp(ll, *

* * 3

lk,

n),

/k

is (11.10)

!1,....!k*1

where P(ll, the bgf

. . . ,l k , n) is the probability in (11.7);thus we want to apply @,

1 k-1 1 T eu(wz)/w , 2 k 4 ( k - l ) ! r n ('"1-T)

to

(11.11)

where T = T(wz). And it should be clear from the calculations in Sections 5 and 6 that the principal effect of each additional factor In 1 / ( 1 - T ) is to multiply the inner integral by In 1 / ( 1 - e-") = (3 Inn + O(ln(1 + Ip1))(1+ O(n-4)). Therefore the result is 8(logn)k-' times the result of Theorem 3. 0 In this proof we have defined the random variable lk to be zero if the first k cycles are not well separated, i.e. if they do not fall in distinct components. This seems reasonable because the concept of kth cycle becomes murky when many cycles are formed simultaneously. [See the Addendum following the References.] A somewhat paradoxical situation arises if we ask for the conditional expected length of the first cycle, given that the first k cycles appear in different components. For example, suppose k = 2. Let a, be the unconditional expected length of the first cycle; let b, be the probability that the first two cycles are well separated; and let cJb, be the conditional expected length of the first cycle given that the first two cycles are well separated. Then we find

P. Fi'ajolet et al.

208

where T = T ( w z ) . Since T2/(1 - T)'= T/(1- 7 ')' - T/(1- T), we have c, = f(a, - 1) exactly. Thus the expected value 4 6 , is asymptotically only half of a,, although both quantities represent the expected length of the first cycle, and although we are conditioning on an event that almost surely occurs! The reason is that the distribution of first cycle lengths has a tail that decays very slowly; and cases when the first cycle is extremely long are much more likely to attract the second cycle into the same component. Similarly, it can be shown that the conditional expected length of the first cycle, given that the first k cycles appear in separate components, is asymptotic to ~l-~a,,.

12. Concluding remarks We have shown that a combination of generating functions and contour integration can resolve problems that apparently could not be treated successfully with the techniques that have previously been applied to random graphs. Many of the previous techniques, like the laws of large numbers, can be based on special cases of contour integration with the saddle point method; the approach in this paper may have succeeded primarily because we were free to use the saddle point method in a more general context. It would be interesting to push the techniques further, for example by determining the asymptotic value of L, - Knd when Ln denotes the expected first cycle length.

Appendix. Evaulation of integrals Let us complete this discussion by studying the behavior of the integral in Lemma 3, Eq. (4.19), and by finding a numerical estimate for the constant K in (6.8). This proves to be interesting exercise in the theory of functions. First let's warm up by discussing some simplfied functions that will help us get to know the territory. If x is a real number, we define f ( x ) = I-:exp(-it

- xt2 + ir3/3) dt,

exp(irt + it3/3) dt.

g(x) = -m

The motivation for f ( x ) comes from the integral in (4.19), which reduces to a multiple of f ( ( a+ a-')/2) under the substitutions s = a - it and p = a - a-',if Since we integrate on a path from a - i a to a + im instead of on the contour pa). our main application of f ( x ) has x = (a+ a - ' ) / 2 2 1, we can assume that x 2 1 in


209

f ( x ) . We have f ( x ) = ]-e:-x’

cos( -t

+ t3/3) dt,

(A.3)

so f ( x ) clearly converges for all x > 0. We will prove that the related function g ( x ) converges for all real x (even through the integrand in its definition always has magnitude 1). If a is any positive number, we have

dt eo3/3+a~x~

dt = O ( K 2 ) ;

a similar bound applies if we integrate from - R to - R + ia. Hence we can shift the path of integration upward, without affecting the value or the convergence of the integral: g(x) =

r

exp(ir(t + ia) + i(t

+ aQ3/3) dt,

a 2 0.

-m

There is now a term --at2 in the exponent, so g ( x ) must indeed converge. In particular, we have m

g(a2 - 1) =

I_, exp(a - $a3- it -

at2 -k 3 l t

- e‘-2.3’3f(a).

1dt (A.5)

Thus f(x) = e””3-xg(x2 - 1). When x is large, we have f(x) = f i x - ; + O(x-$), hence g(xz - 1) must be mighty small. Another formula for g ( x ) can be obtained by rotating the path of integration: g ( x ) = 2%rexp(irr - it3/3) dt 0

exp(ix5t + it3t3/3)dt) exp(irf;t - t3/3) dt) where

P. Fhjolet et al.

210

(The integral on the arc Reie for 0 Q 8 Q n/6 is negligible for large R, because the magnitude of the integrand is exp(-4R3 sin 38 - xR sin O ) . ) Equation (Ah) will be our key to evaluating f(x) via g(x), because we can expand exp(irct) into a convergent power series in t. Then we can interchange summation and integration, evaluating the resulting integrals by using an analog of (4.14):

It follows that

The real part of ck+’ik is cos($k + &c, when k = (0,1,2)mod 3; hence g(n) = 3-4

which is respectively (jfi, - $fi, 0)

2 (3k (3;X)3k ((3k + l)T(k + 4) - 3$xr(k + 3)). + I)!

(A.lO)

ka0

This series converges for all n; hence g ( z ) and f ( z ) are actually analytic functions in the entire complex plane. We can write (A.lO) is a difference of two hypergeometric series of type g ( x ) = 3-$r(f)F(; f ; 4x3) - 3h-(f)F(;

3; 6x3).

(A.ll)

This representation allows us to deduce that g ( z ) can be expressed as an Airy function, hence as a modified Bessel function of fractional order: 22 f. g ( z ) = M i ( z ) =I 32 K&$).

(A.12)

Equation (4.27) follows from the fact that f(l)=e-fg(0). In general, our derivation leads from (4.19) to the asymptotic formula (A.13) if we assume that the 0 term in (4.19) is of lesser order. A somewhat different approach appears to be necessary if we want to evaluate the constant K numerically. Let us consider the value of the inner integral in (6.8),


-3.0

-2.0

-1.0

0.0

21 1

2.0

1.0

3.0

p

Fig. 4. The function K ( p ) whose integral yields the first-cycle constant K .

for fixed p ; this is the quantity that yields K when integrated over the range - w < p c 00. It is plotted for -3 S p S 3 in Fig. 4. We can argue, as we did following Theorem 5 in Section 10, that K ( p ) is exponentially small when p+ -w, and that K ( p ) is of order p-j when p-, +m. Our strategy for evaluating K will be to find a reasonable way to compute K ( p ) when 1p1 is small, together with a precise asymptotic estimate of K ( p ) when p is large. First let's assume that p is near zero. We have, by definition, (A.15)

where the contour r begins at me-'d3 and ends at wem after crossing the positive real axis. Then the quantity u = s3/3 describes a contour that starts at -a just below the negative real axis, hugs the bottom edge of that axis and circles the origin counterclockwise, then returns (just above the axis) to -m. This is a contour C for which we have Hankel's well-known formula 1

1

(A.16)

Hence we can use the substitution s = 3fuf to write

2ni =-x 3

kaO

(-33x)k

k! r(l- 2k/3).

(A.17)

P. Fkajolct ct al.

212

(The series are absolutely convergent.) This is the desired formula by which we can compute K ( p ) when Ip( is not too large. It can be expressed hypergeometrically in the form

Incidentally, it is interesting to apply the same idea to the integral (4.19). We obtain a formula that looks rather different from ( A . l l ) and (A.13), namely

The quantity in parentheses is a confluent hypergeometric function,

wi, f , -W). 2f3in:

’

equating (A. 19) with (A.13) yields a known identity between Airy functions and confluent hypergeometrics [ l , Equations 13.1.29 and 13.6.251. We can also prove equality between individual “halves” of (A. 11) and (A.19), using the hypergeometric identity e-”’F(u; ZZ; z ) = F(;a + 4; f6z2).

(A.W

Now let’s consider K ( p ) as p + m . Our experience with the similar integral (10.13) in the discussion following Theorem 5 suggests that we try integrating along the path s = p + iy/fi. (The contour fi*) can be “straightened out”, as we found in (A.6). as long as the tails remain exponentially small.) O n this straight line the integral reduces to (A.21) and we can obtain an asymptotic formula by expanding the real part of the integrand as eCvzRtimes a power series in y2 and Up3. Namely, if we set u = p - ; for convenience, we have I exp(-ivy3/3) exp(+iuy’/3) -i + 1 - ivy ) = 1 2 1 +ivy

18yz + 6y4 + y6 18 U2

1.

1 0 8 ~+ ’ 12y’” + y + 1944y4+ 648y6 +1944 u4 5 2 4 8 8 0 ~+~ . + Y’’ u6 + - . (A.22) l2

*

*

524880

Placing this inside (A.21) and applying (4.14) gives

(A.23)


where

+ (1

213

+

3)6 (1- 3 - 5)l - _ - 17 18 6’ (1 *3)1944 (1 - 3 * 5)648+ - * - + (1 * 3 * 5 * 7 * 9 - 11)l - 1801 -c2 = 1944 72 ’ c1=

(1)18

*

+

and so on. However, we need to justify this expansion carefully because CkaO(- l ) k ~ k / p 3 k is divergent. The key is to show that (A.23) is a strictfy enveloping series, in the sense that its partial sums alternately overshoot and undershoot the true value of K ( p ) . The enveloping property is not difficult to prove, because we can show that series (A.22) is enveloping with respect to v2.If we remove all terms on the right side of (A.22) that have degree greater than 2k in v, the resulting sum is an upper bound or a lower bound for the function on the left side, according as k is even or odd, for all real values of v and y. This property holds, because the left side is cos(vy3/3) - vy sin(vy3/3) 1 v2y2

+

(A.24)

and because the power series for cosine, sine, and (1 +v2y2)-l are strictly enveloping [14, Problem, 1.1421. Incidentally, one can readily verify that the coefficient c,, of (A.23) can be expressed as a rathersimple sum, 2n

cn=

C

(2n+2k)! + k ) !k! ’

k=O 2”+&3&(n

(A.25)

because each term y i d of (A.22) arises from precisely one term in the expansion of (A.24). The denominator of c,, turns out to be exactly 22n--z(n)3(3n-v3(n)))n,

(A.26)

where vr(n)denotes the sum of the digits of n in radix-r notation. Since the series (A.23) for K ( p ) is enveloping, we can integrate it term by term to get an enveloping series for the tail of the integral, (A.27) For any fixed p this series is divergent, but we can find a “best” place to stop it (where the terms begin to increase in magnitude). For example, when p = 5 , the sum of the terms involving ck for k 6 2 1 on the right of (A.27) is 0.4458165587745; and the partial sum for k S 2 2 is 0.4458165587784. So we know that 0.4458165587745
q2 - q we obtain that t*(H) > q - (l/q). Apply Theorem 2.1. Hence H is a multihypergraph obtained from the truncated projective plane P. Denote the multiplicities of the edges E E E(P) by m ( E ) . We claim that

4 - 93 r D k l

(3.5)

holds for every edge E. Indeed, if m(Eo)> [ D / q l , then

This is less than the right hand side of (3.4), thus (3.5) follows. Let t ( E ) = [ D / q l - m(E). Then t is an i-cover of the dual of P, that is C t ( E ) a ti(A) 5 ti(q).Finally,

IEWI = q 2 [ D / q 1- EEP Ct

( ~s)q 2 [ D / q 1- Z i ( q ) *

0

The case D = q2 - q also follows from the above argument.

Acknowledgement The author is indebted to E. Boros for the Example (1.4).

Note added in proof The main result of the second part (Theorem 3.4) verifies a conjecture of Bierbrauer [ll]. He also conjectures that GyBrf6s’ lower bound [4] for f ( D , q + 1) coincides with the value given in Theorem 3.4. Moreover, he determines f(D, 6) = 5D - 3 for D 3 89, D f (mod 5 ) .

References [l] B. AndrBsfai, Remarks on a paper of GerennCr and GyBrfBs, Ann. Univ. Sci. Eotvos, Budapest 13 (1970) 103-107.

226

2. Fiiredi

[2] J. Bierbrauer, Ramsey numbers for the path with three edges, Europ. J. Combin. 7 (1986) 205-206. [3] J. Bierbrauer and A. Brandis, On generalized Ramsey numbers for trees, Combinatorica 5 (1985) 95-107. 141 J. Bierbrauer and A. GyBrfb, On (n, k)-colorings of complete graphs, Congressus Numer. 58 (1987) 123-139. (51 F.R.K.Chung, Z. Fiiredi, M.R. Garey, and R.L. Graham, On the fractional covering number of hypergraphs, SIAM J. Discrete Math. 1 (1988) 45-49. [6] P. Erd6s and R.L. Graham, On partition theorems for finite graphs, in Infinite and finite sets (A. Hajnal et al., eds.) Proc. CoUoq. Math. Soc. J. Bolyai 10 (Keszthely, Hungary, 1973, North-Holland, Amsterdam, 1975) 515-527. [7] P. Frank1 and Z. Furedi, Finite projective spaces and intesecting hypergraphs, Combinatorica 6 (1986) 335-354. [8] Z. Fiiredi, Maximum degree and fractional matchings in uniform hypergraphs, Combinatorica 1 (1981) 155-162. [9] L. Geren&r and A. GYMAS,On Ramsey-type problems, Ann. Univ. Sci. Eotvos, Budapest 10 (1%7) 167-170. [lo] A. GyBrfAs, Partition covers and blocking sets in hypergraphs, (Hungarian), PhD thesis, MTA SzTAKI Tanulmhyok 71 (Budapest, 1977). [ l l ] J. Bierbrauer, Weighted arcs, the finite Radon transform and a Ramsey problem, Graphs and Combinatorics, submitted.


227

A DENSITY VERSION OF THE HALES-JEWElT THEOREM FOR k = 3 H. FURSTENBERG and Y. KATZNELSON Dept. of Mathematics, Hebrew University, Jerusalem, Israel

0. Introduction

For k 3 2 and n > O we denote by Q n = Q n ( k ) = ( 0 , . . . ,k - 1 ) “ the set of all the words of length n on k digits. The well known theorem of Hales and Jewett ( [ 5 ] ,Theorem 1.1 below) states that for every k and I > 1, there exists an integer N(k, I) such that if n > N ( k , I), then for any I-coloring of Qn(k) there exists a monochromatic subset of k points forming a “combinatorial line” (see Definition 1.1 below). This extends the famous van der Waerdon theorem as well as its multi-dimensional version (Gallai’s theorem). A “density version” of this theorem would state that such combinatorial line exists in any subset A c SZ,(k) with relative density bigger than E, provided n > n(k, E ) . This would be a far reaching extension of SzemerCdi’s theorem, and of the various extensions thereof ~ ~ 3 1 . For k = 2 a stronger result was known years before the Hales-Jewett theorem, namely Sperner’s lemma. Here we deal with words with only zeros and ones as digits, i.e. indicator functions of subsets of [0, . . . ,n ] and the two points of a combinatorial line are the indicators of sets A, B such that A c B. Sperner’s lemma states that a collection of subsets of [ l , . . . , n ] which has more than (&,) elements does have pairs A, B as above. This is clearly best possible since the set of all subsets of exactly [ n / 2 ]elements does not. The purpose of this note is to announce the density version of the Hales-Jewett theorem for k = 3 and to outline the main elements of the proof. The method we use is “ergodic” in the spirit of [2] and [ 3 ] ,and the first step is to show that the theorem in question can be formulated as a statement regarding a certain family of measure preserving transformations acting on a (probability) measure space. The phenomenon that appears here as well as in our treatment of SzemerCdi’s theorem, is that when a family of measure preserving transformations having a certain structure acts on a measure space, a set of positive measure will necessarily return to itself (“recur”) under certain combinations of transformations. This recurrence phenomenon for sets of positive measure is then translated into the appearance of certain patterns in subsets of a sufficiently large structure, provided the density of the subset is bounded below. Research was partially supported by NSF Grant No. DMS86-05098. 0012-365X/89/$3.500 1989, Elsevier Science Publishers B.V. (North-Holland)

H. Furstenberg, Y. Katznelson

228

Having converted the problem into one regarding measure preserving transformations of a measure space, we can use the machinery of ergodic theory, and, in particular, we can associate to each measure preserving transformation the unitary operator induced by it on the L2-space of the measure space. We now have available the methods of functional analysis. An important tool will be the fact that in a family of unitary operators with a certain minimal multiplicative structure one can find sequences converging weakly to projection operators. In developing this tool there is an interplay of functional analysis and combinatorial theory, and, not surprisingly, we will need an extension of the Hales-Jewett theorem to “infinite patterns” in the spirit - but slightly stronger than - results of Carlson and Simpson (cf. [l]). This result will also generalize the theorem of Hindman, whose significance for SzemerCdi type results we have met with already in [3]. There, Hindman’s theorem enables one to develop ergodic theory for “IP-systems” of measure preserving transformations. The new feature in the ergodic approach to the Hales-Jewett theorem, compared with our previous work, is that the system of operators obtained from converting the combinatorial problem to a measure theoretic one is no longer commutative. With the lack of commutativity we shall have to work harder in order to exhibit the multiplicative subsystems which lie at the heart of our analysis. In this exposition we have tried to highlight the main features of the proof by presenting various simplified versions of our actual statement. We hope this can also serve as an introduction to the detailed proof which will appear elsewhere. After introducing the basic notation and stating the main result, Theorem A, in the very short Section 1, we devote the following section to a statement of the Ramsey-type theorems, or coloring theorems, that we shall need. We use these first of all in Section 3, where we establish the equivalence of Theorem A and its measure theoretic counterpart, (which we restate again in Section 4, after describing a workable setup for it, as Theorem B.) In Section 5 we show how Theorem B involves the behaviour of various multiplicative sets of unitary operators and prove Theorem 4.1, a weak imitation of Theorem B. Finally in Section 6, we prove the simplest cases of Theorem B introducing thereby most of the ideas that go into its proof.

1. Notation and statement of the main result Notation: Q,, = Q,,(k) = (0, . . . , k - 1)“ the words of length n,

52‘= U a,,= (0, . . . , k - 1} 1 , there exists an integer N(k, 1 ) such that if n > N(k, I ) , then for any 1-coloring of SZ,(k) there exists a monochromatic combinatorial line (HJ-sequence). The density version, ( k = 3), is our

Theorem A. For every E > 0 there is an integer N = N ( E )such that if n > N ( E ) , every subset A c 52,(3) whose relative density is contains a combinatorial line (HI-sequence).

2. Coloring theorems A “coloring” of a set E is a function c from E into a space C, the space of colors. The coloring is finite if C is finite; it is an 1-coloring if C has 1 elements; finally, it is compact if C is a compact metric space. A “coloring theorem” for a finite coloring is a theorem guaranteeing that at least one element of the partition of E according to color, {c-’(x);x E C } contains a subset with a certain structure (e.g. Ramsey’s theorem or van der Waerden’s). Another way of saying it is that we have a monochromatic subset with the given structure. For compact coloring the sets c-’(x) may all be singletons or empty and thus contain no-non trivial configurations; however, as we can cover C by a finite number of &-balls,we can look for whatever we are looking for in the preimage of an &-ball. This is interesting especially if we are looking for an infinite configuration and, assuming we can always find one, we can keep refining by using smaller and smaller balls and, with appropriate notion of “filtering to infinity”, get eventually either convergence or, more generally, uniform continuity of the coloring function along some subconfiguration. This may appear vague as stated but should become clearer with the concrete examples listed below (Theorems 2.2, 2.4, 2.5) and the applications which we give in the following sections. We mention also that the context of compact coloring is not only the form in which the coloring theorems are often useful but also the context in which it is often easiest to prove them. Throughout this paper we shall be dealing with coloring of finite words based on a give finite alphabet, namely a,, and 52‘ introduced in Section 1 . The following theorems deal with coloring of 52‘ and provide infinite monochromatic configurations. We begin with the description of the configurations involved.

H . Furstenberg, Y.KatzneLron

230

Divide the natural numbers into disjoint consecutive intervals 4 = [nj-,+ 1, nil and let { E j , l$}be a partition of 4 with F;- nonempty. Now fill the places in Ej with fixed digits, and those in 8 with a single variable digit (so that as the variable ranges over (0,.. . ,k - l } we get an HJ-sequence on 4) Denote by yj the variable which takes as values the words just defined. The set a* of all finite words y,y2 * - .y, will be referred to as a combinatorial infinite-dimensional subspace of @ (usually shortened to "subspace"). a* has the same structure as 52'; in fact it is a homomorphic image of sz' under the injection which assigns to the word w E s;z' the word yl(w)y2(w)- - .y,,,(w), where rn is the length of w and yj(w) is the word described above with the variable in it replaced by w ( j ) , i.e. yj(w)takes the constant value w ( j ) on 4. The filling in Q* is made up of the parts of y j carried by Ei. Later on, we shall consider subspaces for which some restrictions have been imposed on the filling.

Theorem 2.1 (Carlson-Simpson). For every finite coloring of szf there exist monochromatic combinatoriaf infinite dimensional subspaces.

There is a natural metric on @, where two points are close if they have a long common beginning; specifically we can write p ( w l , w2)= 1 / 1 if the first f - 1 digits of the two words agree and the l'th digit is different (that includes the case that one of the words has no f'th digit) and check that this is a proper metric on szf (relative to which it is precompact, and its completion can be identified with

@ u a).

Assume now that @ is a function from 52' into a compact metric space C. Partition C into a finite number of subsets Ci of diameter less than E , fix some integer m l , enumerate all the words of length m, as {w;}and write the portion of @consisting of all the words starting with a given word w as w x sz'. Restricting @ to w;x @we obtain a partition of w,X st' into { @-'(Cj)}which we can view as a partition of sz'. Taking the join of all these partitions for the various w; and applying Theorem 2.1 we obtain a subspace 52, of 52' such that the variation of @ on w,.x a, is less than E for all i. Now the union Q(') of wix B1for all i is again a subspace of 52' where the first ml digits are the original ones, and the others are the "new digits" given by Theorem 2.1; the restriction of @ to this satisfies the condition: p(u, v ) < l / m l

+dist(@(u), @ ( v ) )
0 there exists j ( E ) so that if il , i2, . . . ,ik > j ( E ) then dist($(a,, U cyi2 U

--

U aik), xo)< E.

This notion of convergence can also be regarded as a form of convergence of sequences. Namely we may speak of an $-sequence in M as a sequence {x,} indexed by cy E 9 (instead of a function $ : $t-+ M ) , and we write IP-lim x, = x' E M a

if for E > 0, 3 j ( E ) such that if cy = { i l ,i 2 , . . . , i k } with all its entries > j ( ~ ) ,then dist(x,, x ' ) < E. Theorem 2.5 then asserts that given any $-sequence in a compact metric space, there exists a convergent $-subsequence.

3. Tbe various equivalent forms Proposition 3.1. The following statements are equivalent: (a) For every E > O there exists n(E) = n ( ~k,) such that if n > n ( ~ ) and , A c a,,, (A1> E k " , then A contains an HJ-sequence. (a*) If A c s;r' and lim supn k-" IA n sZnl > 0, then A contains an HJ-sequence. (b) For every E > 0 there exists n(E)= n(E, k ) such that if n > n(E), and for every w E Q,, there is given a measurable set B, in some fixed probability measure space { x , 3,p } , and p ( B , ) > E , then there exists an HJ-sequence {w,} in Q,, such that p(nB,,) > 0. (b*) If for every w E @ there is given a measurable set B , in some fixed probability measure space { X , 3,p}, and y(B,) > E > 0, then there exists an HJ-sequence {w,} in s;r' (i.e. in some Q,,) such that p ( n B,,) > 0. (c) The statement ( b )above except that the sets B, have the special form that we describe here: On { X , 99, p } we are given an array of invertible measure preserving transformations { U;};It: ;:-', and we form U, = UT"'.. . u,"'"',for w = { w ( l ) ,. . . , w(n)}. Now, for A c X , y ( A ) > E , set B , = U i ' A . (c*) Given an infinite array { t!J;}',&;'-',

we define the transformations U, as

Density version of Hales-Jewett theorem

u.

233

above, this time for w E The claim now is that given any A c X, p ( A ) > 0, there exists an HJ-sequence { wi} in Q.such that U,'A) > 0.

p(n

Proof. The implications (a) (a*) j(b) (b*) 3 (c) j(c*) are clear, and what we propose to prove here is that (b) j(a) and that (c*) .$(b)

Proofof (b) I$ (a). The conclusion of (a) is clearly valid for E > 1 - l / k . In that range one can take n(E) = 1 and notice that, given A c Q,, one can split A to the set Ai = {w E A ;w(1) = j } and the sets A,! c Qn-, which are the projections of Ai on the last n - 1 digits, have a nontrivial intersection if the measure (density) of A exceeds 1 - l / k . Denote E~ = inf E' for which the conclusion of (a) is valid. We use (b) to prove , the function given by (b). that = 0. Otherwise, if e0> 0, take m > n ( ~ , / 2 )n(E) Take E~ = ~ ~ ( k-'"-*) 1 so that e2 = E~ + (eo/2)k-"' > c0. Let M be large enough so that the conclusion of (a) is valid for n > M and sets of measure > E ~ . We claim and n > m M which that the conclusion is still valid for sets of measure contradicts the definition of E ~ . Let A c Q" have measure k-" IAI > E ~ and , define for every w E Q, the set Ah = { u E Q,,-,,,; wu € A } . If the measure of every A: is at least ~ ~ we/ can 2 invoke (b) and obtain an HJ-sequence {mi} c Qmsuch that the corresponding A;, have a nontrivial intersection. If u is in that intersection then { w p } is an HJ-sequence in A . On the other hand, if the measure of one A; is less than eo/2, then, since the average of the measures of A: for w E a,, is the measure of A , some Ah, has measure exceeding E~ and we have our HJ-sequence in it. 0

+

For the proof of (c*)+(b*) we need the following lemma whose proof is straightfoward: Lemma 3.2. Let B, and be finite algebras of measurable sets in a probability space { X , 9, p } . Assume that there is a measure preserving isomorphism U : %H 9,. Then there exis& an invertible measure preserving transformation U on { X , 9,p } which induces 0, i.e. U ( B ) = U-'(B). Remark. It is often more convenient to talk of a measure preserving mapping of the partition (into atoms) of Pd0rather than of 9B0 itself. The two are equivalent. We refer to the data given in (b*), namely the set {B,,,}, as an array and we define a sub-array to be the restriction of an array to a combinatorial subspace of the index space sz'. Since a combinatorial subspace has the same structure as sz' except that it is built on "new digits", which are words in the original space, a sub-array is an array and we can use the various coloring theorems that are stated for L$ in the context of arrays and sub-arrays (as we did in the proof of Theorem

2.2).

H. Furstenberg, Y. Katznclson

234

What characterizes arrays of the form {UG'A} among the more general {B,} is the stationarity of their joint distribution, namely the function m(Z) defined on the set of finite subsets of sz' by

If {B,} is of the form {UL'A}, then its distribution is stationary in the fallowing sense: if I is a subset of SZ,, and u E Q, then Iv = {wv; w E I} c SZ,,+, and, writing

u(n,v ) - ~ (,€I n U G ~ A=)wn u a so that m(z) = m(zU). uslu On the other hand, suppose that we have an array {B,} with stationary distribution. We can define the transformations Ui by applying Lemma 3.2 to the algebras B,,spanned by {B,,.}; w E Qi-, and 94 spanned by { B w i } ;w E Qj-, with the obvious correspondence which, by the stationarity, is measure preserving. There is no reason to expect that the array { B , } given in (b*) is stationary, but we can invoke now Theorem 2.1 and obtain a sub-array which is almost stationary in the following sense:

Defialtion 3.1. For q > 0, the array {B,} is q-almost stationary if p(Z3,) is constant within 7, and for every n > 0, and all finite u,

2

I m ( I ) - m(Iv)l< q2-".

ICQ.

When we consider sub-arrays, the reference to SZ,, will be in terms of the "new" digits which define the sub-array. Lemma 3.3. Given an array { B , } and q > 0, there exist an q-almost stationary sub-array.

Roof. We invoke repeatedly Carlson-Simpson. There exists a sub-array on which p ( B , ) is constant with q/4,and, freezing the first digit y, we consider the functions m ( I u ) = p ( n y l s l B l u on ) the words v of the digits y2, y 3 , . . . , and choose a sub-array on which each of these is constant within q/4.We now fix the first of the new digits and attach it to y , and this is the first digit of our final sub-array. Now take all the subsets I of words on the first two digits (the first permanent, the second temporary), choose a sub-array on the other digits so that ~ ( I u )are ql8-almost constant, fix the first new digit and attach it to the temporary second digit of the complete sub-array (i.e. including the first two digits) thereby making it permanent, etc. 0 Proof of (c*)

+(b*).

The collection of all possible joint-distribution-functions


235

m(Z) (with Z ranging over the finite subsets of @, containing words of the same length) is clearly a compact space under pointwise convergence. One needs to check that if rnj(Z) is the joint-distribution-function of an array { B i } and if m(Z) = lim mj(Z), then there exists an array {B,} on an appropriate probability space, whose distribution function is r n ( Z ) , and we leave this as an exercise to the reader. The next remark is that if m(Z) = lim mj(Z) and rnj(Z) is qj-almost-stationary with qj+ 0, then r n ( Z ) is stationary. By Lemma 3.2 above, any array contains sub-arrays which are arbitrarily almost stationary. Denoting by mi the corresponding joint-distribution-functions and by rn a limit point of these, it is clear that rn is stationary and the measure assigned to singletons is bounded below by E. By (c*) there exist HJ-sequences Zo such that m(Zo)> 0 and it follows that rnj(Zo) > 0 for all mi sufficiently close to rn. These correspond to HJ-sequences in the original array. 0

4. The operator setup

We now limit ourselves to the case k = 3. The context is that of statement (c*) in Proposition 3.1; we have a probability measure space {X,58, p } and an array of invertible, measure preserving transformations which we denote here as {R,, Sj, q}. We shall find it convenient to work with szf(4), i.e. we allow the digits 0, 1 , 2 and 3, and define the following transformations: for 1 = 0, 1 , 2 write

Rj if 1 = O p ( j , 1) = a(j,1 ) = z ( j , 1) =

I;. i f 1 = 2 p ( j , 3) = Rj

u(j, 3) = Sj

z ( j , 3) = q

and for w E Q,, we set

With this notation, the statement (c*), which we have just seen (Proposition 3.1) to be equivalent to Theorem A, becomes

Theorem B. Given a set A of positive measure, there exists a word w in which the digit 3 occurs, such that p(p(w)-'A n u(w)-'A f l t(w)-'A) > 0.

H.Fwstenberg, Y.Katznelson

236

Equivalently, i f f denotes the indicator function of A , there exists w as above such that JX

p(w)-'fa(w)-'fr(w)-'f d p > 0.

(2)

We use the symbols p, u and p and their inverses both as measure preserving transformation and as operators on L z { X , 9, p } in the standard way of associating with any invertible measure preserving transformation Q the unitary operator, denoted by the same letter and defined by Qf ( x ) =f (Q-'x).' We shall not prove Theorem B here, but set ourselves the modest goal of proving that for some such w, the three double intersections p ( w ) - l A n a ( w ) - ' A , p ( w ) - ' A n t ( w ) - ' A , and a ( w ) - ' A n r(w)-'A are ail of positive measure. The equivalent statement in terms of integrals is

Theorem 4.1. Assume f = lAwhere A is a set of positive measure. Then there exist words w, with at least one occurrence of the digit 3, such that

Remark. Since p is measure preserving (resp. unitary) we can rewrite (2) as IxfP(w)o(w)-'fp(w)r(w)-'f

dP > 0

(6)

with similar forms for (3), (4) and ( 5 ) . The basic (formal) features of our setup, given by Proposition 4.2 below, namely the rnultiplicativity in certain situations, is crucial in what follows. Proposition 4.2. The operators p(w)a-'(w), p ( w ) t - ' ( w ) and a ( w ) r - ' ( w ) have the following properties: 1. If w1 and w, are finite words, of possibly different length, which agree at least until the last occurrence of the digit 3, then p(wl)u-'(wl) = p(wz)a-'(wz) and similarly for pt-' and for or-'. 2. If w E Q,, is a word without occurrence of the digit 3, write, for 1 = 0, 1, 2, A, = m-'(l). Let a and @ be subsets of (1, . . . , n } such that every element of a is This definition guarantees that the unitary operator associated with the measure preserving transformation QlQ2 is the product QIQzof the corresponding unitary operators.


237

smaller than any in /3, and assume that a c A*. Set wa(j) = w ( j ) for j $ a and w, = 3 on a;w s ( j ) = w ( j ) off/3 and ws = 3 on /3 and similarly for waus. Then, if l=O and tp-’(w,)zp-’(ws) = tp-’(waUs). a p - ’ ( w , ) ~ p - ~ ( w=~ap-’(waUs) ) Similarly, if I = 1 we get the corresponding formula (i.e. multiplicativity) for zu-‘ (and pa-’; and $ 1 = 2, for pz-’ and az-’).

Proof. This is purely formal; check that things cancel out properly.

0

5. Limits of multiplicative sets of unitary operators The set of linear operators, of norm bounded by 1, on Lz{X, 93, p } if endowed with the weak operator topology, is a compact space, and if we have an array (in the sense of Section 1) of, say, unitary operators, we can invoke the “compact coloring” theorems and obtain sub-arrays which are uniformly continuous. Consider for example the following. Recall that an %-sequence in a space M consists of elements of M indexed by a E S.Take M to be a set of operators on some other space. We say that an %-sequence { U,} in M is an IP-system if for a,/3 E 9with every element of (Y less than every element in /3, we have uaufi=

uau~~

Note that an %-subsequence of an IP-system is an IP-system since UalUn2U-..Uak=

UalUp, . +

* uak

when L Y ~ < ~ ~ < - - - . Now let {U,} be an IP-system of unitary operators. By Theorem 2.5 there is a sub-IP-system which converges in the weak operator topology. A fundamental fact for our analysis is that the limit operator is rather special, namely an orthogonal projection. This merits an explicit formulation:

Lemma 5.1. Assume that {U,} is an ZP-system of unitary operators on a Hilbert space X,and that P = IP-lim U, exists, then P is an orthogonal projection. For the proof we need only to check that P is IdemPotent since its norm is bounded by 1, and an indempotent of norm 1 is clearly orthogonal. The idempotency is obtained by making precise the heuristic argument that for a far aways and /3 farther away P Uaus = UaUs- P2. Returning to the setup of Section 4 above, we invoke our coloring Theorem 2.4 and obtain a subspace without occurrence of the digit 3 in the “filling”, such that restricting the indices to that subspace, the operator valued functions up-’(w),

-

H. Furstenberg, Y. Katznekon

238

p t - ' ( w ) , a t - ' ( w ) , are uniformly continuous and have an extension by continuity (in the weak operator topology) to Q = Q(4). We extend the notation as well, thus we denote lim,,,-+m p a - ' ( w ) by pa-'(o), etc. Notice that there is no claim that pu-'(w) is a product of p(w) by a-'(o)

nor that either exists. Although the (finite) words w have been taken in @(4) we will not make use of infinite words in which the digit 3 occurs, and we restrict our attention to 52(3), and in fact we need just one point w E Q(3) with infinite occurrence of all three digits. At such points we have

Proposition 5.2. The operators p a - ' ( w ) , p t - ' ( w ) , a t - ' ( w ) , t a - ' ( w ) , are all orthogonal projections. Proof. This is a consequence of Lemma 4.1 and of Proposition 3.1. In order to make it clearer we modify the notation somewhat as follows: If a c N is finite and non-empty, we write w ( a ) for the word obtained from o by replacing the digits occurring at indices in a by 3, and truncating after the last element in a.We also write pa instead of p ( w ( a ) ) , and similarly for u and t. If we restrict a to be subsets of the set of indices where the digit in w is zero, then, by Proposition 3.1 {p,a;'} and {par;'} are adjoint to IP-systems (of unitary operators) and by Lemma 4.1 their limits are orthogonal projections. For at-'(o), and up-'(w) we obtain IP-systems if we restrict a to be subsets of the set of indices for which the digits of w are equal to 1, etc. Notice that since the limit exists in the context of 8(4), we are free to choose the mode of approach to o appropriately for each of the operators.

Lemma 5.3. With o as above we have: az-'(o) = t a - ' ( o )

p o - ' ( w ) = up-'(w)

p t - ' ( o ) = zp-'(w).

Proof. The limit of the inverses of a sequence of unitary operators in the adjoint of the limit, which here is self-adjoint. 17 We shall denote Po, = a t - ' ( w ) and similarly Pp, = p t - ' ( w ) and Pep= ap-'(w). Thus we have

P,,

= lim sat;' = lim tau;' a

a

the limit as a goes over the eastern horizon.

Proposition 5.4. P , is a conditional expectation to a factor { Z , $3) of {X,8,p } . An element f E L2 is in the range of P,,, that is measurable B1,if, and only if Ilo;'f - Z,'fl+O as a--,=J.

Demity version of Hales-Jewett theorem

239

Proof. If Pf = f we have u,z;'f-tf in the weak topology and since there is no loss of norm, the convergence is in norm and we may multiply by u;'. This characterization and the fact that we are dealing with measure preserving transformations imply that the range is a sub-lattice of L2 which means that it is L2of a factor. 0 We clearly have the analogous results for Pupand Prp, and are finally ready to prove Theorem 4.1.

Proof of Theorem 4.1. We need to show the existence of a point w in which the digit 3 occurs for which the three integrals appearing in (l), (2), and (3) are positive. We claim that any w sufficiently close to w will do. In fact

provided w-+ w. In our case f = lAand its conditional expectation relative to any subalgebra is positive a s . on A. Similarly for the two other integrals. 0 6. Triple intersection

The real objective, namely the proof that the triple intersection is non-empty, at least sometimes, still requires some work. We do not propose to do it here in full detail, but would like to give some idea of what is involved in the proof. This we do by showing the complete picture in two extreme cases. The first, trivial, case is when one of the projections, say PUT,is the identity. By Proposition 5.4 that means that (along our subspace) 1lu;'f - r;'fII-,O for all f and in particular for f = lA,and our triple intersection becomes really a double one, and hence nonempty for many values of a. The second, more interesting, case lies at the other end. If the range of P,, is just the constants, (only the constants behave "in the same way" under a;' and r;') or, in other words, P U f = for all f E L2, we obtain a situation which, while being simpler than the general case, requires already another tool, namely the "weak-convergence lemma" formulated below. We state it in our present context, namely that of IP-convergence, and just mention that it has corresponding versions for other modes of convergence or summability.

If&

Lemma 6.1 (Weak Convergence Lemma). Suppose { x ~ } ,is~ a~ weakly convergent Ssequence of vectors in a Hilbert space X.If

H. Furstenberg, Y.Katznekon

240

then the weak limit IP-lim x, = 0. aa9

For the (easy) proof see [3]. The relevance to our problem becomes clear once we realize that one way to obtain (6) would be to find the weak-limit of p,u,lfP,t:'fand show that it does not vanish on the support off (which we assume nonnegative). The weak limit may be assumed to exist (by Theorem 2.4). In our present (special) situation we can describe the weak limit in question quite explicitly. Begin by writing (for any pair of bounded measurable function @ and q ) x, =x,(@, q )= p,u;'@p;'V; then, (remember the multiplicativity given by Proposition 4.2)

and as /3 + this converges to

/($p,a,'$)P,r(Wp,r.lq)

dp = /(@paail@)~ P / ( W ~ D ~ , 'dp V)

by our assumption on Po,.As L Y - - , ~ Jthis converges to llPpu@I12Ilf',,q~))~. It follows that the weak limits of pua;'+p,t,'q? and p,a,'@p,t,'q are one and the same if PP0(@- @) = 0; similarly we can replace, without affecting the weak limit, q by Y provided P p r ( q - Y) = 0. This is true in particular if we take @ = PpOCp and Y = P,&. In this case we can identify the limit since, by Proposition 5.4 pus,'@,-, @ and put;lY--, Y in norm which implies (for bounded Cp and v ) IP-Iim p n a : l @ p a t : ' ~ = P,,~P,,v and taking @ = q =f ( = l A )we obtain that the limit that we are after is JfP,,fP,,f,and this is positive since the integrand is strictly positive on A. This completes the proof of Theorem B in the case that at least one of the projections Po,, P,,. or Ppo is trivial. 0

For the general case we take one of these, say P,,, and study the behavior of relative to the factor {Z, Ed} =the range of P,,, (see Proposition 5.4). Once again we make basic use of Lemma 6.1 above and show that one may replace f, without affecting the limit we are studying, by its projections f,, f, on appropriate extension of { Z, 9}and then show that these have special behaviour which enables us to arrive at our conclusion. The details will appear elsewhere. p,t;'f


241

References [l] T.J. Carlson and S.G. Simpson, A dual form of Ramsey’s theorem, Advances in Mathematics 53 (1984) 265-290. [2] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory (Princeton Univ. Press, Princeton, New Jersey, 1981). [3] H. Furstenberg and Y. Katznelson, An ergodic SzemerBdi theorem for IP-systems and combinatonal theory, J. Analyse Math. 45 (1985) 117-168. [4] H. Furstenberg and Y. Katznelson, Idempotents and coloring theorems, to appear. [5] A.W. Hales and R.I. Jewett, Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963) 222-229.



243

ON THE PATH-COMPLETE BIPARTITE RAMSEY NUMBER Roland m G G K V I S T Matematiska Institutionen, Stockholm Universitet, Box 6701,113 85 Stockholm, Sweden

Let r(Pk, K,,,) denote the (mixed) Ramsey number between a path Pk on k vertices and a Kn,m. Thus r(Pk, KnIm)is the minimal number such that every graph G on r(&, K,,J vertices either contains a Pk, or else contains a KnPmin the complement G. Theorem. r(Pk, &),

=sn

+ m + k - 2.

Proof. The theorem is trivially true for n + m + k =4. Assume to obtain a contradiction that

r(Pk, Kn,,,J > n + m

+ k - 2.

(1) This implies the existence of a graph on n + m + k - 2 vertices which does not contain any Pk, and whose complement contains no K,,m. Let G be an example of such a graph with the minimum number v ( G ) of vertices. Since the theorem clearly holds for all k when n = m = 1, we know that v ( G ) 3 k. We next prove that G is connected.

(2)

Proof of (2). Assume to the contrary that G is separated into disjoint graphs GI and G2. Put v1= v(G1) and v2 = v(G2). Since Cl and G2 are disjoint, G contains a K,,,,,. This implies that either v1C n or v2< m, let’s say the former. We have Y ( G ~=) II

- V1+

m

+ k - 2 3 T(Pk, K,,-,,,,),

since G is a smallest counterexample. Hence G2contains a K,-,,,,, (we know that G2 $ Pk). But then G contains a K,,,, contradicting the choice of G. Hence (2) holds. By the choice of G, r(Pk-l, K,,,J S n + m + k - 3; hence G contains a Pk-,. Let P : z, - zz - - - - z k - 1 be such a path in G . Put H = G - V ( P ) and let H,, H,, . . . ,H’ be the connected components of H with indices chosen such that v ( H J 3 v(Hi+,) for i = 1,2, . . . ,p - 1.

Put

0012-365X/89/33.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

R. Haggkvist

244

and q = min{t: z, or Zk-1 is joined to a Hi with i

s}.

Without loss of generality we may assume that zy is joined to a component H, with r 6 s. We note that q > 1 since P is maximal. Put

Q={

~ i 22, ,

.. .

zq-l}

and R = {Zk-q+l, zk-q+2, . .

-

3

zk-1).

By definition Q and R are nonadjacent to every vertex in N = uf=lH,, and so is every vertex in M = UiP_s+I H,. Hence 2q - 2 v ( M ) < rn, since otherwise G 3 Kn,,, Put

+

c v(H,) S

n, =n

-

=n - v ( N )

+ V(H,),

I=]

rfr

m1= rn

-2q

+2-

v(M).

Note that n , > 0 since n > v ( N ) - v ( H s ) 3 v ( N ) - v ( H , ) . Moreover, v ( N ) v ( M ) = v ( G )- v ( P ) = n rn - 1. Hence

+

+

v ( H , ) = n + rn - 1 - v ( N ) - v ( M ) + v ( H , ) = n , + rn, + 2q - 3, which gives

v ( H r ) 3 r(P~q-19Kn,.m]) since G was a smallest counterexample. This means that either H, contains a Pa-, or else B, contains a K,,,,,. Both these cases lead to a contradiction as follows. If H, contains a Pa-lr let S : x1- x 2 . . .x a - l be one such path. There exists a path T: zq - u1- u2 . . u, - x u in G, all of whose interior vertices belong to H, - V ( S ) . We know that either l{xl,x2, . . . ,x,}l -'q or l { x u , x u + , , . . . , x y } (2 q (since v ( S ) = 2q - l), let's say the former. Then x 1 - x 2 . * . -xu - uf - uf-, . . -v, - zq - z,+~ . . zk-1 is a path of length at least k - 1; i.e. a P k . Since G contains no Pk, H, contains no P%-,: we deduce that fir contans a K,,.,,. In other words there exist disjoint sets of vertices X = {xl, . . . , x,,} and Y = { y,, y 2 , . . . , ym,} in H, such that no vertex of X is joined to any vertex of Y. But then no vertex of the n vertices in X U (N - V ( H , ) ) is joined to any of the rn vertices in R U S U Y U M ; i.e. G contains a K,,,, contrary to the choice of G. This final contradiction proves the theorem. 0

-

-

Corollary. r(Pk, K,,*) = n + r n

+ k - 2 ifn

= m = 1 (modk - 1).

Proof. The graph ( n + rn - 2 + k - l)/(k - l)Kk-l has n + m and contains no P k while its complement contains no Kn,m. 0

+k -3

vertices

Path-complete bipartite Ramsey number

245

Remark. It seems likely that the determination of r(Pk, KnJ for all k, n, m will be tricky. Parsons [2] has determined r(Pk, K1,,)for all k and m ;the results are slightly surprising. The reader is referred to [2] for details. For background material on generalized Ramsey numbers see 111.

References [I] S.A. Burr, Generalized Ramsey theory for graphs - a survey, Graphs and Combinatorics, Springer Lecture Notes 406 (Springer Verlag, Berlin, 1974) 52-75. (21 T.D. Parsons, Path-star Ramsey numbers, J. Combinat. Theory B, 17 (1974)51-58.



247

TOWARDS A SOLUTION OF THE DINITZ PROBLEM? Roland HAGGKVIST Matematiska Institutionen, Stockholm Universitet, Box 6701, 113 85 Stockholm, Sweden

An r x n latin rectangle is an r X n array filled with m symbols, say, such that every cell contains one symbol and every symbol occurs at most once in each row and column. The purpose of this paper is to prove the following result.

Theorem. Let L = (Li,j)be an r X n array of n-sets with r G 3n. Then L contains an r x n latin rectangle. This goes some way towards a positive answer to a well-known problem by J. Dinitz, who in 1978 conjectured that the theorem could hold even for r = n in place of r G $n. For background see for instance my paper written in collaboration with Amanda Chetwynd [3] and the references therein, in particular the papers by Bollobfis and Harris [2] and ErdBs, Rubin and Taylor [4]. Further related problems can be found in the extensive literature on the partial latin squares completion problem (see for instance [l]for some fifty references). Before proving the theorem let us give a couple of definitions and a lemma. For a graph G and subsets A , B of G, we denote by NG(A) the set of neighbours of A in G, by EG(A, B) the set of edges between A and B in G and by 6G(A) the minimum vertex degree in G among vertices from A . If B = B(S, T ) is a bipartite graph (with bipartition ( S , T ) ) and A c S then we denote by N i ( A ) the set of neighbours of A joined to A by at least IS( - 6,(S) + 2 edges. An S-matching in B is a set of independent edges which covers all of S, and when p is a natural number then a p-matching is a set of p independent edges. For a matching M we denote by V(M)the set of vertices incident with M. Other unexplained graph theoretical notation should hopefully be standard. With this terminology we have the following lemma.

Lemma. Let F be a matching in the bipartite graph B = B(S, T ) and let A c S be a given set of vertices. Add to B - F an (IS1- INi(A)l)-matching M between S and T - N;(A). Then the resulting graph B* has an S-matching. Proof. Consider a set A c S for which IAl3 S,(S). IS1 - a,($) whence

Then in particular IA - A1 s

N;(A) c N,.(A). 0012-365X/89/$3.500 1989, Elsevier Science Publishers B.V.(North-Holland)

R. Haggkvist

248

Consequenti y INs*(A)l* I N W ) l + IV(M)

n A1

3

t N X A ) l + (IS1 - "A)O

3

IAl.

- IS - Al

The same inequality obviously holds for every set A c S for which

< dB(S) - 1(s6B*(S))j and therefore Hall's theorem guarantees that B* has an S-matching.

Proof of the theorem. For i = 1,. . . , r, let L' denote the ith row of L, and L - L' the array obtained by deleting the ith row. We may and shall assume that L fails to contain a latin rectangle, but that L - L' contains at least one latin rectangle, for i = 1, . . . , r. Let S be the set of columns of L, and T the set of symbols. For each i and each latin rectangle R' in L - L' we form a bipartite graph B(R') with bipartition ( S , T) as follows. For every column s and every symbol t in L we let (s, t) be an edge in B(R') if and only if t belongs to Ll,5but not to any cell in the sth column of R'. It is clear that B(R') contains no S-matching since otherwise R' can be extended to a latin rectangle in L. Consequently, by Hall's theorem, for each R' there exists a smallest set A ( R ' ) c S such that we have lNB(R1)(A(R'))I

< lA(R')l*

Let C(R') be the bipartite graph with bipartition (S, T) for which (s, t) E E ( C ( R ' ) )if and only if t E Lls,and some cell in the sth column of R' contains the symbol t. Then B(R') and C ( R ' ) are edge-disjoint and their union D' has degree n at every vertex of S since (s, t) is an edge in D' if and only if t E Ll,,. Moreover, C(R') has a natural proper (r - 1)-edge-colouring where the edge (s, t) has colour j if and only if t = Rf,,. From now on let Rk be fixed such that (N*(A(Rk))lis minimum over all possible choices of Rk. Put A = A ( R k ) , B = B ( R k ) , C = C ( R k ) ,D = Dk,N ( A ) = N,(A) and N*(A)= N i ( A ) . Then we claim that &-(A, T - N*(A))I S ( r - 1)(2(n - IN*(A)I)- 1).

(1)

Proof of (1). If (1) is false, then at least one of the (r - 1) colours in C is present on at least 2(n - IN*(A)I) edges incident with T - N * ( A ) . Let these edges have colour r, say, and form the matching M*. An application of the lemma gives immediately that B + M * has an S-matching Mk.

Let G ( R k )be the graph where (s, t) is an edge if and only if the symbol t occurs in L , , but not in any cell except possibly Rf,sin the sth column of Rk. Let Q be the latin rectangle obtained by deleting row r from R k , and inserting kth row with

The Dinitz problem

249

Qk,,= t if and only if (s, t) belongs to Mk. Then G(Rk)- Mk is the bipartite graph B ( Q ) . By assumption B ( Q ) fails to have any S-matching and moreover IN&,(A(Q))l Z= IN*(A)I

by the choice of Rk. However, G ( R k ) has an S-matching (with edges (s, t) for which t = RE,), and indeed at least one such S-matching F satisfies

IM*n FI d n - IN*(A)(. To see this we consider two cases as follows. Case 1. At least n - JN*(A)Jedges in M * are incident with N,&k)(A(Q)). Deleting these edges from G ( R k ) gives a graph H which still has an S-matching F, since every set A c S of cardinality at least 6,,,*,(S) has the same set of neighbours in G ( R k ) and H . This implies, since G ( R k ) is known to have an S-matching, that (A(d INH(A)I whenever /A( &(A) + 1, and the same inequality obviously also holds when IAl s a&). Therefore Hall’s theorem guarantees the existence of F.

Case 2. At least n - IN*(A)I edges in M * are incident with T - N:(Rk)(A(Q)). In this case we let A? be one such set of exactly n - JN&,t)(A(Q))I edges from M * and consider G ( R k )- M * + M. By the lemma this graph has an S-matching F which works. Another application of the lemma, this time to the graph B - F + (M* - F), produces one more S-matching P. It is clear that contrary to assumption, L contains the latin rectangle U obtained from Rk by deleting the rth row and adding rows r and k by

Ur,,= t

for every edge (s, t) in F, u,,s= t, for every edge (s, t) in P. this contradiction establishes (1). Next let us show that if r 6 $n then IN*(A)I 2r + 1. Proof of (2). First we note that, since &(A)

8n

- r and

(A1 IN*(A)I + (IAI - 1- IN*(A)l)r 3 IA( &,(A),

then

Now, assuming that (N*(A)I s 2r, we get

n -2r

2r 8-

IAI - r

PI,

R. Haggkvist

250

or in other words (4r - n ) IAI

3 2r2.

+

However, (4r - n)n < 2r2 when r < 1/(2 \/Z)n (and also when r > 1/(2 - f i ) n , but that is irrelevant here), whence in particular, (4r - n ) [ A [< 2r2. This contradiction establishes ( 2 ) . It now only remains to show that ( 1 ) and ( 2 ) are contradictory (when r s $ n that is) to prove the theorem. For this, we note firstly that IEc(A, T - N*(A))I = IED(A,T)I - IED(A, N*(A))I - IEB(A,T - N*(A))I

IAI - lN*(A)I

IAI -

- d B ( s ) + 1)

-

IAl - IN*(A)I IAI - ( I W ) I - IN*(A)l)r, since 6 , ( S ) 3 n - r + 1. Using the fact that "(A)( = IAI - 1 and simplifying slightly we get [&(A, T - N*(A))I 3 ( n - r - IN*(A)I)IAI

+ r(lN*(A)I + 1).

(3)

We consider two cases.

Case 1. IN*(A)I > n - r. In this case the first term in (3) is negative whence we have

+

+ 3 (n - r)(n - IN*(A)I)+ r

IE,(A, T - N*(A))(2 ( n - r - IN*(A)()n r((N*(A)( 1 )

k 2(r

- l)(n - JN*(A)J) - r + 2,

using the assumption that r S $n < f ( n + 2). This contradicts (1). Case 2. ( N * ( A )S( n - r.

Here we get, using only the crudest of estimates, that

+ + r(lN*(A)I+ 1)

IE,(A, T - N*(A))I3 (n - I - IN*(A)I)(IN*(A)I 1) 2 (IN*(A)I 2

+ l)(n - IN*(A)I)

(2r + 2)(n - IN*(A)(),

by (2). This blatantly contradicts (l), and finishes the proof.

0

Remark. It is clear that the last part of the argument can be sharpened by working directly with (3) rather than ( 2 ) . However, it is not possible to get a theorem valid for, say r S f n that way.

The Dinitz problem

251

References [l] L.D. Andersen, Completing Partial Latin Squares, manuscript currently available as research report R-85-2,from the Institute of Electronic Systems, Aalborg University Centre, Strandvejen 19, DK-9000Aalborg, Denmark, to be published in the volume issued by the Royal Danish Academy of Sciences and Letters on the occasion of the centenary of the birth of Niels Bohr. [2] B. Bollobh and A.J. Harris, List-colounngs of graphs, Graphs and Combinatorics 1 (1985) 115-127. [3] A.G. Chetwynd and R. Haggkvist, A Note on List-Colourings, manuscript available in preprint form as Research report no. 17-1986, Department of Mathematics, University of Stockholm, to appear in J. Graph theory. [4] P. ErdGs, A. Rubin and H. Taylor, Choosability in graphs, in; Proc. West Coast Conference on Combinatorics, Arcata (Humboldt State University 1979) Congressus Numeranturn 26 (1979)

125-157.



253

A NOTE ON LATIN SQUARES WITH RESTRICTED SUPPORT Roland HAGGKVIST Matemahka Institutionen, Stockholm Universitet, Box 6701, 113 85 Stockholm, Sweden

The purpose of this note is to give a simple theorem which hopefully will inspire some reader to more profound explorations. First we give some definitions. A partial n X n column-latin square L on 1, 2, . . . ,n is an n x n array filled with the symbols 1, 2 , . . . , n in such a way that every cell contains at most one symbol, and every symbol occurs at most once in every column. The array L is a latin square if, in addition, every symbol occurs exactly once in every row and column.

Theorem. Let n = 2k and let L be a partial n x n column-latin square on 1,2, . . . ,n with empty last column. Then there exists an n X n latin square A on the same symbols which differs from L in every cell. Proof. We use induction on k. The theorem is obviously true when k =O. Assume that the theorem has been proved for order m and let n = 2m. By rearranging rows if necessary (and filling in some empty cells perhaps), we may assume that the mth column of L has the entries 1, 2, . . . ,2m in that order. If we suppress the symbols 1,2, . . . , m in the upper left m x m quadrant B and the lower right m X m quadrant E in L, we find ourselves with a pair of partial column-latin squares H and I on m + 1, m + 2, . . . , 2m which both have empty last columns. Therefore we can find a pair of latin squares F and G on m + 1, m 2, . . . ,2m, without any entries in common with H and I respectively, and certainly not with B and E either. Similarly, by suppressing the symbols m + 1, m 2, . . . ,2m in the upper right m X m quadrant C and lower left m x m quadrant D in L, and applying the theorem, we find a pair of latin squares J and K on the symbols 1,2, . . . , m, which fit into the upper right and lower left corner of L respectively, without any entries in common with C and D. Together F,J , G and K make up A. 0

+ +

The theorem is not valid for every n as seen by example below.

1 1 * L=3 2 * 2 3 * 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

254

R. Haggkvisi

However, this is likely to be the only exception. In general perhaps the following is true for some positive constant c, which could be as large as 4, say.

Conjecture. Let L be an n X n array of m-sets from a set of symbok 1,2, . . . ,n where wery symbol ki used at most m S cn times in each row and column. Then there exkits an n x n latin square A on 1, 2, . . . n with entries in the complement of L. A positive answer could have some impact on the following question.

Dinitz’ problem. Given an m x m array of m-sets, is it always possible to choose one element from each set, keeping the chosen elements distinct in every row and column? For some related material see the references.

References [I] B. Bollobb and A.J. Hams, List-colourings of graphs, Graphs and Combinatorics 1 (1985) 115-127. [2] A.G. Chetwynd and R. Haggkvist, A note on List-colourings, to appear in J. Graph Theory. [3] R. Haggkvist, Towards a solution of the Dinitz problem?, this volume, 247-251.


255

PSEUDO-RANDOM HYPERGRAPHS Julie HAVILAND and Andrew THOMASON Department of Pure Mathematics and Mathematical Statistics, 16, Mill Lane, Cambridge CB2 lSB, England

1. Introduction The study of random graphs has proved very successful in showing the existence of graphs which are extremal with respect to certain properties (see Bollobh [l] for a detailed exposition). Typical of the problems to which they have been applied are subcontractions [ll], Zarankiewicz’s problem [10) and Ramsey’s theorem [6]. Random graphs also offer us examples of graphs with particular properties, giving us expanders [4], graphs of small diameter [3] and parallel sorting algorithms [2]. In most cases the difficulty remains of constructing explicit extremal graphs, or of checking whether a given randomly-generated graph is extremal. As an initial approach to this problem, a simple criterion was proposed in [12], whereby any graph satisfying it might be regarded as a pseudo-random graph ; that is, it would possess certain desirable properties of random graphs. The criterion was stated in the following terms: a graph G is said to be ( p , a)-jumbled if p and a are real numbers satisfying 0 < p < 1G (Y, and if every induced subgraph H of G satisfies

where e(H) is the number of edges in H. Other possible definitions of pseudo-random graphs are offered by Chung, Graham and Wilson in [ 5 ] ; the definitions turn out to be roughly compatible, but for our purposes the definition given will prove the most satisfactory. A ( p , &)-jumbled graph could be regarded as behaving rather like a random graph of edge-probability p , the parameter a determining the closeness of this resemblance. In fact a modification of a theorem of ErdGs and Spencer [8] shows that a must be at least of order lG14, and subject to this constraint it is not hard to verify that almost all random graphs with edge probability p are (p, a)jumbled. In [12] and [13] it was shown that (p, a)-jumbled graphs possess some of the desirable properties of random graphs, at least for reasonably large values of p (p >> n - f , and usually p >> n-4). Moreover two sufficient conditions were found for a graph to be ( p , a)-jumbled. One, stated in Proposition B below, is a 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V.(North-Holland)

256

J. Haviland, A . Thomason

‘global’ condition on all induced subgraphs of a fixed size (rather than all sizes as in the definition). The other, stated in Proposition A below, is an easily checked ‘local’ condition on the degrees of the vertices and of vertex pairs. This latter theorem is a means whereby a specific graph can be shown to be (p, &)-jumbled. The remarks about the value of random graphs equally apply to random hypergraphs, and our aim in this paper is to try to extend the work on (p, a)-jumbled graphs to r-uniform hypergraphs. A specific motive for doing this is to answer a question of Erdos and S ~ Sposed , in [7], concerning the number of K:s (complete 3-uniform hypergraphs on 4 vertices) in 3-uniform hypergraphs. A solution to their question would follow immediately if the obvious generalisation to 3-hypergraphs of a theorem in [12], giving the number of cliques in a jumbled graph, were true. It turns out, rather surprisingly, that this particular generalisation is false. Nevertheless, we are able to prove characterisation theorems analogous to Propositions A and B, and to establish some of the basic properties of jumbled hypergraphs. To begin, we shall define a (p, &)-jumbled hypergraph in the obvious way.

Definition. An r-uniform (r 3 3) hypergraph G is said to be (p, &)-jumbled if p, a are real numbers satisfying 0 < p < 1=z a, and if every induced (r-uniform) subgraph H of G satisfies

where e ( H ) is the number of edges in H. Adding extra conditions in this definition might permit stronger theorems to be proved. However we shall not do this, since it would defeat the object of the exercise, which is to see what consequences follow from just this simple definition.

Notation We shall employ throughout the following notation. If x is a nonnegative integer, then B ( x ) will denote any real number y of absolute value at most x . Hence y = B ( x ) means lyl S x , and 0 S z S x implies B ( z ) = B ( x ) . In this sense the notation behaves like Landau’s O(x) notation. Therefore we may rewrite the definition of a (p, &)-jumbled r-uniform hypergraph G as

for all induced H c G. Further, all hypergraphs will be r-uniform for some t, and we shall often refer to a hypergraph as jumbled if it is (p, &)-jumbled for some p and a whose actual values are not of specific interest. In fact the extension of

Pseudo-random hypergraphs

257

Erd6s and Spencer's theorem to hypergraphs shows a must be of order at least lGl(r-l)n. (A full proof of this fact, together with the less illuminating details of other later proofs, is presented in [9]). We use X")to denote { Y c X;IYI = t}, and if x is a real number, (x)' denotes the falling factorial x(x - 1) ( x - t + 1). We shall say that u E V(G)(,-') and x E V(G) are neighbours if u U { x } E E(G). If H is an induced subgraph of G, we write d&), the degree of x in H , for I{ t E V(H)@-'); t U { x } E E(G)}I and d H ( u ) , the degree of u in H, for l { y E V(H); u U { y } E E(G)}I. If H = G the subscript may be omitted. Finally, if S c V(G) and T c V(G)\S, then the set of edges of G containing i vertices of S and r - i of Twill be denoted by E,(S, T), and we write ei(S, T) for IEi(S, T)I. If H and F are induced subgraphs of G, we may write ei(H, F) instead of ei(V(H), V(F))*

---

Examples of pseudo-random hypergraphs We now give just a few examples of pseudo-random hypergraphs, some of which generalize examples of [12];another appears later in the paper. Verification of the examples can be found in [9]. ( 1 ) Almost all r-uniform hypergraphs G, having edges chosen independently with probability p , are ( p , 0(~G~~'-'~"))-jumb1ed.This is a straightforward exercise in random hypergraph theory. Alternatively, Theorem 1 below can be used to show G is ( p , O(IGI'-j))-jumbled. (2) Let q be a prime and let FI , be the field of order q. Consider the hypergraph G where V(G) = FI, and {xl, . . . ,x,} E E(G) if and only if x1 - - . + x , is a square (modq). Elementary theory of characters over finite fields shows that for this graph, each vertex appears in a number of edges in the range j(' - f r(q Moreover, for each pair x , y E V(G), their number of r- 1 2(r-l)! 3(r + l)(q - 2)r-Z. It will follow common neighbours lies in the range 4 (r 4(r - l)! from Theorem 1 below that G is (4,2 IG('-*)-jumbled. (3) Let the vertices of G be the 4% vectors in a vector space V of dimension 2k over FI, and let f be a non-degenerate quadratic form on V. Let {xl, . . . ,x,} E E ( G ) if and only iff (xl + - + x,) = 0. Again Theorem 1 below can be applied to show G is ( l / q , 2 IGl'-3)-jumbled.

+

')

--

2. Conditions implying a hypergraph is jumbled In [12], the two propositions stated below provided local and global tests respectively for determining whether specific graphs were (p, &)-jumbled.

1. Haviland, A . Thomason

258

Pqosition A ([12]). Let n be an integer, and let O
0 there is a c ( q ) , so that for


271

every i > io(q) every induced subgraph of Giwith m > qi vertices has (c(r])+ o ( l ) ) ( : ) edges. Is it true that there is no sequence of extremal graphs belonging to H which is uniformly distributed? (In particular, is it true for the case H = K:, the complete 3-uniform hypergraph of order 4?) The proof of Proposition E (with F = K k ) goes roughly as follows. Select a vertex x1 and let Hl be the subgraph spanned by its neighbours. For most choices of xl, IHII =pn. Select a vertex x2 of HI and let H2 be the subgraph spanned by the neighbours of n2 in Hl. Again, for most choices of x2, lH21 =p2n, and so on. In this way ordered k-cliques (xl,. . . ,x k } are counted. To be able to count cliques in a jumbled hypergraph, we would need something to the effect that for each vertex, the (r - 1)-uniform hypergraph induced by its neighbours was jumbled, and that this (r - 1)-uniform hypergraph was in some sense ‘independent’ of the original hypergraph. Such properties will not hold for the general jumbled hypergraph, though even if they do, the analogue of Proposition E may still fail; here is a class of examples. A divkion of the set X = (1, . . . ,n } will be a collection 9 of functions fs, 5 E fl-’),such that fs :(X- 5)+ { -1, 1). The set of all divisions is given the uniform probability distribution, so each division has probability 2-“, m =(n-r+2)(,!!2). Let p and 6 be real numbers, O < p , 6 < 1 . For each f E fl-2)and { x , y } ~ ( X - - ) ( ~ )we define the random variable ~ (y ) k ; f ) = -fs(x)fs(y)6. Thus ~ ( b Y ),; E ) equals 6 iff&) Zfs(y) and equals -6 otherwise. A given division 9 induces a probability distribution on the set of r-uniform hypergraphs with vertex set X as follows: the edges appear independently, and for a E A?’), Pr(a is an edge, given

a=1 -

n

(1 - p ) ” ( ; ) ( l + &(a-f ; 5))= g ( u , 9).

Eed‘-2)

Observe if 6 = 0 this probability equals p . We can think of a as being an edge as a result of at least one success among a set of Bernoulli trials, one for each 6 E a(‘-’), each with probability of failure (1 -p)”(;)(l+ E ( U - f ; f ) ) . (We will assume 6 is sufficiently small that, say, (1 - p ) ” ( ; ) ( l + 6) 6 1). We define the space %,(n,p, S), which is the set of r-uniform hypergraphs with vertex set X, wherein, for a given set A cx’r), Pr(A = E ( G ) )=

c. P4.W n 9F

oeA

g(a,

w.

Hence the probability of generating a given hypergraph G is the expected value of the probability of G given 9.More wieldy expressions for Pr(A c E(G)) are given by the next lemma; prior to stating it, we require some more notation. For A c x’r) and 5 E A?-’), the graph A, has vertex set X and edge set {a - fj; 0 3 5 and a E A}. Further, let T&A) denote the set of eulerian subgraphs of A, (those in which all the vertex degrees are even). Finally, let T ( A ) = T,(A), and for t = r, E T(A) let #t = IUs{a E x’”; u - 5 E E(rE)}J.Thus

n,

n,

212

J. Haviland. A. T h o m o n

#t i s the number of (I in A needed to construct all the eulerian graphs t5c A, which form the components of r.

Lemma 9. Let A c A?). I n the probability space Xr(n,p , a),

Proof. We have

and where E denotes expectation. Define Q(A) to be {(a, 5);o E A, 5 E for each R c Q(A) define R , to be R n (x")x { E } ) for each 5 E X(r--2),and # R to be [{a;(a, 5) E R for some a}l. Then Pr(A c E(G ))= E(

2

n n &(a - 5'; Q )

p'A'-*R(p- l)*R

5 OGRE

RcQ(A)

because the values of E ( * ; 5 ) and E ( * ; 5') are independent if 5 # 5'. Now R, corresponds in an obvious way to a subgraph B, of A,. Given a vertex v of this graph, we may partition the divisions into pairs (9, S'),such that f 5 ( u )= -fi(v) and 4 and 9' otherwise agree. If the degree of u is odd, the value of flofReE ( U - 5; E), given 9,will be minus one times its value under 9'. Hence the final expectation will vanish unless B, is eulerian. On the other hand, if BE is eulerian, then for every f5,

n

- 5; 6) =

&((I

O E R ~

n

UU€E(Be)

-fS(4t&J)6

=

(-WBE),

since the number of edges between fil(-l)and fF'(1) is even. Now, because every RE corresponds to an eulerian subgraph, we see that R, corresponds to t E T ( A ) and # R = #t. Thus


273

Finally, on writing a(t)=Ug{ u E ~ )a -; E ~ E ( t e ) } so , that #t= Ia(t)l, we see

o(t)cB

We are now in a position to show that the graphs in Xr(n,p, 6) are most surely jumbled; in fact a considerably stronger statement is true.

Theorem 10. Almost every hypergraph in Xr(n,p , 6 ) has the property that, for each 1,O 6 1 6 r - 3, and for Y EX"),the (r - 1)-uniform hypergraph induced on X

-Y

by the edges containing Y b (p, 0(nr-'-j log n))-jumbled.

Proof. It is sufficient to prove the statement for 1 = r - 3, for then, if Y E X(')and H is a subgraph of the induced subgraph on X - Y , where IHI = k, consider the 3-uniform hypergraph Hp induced on V ( H )- p by Y U p, where p E V(H)(r-3-'). We have

So, let G E Xr(n,p , a), let Y E fi-3), and let G, be the 3-uniform hypergraph induced on X- Y by Y. Let x , y E X - Y and 2 c X - Y - { x , y } . Setting A = {Y U { x , y , z}; z E Z } , we see that for every 5 E fl-2), A, is empty or is a star. Since A, contains no non-empty eulerian subgraphs, Lemma 9 implies Pr({x, y, z} E E(G,);z E 2) = Pr(A c E(G)) =pIA1 =pizI,

so the occurrence of edges of G, containing { x , y} follows a binomial distribution. By standard estimates, the number of edges containing { x , y } lies in the range

I. Haviland, A. Thomason

274

pn f ni logn with probability 1 + O(n-'Ogn),so with probability 1+ O(n2-'"g") every %-tupleof V(G,) is contained in p n f ni log n edges of G,. We now estimate the number of common neighbours of x and y. As the graph Kn-,-l has edge chromatic number at most n - r - 1, the set ( X - v - { x , Y } ) ( ~ ) can be partitioned into sets MI, . . . , Mn-r-l such that for each i and A, p E Mi,A f l p = 0 holds. Moreover [(n - r - 1 ) / 2 ] G lMil s [(n - r - 1)/21. Let W Mi and let A = { ~ U { ~ } U A , ~ U { ~ } U A ; A E W For } . any l j ~ X ( ' - ~ A ) ,, is empty, a path, or a set of independent edges, because the A E W are disjoint. Hence, once again, Pr({x} U A, {y} U A E E(G,); A E W) = Pr(A c E(G,)) =plAl =p2Iw', so the number of A E W with { x } U A and { y } U A in E(G) follows a binomial distribution with probability p 2 . Thus the number of A in Mi with this property is at most p2n/2 + n f log n with probability 1+ O(n-logn),and summing over all Mi we see the number of common neighbours of x and y is at most p2("2') + n4 log n with probability 1 O(n'-'"gn). The same holds for all pairs { x , y } E (X- Y ) ( ~ ) with probability 1 + O(n3-IoK"). Applying Theorem 1 to G,, we see that G, is ( p , U(nf 1ogn))-jumbled with probability 1 + O(n3-'Og"),and this will hold for every v E f l F 3 with ) probability 1 O(n'-'osn)= 1 + o(l), as claimed. 0

+

+

Theorem 10 cannot be extended to I = r - 2. For let v E fl-2), and consider G, and { x , y } ~ ( X - v ) (as ~ )before. If Z c X - v - { x , y } and A = { v U { x } U z , v U { y } U z ; z E Z}, then A, is a complete bipartite graph K,,,,,. From this it follows that IAi

Pr(AcE(G))=piA' j=O

I even

(IAlI. )(-)p -P 1 'J(-6)2J

=+{(p2+ (1 -p)262)'Al+

(p2-

(1 -p)262)'Al}.

Hence the distribution of common neighbours of x and y is bimodal, being the average of two binomial distributions with probabilities p 2 + (1- p ) 2 6 2 and p 2 - (1 - p)262. Certainly the number of common neighbours will almost surely not lie close to p 2 n , so G, will not be ( p , o(n))-jumbled. To some extent, this may explain why the proportion of (r + 1)-cliques in G E X,(n, p , 6 ) is not that found in a random hypergraph, namely pr+', as we proceed to demonstrate.

Theorem 11. Let K c X, lKl= r + 1. Then for G E Xr(n,p,

=p r + l + f

- 2 (

1- p)3a3 + O(b6),

a),

275


the last term signifiying p constant and 6 +0. In particular there are values of 6 for which Pr(K(') c G) #p'+l.

Proof. Let BcK"), IBl=j. Then B = { K - { ~ } ; ~ E Y }for some set Y with JYJ=j . Let l j E X(r-'). The graph BE is empty unless l j E K(r-2), in which case p = K - l j E K(3)and BE has one edge for each element of Y n p. So BE contains no non-trivial eulerian subgraphs unless p c Y,when BE is a triangle. There are (4) such p, and hence f , for which this holds. By Lemma 9,

)(p

- 1)'(1-

63)(4). 0

Although we cannot prove that the proportion of ( r + 1)-cliques in a (p, o(n'-'))-jumbled hypergraph is around pr+', it may yet be possible to establish that the number of (r + 1)-cliques in non-zero. This would be enough to answer the above-mentioned question of Erdiis and S6s.

Appendix Here we establish the identities employed in the proof of Theorem 1.

Identity 1.

Proof. We start by demonstrating that the first two expressions are equivalent. Taking the left-hand side, we have (n - l)(n - 1)-lr$l k (k-r+1) r-1 k-1 (r; 1)' -1 r-1

i=l

r-1 i=l

'

-1 r-1

r-1 j

n-r+l k-r+i

'

=(r - 1 )("-') k-r ]el( ) 1 4 1 - 1 h ( - 1 ) i - 1 [ ( k - ~ ~ ~ - l ) + ( k ~ ~ ~ ~ ] i=l

k-r

-r+j

J . Haviland. A. l7wmason

276

To prove that the second two expressions are equivalent, we observe, by expanding binomial coefficients and rearranging,

=

5' (k

r;j

+ 1)(

/=o

n-k-j n-k-r+l

n-r+2 r-1

>=(

1'

the last step following by counting the number of geodesics in ' 2 from (0,O) to (r - 1, n - 2r + 3) passing through ( j , k - r + 2). 0

ldentity 2.

identity 3. r- 1

2 ('

/yo

7

1)2S(j)2IX,[-' - (k - r

= k(k - l)(n - 2 )

r-1

+ k ( nr -- 11

+ 1)2

).


277

Proof. Taking the left-hand side of the expression, expanding the binomial coefficients in the first term, and multiplying its numerator and denominator by ( k - r + l ) !(n - k - r + l ) ! ,we obtain

'2'

,:

( k - r + l ) !( n - k - r + l ) !k ! ( n - k ) ( k - r ; j + 1)( n - k - j (n - r)!2 j=O n-k-r+l

(2 - [il x

-'+ '))2 - ( k - r + l ) f rf l ) ( n

(-l)i(n k-r+i

i=l

(k

r ; j + 1)(

j=O

-

(" -r -r1+ 2 ) ( n --k

[i'( k - r ;

j

j=O

:)-'

n - k -j n-k-r+l

r)2}(

+ 1)(

:')(:I

r-1

)(k -

+

n - k -j n - k - r + l )(-')'(k

- k)-l( - ')-' r-1 n-k n-r -r

n-r

+i)- ( k -A)

Note that we have the identity Cl~~((k-rf'+l)(nlkk;il)=( n -rr-+l 2 ) from the proof of identity 2, so making this substitution, our expression becomes

)(n -nk- -r j )((n -nk- -rj

( k * ( k - r ; j + l ) ( n -nk--kr-+j l j=O

(r

n-k

ff

"'(1

-1)

-1

n-r

) - 2 ( - l ) j ( nn- r- )k) ]

-2

( n -k )

1 r - l ( k - r + j + 1) 2( -1)' =k!(n-k)!z . ( n - k ) !( k - r ) ! i=o I ! ( r - j - l ) ! (n - k - j ) ! ( k - j + r ) ! ( k - r +j + 1) 2 . j = o J ! ( r - j - l ) !(n - k - j ) ! ( k - j + r)! r-1

=k! (n - k)!

')

(-1)' ( r : ' ) ( k - r + j + 1 ) J r-1 ( k - r + j + 1) = k ! (n - k ) ! . j = o l !(r - j - l ) !(n - k - j ) ! ( k - j

-2k(k-

r-1

j=o

z

- ( rk- !l ()n! -( kn)-!r ) r-'! z ( r -j 1 ) ( n -nk- -r j -

k!(n-k)! (r-l)!(n-r)!

[z( r-l

+ r)!

)(k-r+j+l)

r-1 n-r-1 )(n-r)+z(r:l)( n-r j )(n-k-j j=o J n-k-j

>1

J. Haviland, A. Thomason

278

k! (n - k)!

- ( r - l)!(n - r ) ! [(n

-.)(,"I): (: I:)I

= k ( k - 1 ) ( n - 2 )+k( n - 1 r-1 r-1

+

).

0

References [l] B. Bollobh, Random Graphs (Academic Press, London, 1985). [2] B. Bollobhs and A. Thomason, Parallel sorting, Discrete App. Math. 6 (1983) 1-11. [3] B. BollobBs and W.F. de la Vega, The diameter of random regular graphs, Combinatorica 2 (1982) 125-134. [4] F.R.K.Chung, On concentrators, superconcentrators, generalisers and non-blocking networks, Bell Syst. Tech. J. 58 (1978) 1765-1777. [5] F.R.K. Chung, R.L. Graham and R.M. Wilson, Quasi-random graphs (preprint). [6] P. E r d h , Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947) 292-294. , Ramsey-TurBn type theorems for hypergraphs, Combinatorica 2 [7] P. Erdiis and V.T.S ~ S On (1982) 289-295. [8] P. Erdos and J. Spencer, Imbalances in k-colorations, Networks 1 (1972) 379-385. [9] J. Haviland, Ph.D. thesis, University of Cambridge (in preparation). [lo] P. Koviki, V.T. S6s and P. TurBn, On a problem of K. Zarankiewin, Colloq. Math. 3 (1954) 50-57. [ll] A. Thomason, An extremal function for contractions of graphs, Math. Proc. Cambridge Phil. Soc. 95 (1984) 261-265. [12] A. Thomason, Pseudo-random graphs, in Proceedings of Random Graphs, Poznad 1985, M. Karonski, ed., Annals of Discrete Math. 33 (1987) 307-331. 1131 A. Thomason, Random graphs, strongly-regular graphs and pseudo-random graphs, in Surveys in Combinatorics 1987, C. Whitehead, ed., London Math. Soc. Lecture Note Series 123 (1987) 173-195.


279

BOUQUETs OF GEOMETRIC LATTICES: SOME ALGEBRAIC AND TOPOLOGICAL ASPECTS Monique LAURENT* CNRS, Lamsade, Universilk Paris Dauphine, Place du Markchal de Luttre de Tassigny, 75775 Paris Cedex 16, France

Michel DEZA CNRS, UA 212, UniversitP Park 7, 2, Place Jussieu, 75251 Paris Cea'ex 05,France

Introduction Matroid theory is in the center of Combinatorics, Finite Geometry, Lattice theory and Combinatorial Optimization. During the last decades, extensive search was done in order to find a good degree of generality which still preserves the validity of deep results known for matroids. One of such generalizations is the concept of bouquet of matroids introduced in 1983 by Deza, Frank1 and Laurent and studied in a dozen papers (cf. [7, 11, 14, 171 and references mentioned there). The following matroidal features were extended in a satisfactory way till now: -classical axiomatizations and their equivalence (axiomatizations through flats, independent sets, circuits, rank function, closure operator) (cf. [ l l , 171) -operations and extremal theorems for perfect matroid design case (cf. [11, 12,611 -diagram representation and geometrical aspects (cf. [14, 171) -algorithmic and polyhedral aspects (cf. [8, 91) -orientation (cf. [13]). This paper is a follow-up work in the above series of articles on bouquets and it deals especially with the following features: other operations (contraction, restriction and cuts), strong maps and mapping cylinders, representability , topological aspects and, in particular, shellability of various simplicia1 complexes associated with bouquets and relation with connectivity properties. On the other hand, the starting point of this paper was the important paper of Wachs and Walker 1231. We realized that their principal concepts and results (strong map, mapping cylinder, realization theorem) stated for geometric semilattices could be naturally extended for the broader framework of bouquets. * This work was performed while the author was in CNET, Issy Les Moulineaux, France. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V.(North-Holland)

280

M. hurent, M. Deza

We also give new examples of geometric semilattices; actually, our transversal geometries include all examples of [23]. The paper is organized as follows. Sections 1 to 3 recall briefly generalities on bouquets of matroids: main axiomatizations (through flats in Section 1 and through independent sets and circuits in Section 3), central examples of transversal geometries and d-injection geometries in Section 2, structure of the semilattice 9($) of all bouquets with given independence system 9 in Section 4. In Section 5, we introduce bouquets of geometric lattices as the lattice representation of bouquets of matroids. In Section 6, we consider operations on bouquets: contraction, restriction and cuts and we study their effect on the independence system of the bouquet. In Section 7, we study strong maps on bouquets; we give two new examples of strong maps coming from the closure operator between comparable bouquets having the same independence system (Theorem 7.2) and from the projection map for transversal matroid designs (Theorem 7.6). Then, using the mapping cylinder construction, we prove a realization theorem (Corollary 7.20) which essentially says that every bouquet with M branches can be obtained from a “better” bouquet having only m i. Let % be a non-principal ultrafilter on N, and set F = He/% and V = %/%. Then it is easy to see that F is a field, with char F = 0, and V is a vector space' over F. 0012-365X/89/$3.50

0 1989, Elsevier Science Publishers B.V. (North-Holland)

316

1. Leader

Define @ :M + V by @ ( a )= [(91(a))].We claim that @ is a representation of M over F. Indeed, let {#I,. . . ,a ( ” ) }be a finite independent set in M , and suppose that Ao’&(a(”) =O. Write Lo)= [(A!’))], j = 1, . . . , n. Then { i E I: C Av)q$(aG)) = 0) E %, and so {i E I : A:’) = 0 for all j } E %. It follows that A’) = 0 for all j, as required. Conversely, let {d’),. . . , a @ ) }be a finite dependent set in M. Then, for each i E N, there are A:’), . . . , A?) E I;I such that C A:”&(ao’) = 0, with A?) # 0 for some j . Put A(‘’ = [(A?)], I = 1, . . . , n. Then C Ao)+(ao)) = 0. Moreover, if Ao) = 0 for all j then {i E I : A!” = 0) E 011 for all j , so that for some i we would have A?) = 0 for all j . Thus Lo)# 0 for some j . 0

c

The proof of the following result, which is also from [2], is just as simple.

Theorem 2. Let M be a matroid and F a field such that each finite subset of M is representable over an extension field of F. Then M is itself representable ouer an extension field of F. Proof. Let I denote the collection of finite subsets of M. For each S E I there is a mapping & : M-, V,, where V, is a vector space over an extension field Fs of F, such that $.5 S is a representation of S over Fs. ForSEI, l e t $ = { T E I : S c T } . Thesets$SEIgenerateafilteronI. Let % be an ultrafilter on I extending this filter, and set F’ = Fs/% and V = V,/%. Then F’ is an extension field of F, and V is a vector space over F’.

I

n

n

Define @ : M + V by $ ( a ) = [ ( $ s ( a ) ) ] .As in the proof of Theorem 1, it is easy to verify that c$ is a representation of M over F’. 0 We remark that it is certainly not the case that if each finite subset of M is representable over F then M is itself representable over F. For example, let M be the matroid on an uncountable set whose independent subsets are those with at most 2 elements. Then any finite sybset of M is representable over Q, but M itself is not. However, the assertion does hold if F is finite. Indeed, if F is finite and each Fs = F then we have F‘ = F i n the proof of Theorem 2. We wish to point out that Theorems 1 and 2 can also be proved by making use of the Compactness Theorem for first-order logic. However, the method of making the theory of the representations of a given matroid into a first-order theory is rather unnatural (it involves fixing an arbitrary basis of the matroid), and we feel that the ultraproduct proofs are simpler and more direct.

References 1 1 1 N.E. Fenton, Matroid representations - an algebraic treatment, Quart. J . Math. (Oxford) 35 (1984) 263-280.

Vdrnos theorem on rnatroid representations

317

[2] P. VBmos, A necessary and sufficient condition for a matroid to be linear, in Mobius Algebras-Proceedings of a Conference (Crapo, H. and Roulet, G . , eds.), University of Waterloo (Waterloo, Ontario, 1971) 166-173. [3] D. Welsh, Matroid Theory (Academic Press, London, 1976) xi + 433 pp.



319

AN ON-LINE GRAPH COLORING ALGORITHM WITH SUBLINEAR PERFORMANCE RATIO Lgszld LOVASZ Department of Computer Science, Eotvos University, Budapest, Hungary and Department of Computer Science, Princeton Universily, Princeton, NJ 08544, U.S.A .

Michael SAKS Department of Mathematics and RUTCOR, Rutgers University. New Brunswick, NJ 08903 and Bell CommunicationsResearch, Morrktown, NJ 07960. Supported in part by NSF grant DMS87-03541 and Air Force Ofice of Scient$c Research grant AFOSR-M71.

W.T. TROTTER Department of Mathematics, Arizona State University, Tempe, Arizona 85287. Research supported in part by NSF grant DMS 87-13994. One of the simplest heuristics for obtaining a proper coloring of a graph is the FErst-Fit algorithm: Fix an arbitrary ordering of the vertices and, using the positive integeF as the color set, assign to each successive vertex the least integer possible (keeping the coloring proper). This is an example of an on-line algorithm for graph coloring. In the on-line model, a graph is presented one vertex at a time. Each new vertex is given together with aU edges joining it to previous vertices. An on-line coloring algorithm assigns a color to each vertex as it is received and once assigned, the color cannot be changed. The performance function, p,(n), of an on-line algorithm A is the maximum over all graphs G on n vertices of the ratio of the number of colors used by A to color G to the chromatic numbers of G. The Ffrst-Fit algorithm has performance function n/4. We exhibit an algorithm with sublinear performance function.

1. Introduction One of the simplest heuristics for obtaining a proper coloring of a graph is the First-Fit algorithm: Fix an arbitrary ordering of the vertices and, using the positive integers as the color set, assign to each successive vertex the least integer possible, subject to maintaining a proper coloring. This is an example of an on-line algorithm for graph coloring. In the on-line model, a graph is presented one vertex at a time. Each new vertex is given together with all edges joining it to previous vertices. An on-line coloring algorithm assigns a color to each vertex as it is received and, once assigned, the color cannot be changed. If A is any graph coloring algorithm then, for a graph G, xA(G) denotes the number of colors that A uses to color G. The performance ratio of A on G, denoted pA(G), is xA(G)/x(G),i.e. the ratio of the number of colors used by A to the number of colors in an optimal coloring of G. The perfonnance function of 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V.(North-Holland)

320

L. Lovclsz

et

al.

A , denoted pA(n), is defined for each integer n to be the maximum of pA(G) over all n vertex graphs. Various researchers (see e.g. [1,7]) have shown that no on-line algorithm A has bounded performance function on the class of all graphs; indeed, for any on-line algorithm A, there are trees on n vertices for which that algorithm requires at least 1 +logzn colors. On the other hand, p a ( n ) S n for any algorithm. First-Fit does not do much better than this trivial bound even on the class of bipartite graphs since for any integer k, there exists an on-line bipartite graph on 2k vertices for which First-Fit requires k colors (the graph is the bipartite complement of a perfect matching). Recently, Szegedy [19] showed that for any on-line algorithm A and integer k, there is a graph on at most k(2k - 1) vertices having chromatic number k, but for which the algorithm A requires 2k - 1 colors. Thus the performance function for any on-line algorithm A grows at least as fast as n/(logn)’. It is natural to ask whether there is any on-line algorithm that has a sublinear performance function. In this note we settle this question in the affirmative by proving:

Theorem 1. There exkcits an on-line coloring algorithm Color with pcolor(n)= @/log* n)(l + o(1)). Note that in this paper all logarithms are taken to the base 2, and, as usual, log* n is the smallest k for which the k times iterated logarithm, logtk)n = log . log n is at most 1. The algorithm Color is constructed recursively from an algorithm Partition* that partitions the vertex set into subsets each having clique number strictly smaller than the input graph. This kind of recursive construction was used by Wigderson ([21]) to obtain a polynomial (but not on-line) approximate coloring algorithm with performance ratio n(log log n)2/(log n)*. Previous researchers have considered the behavior of on-line coloring algorithms on restricted classes of graphs. The performance function of an algorithm A with respect to a class G of graphs p A ( n ; G ) , is the maximum of pA(G) over all n vertex graphs in the class G.It is an easy exercise to construct an on-line algorithm that achieves a performance function of O(log, n) on the class of bipartite graphs, which is optimal by the lower bound for trees mentioned above. For the class of interval graphs, Kierstead and Trotter [14] showed that there is an on-line algorithm A with performance ratio 3 and this is best possible. Recently, Kierstead [12] showed that First-Fithas bounded performance ratio on interval graphs, solving a question posed by Woodall ([22]) and Chrobak and Slusarek. GyBrfBs and Lehel ([7]) showed that First-Fit achieves bounded performance ratio on split graphs, complements of bipartite graphs and complements of chordal graphs. On-line algorithms have been investigated in the context of several combinatorial optimization problems, including various problems related to dynamic data structures [2,3,8,10,16,18,20], task systems and server problems

On-line graph coloring algorithm

321

[4,15,5,6], bin packing [9], partitioning a partial order into chains [ll], and representing a partial order by vectors in d-space [13]. We believe that many other problems can and should be analyzed from the perspective of on-line algorithms.

2. An on-line graph coloring algorithm An on-fine graph is an undirected graph G on a totally ordered vertex set V. For any vertex v, let [v] denote the set of vertices preceding or equal to v. The pre-neighborhood of a vertex v, N - ( v ) is its neighborhood in [v].The pre-degree of v, d - ( v ) is the size of its pre-neighborhood. An on-line graph partitioning algorithm is a procedure that constructs a partition n o f the vertex set of G by considering each vertex of G in order and assigning it to one of the previous blocks or creating a new singleton block, without reassigning any of the previously assigned vertices. Such an algorithm is a coloring algorithm if it partitions the vertex set into independent sets. More formally, an on-line partitioning algorithm is a map which associates to each graph G in a class G a partition nGof the vertex set, in such a way that for each graph G E G and each vertex v , the partition of [v] induced by nGdepends only on the graph induced by G on [v]. The on-line coloring algorithm given here is constructed recursively from an on-line algorithm called Partition(n, d ) , where n is an upper bound on the total number of vertices and d is a positive integer bounded above by n . This algorithm partitions the vertex set V of the on-line graph G into sets D1, 4,. . . , Dd, C1,Cz,. . . , C,. Each set 0;is independent and is called a first-fit set, and each set Ciis contained in the neighborhood of some vertex and is called a residual set. Let el = d / n and for i > 1, let ci= &J2. Thus 21-1

.=2($)

.

Say that a subset S of vertices of size s is legal if the intersection of the pre-neighborhoods of its members has size at least esn. Partition(n, d ) is defined as follows. Initially r = 0 and D1, . . . , Dd are each empty. For each arriving vertex v: if v U Diis independent for some i then add v to such 0;. Otherwise if Cj u v is legal for some residual set C j , add v to such a set having maximum size. If there is no such set, increase r by 1 and let C,= {v}. The key property of the algorithm is:

Lemma 2. For n 3 4 and d 5 nllog log n, at most 4n/log log n residual sets are created by Partition(n, d ) on input of any graph having'n or fewer vertices.

322

L. Lovctrz el al.

We begin with a simple combinatorial lemma.

Lemma 3. Let 6 be a constant with 0 < 6 < $. Let S,, &, . . . ,S, be subsets of a set S such that IS,(3 S IS1 for all i and IS, n Sj(< S2 (S(/2, for i # j . Then q < 216. Proof. Suppose, to the contrary that q 3 2/6 and let j = 12/61. For 16 i sj, let Then the sets T,, T2, . . . , I;. are disjoint and = S, - (S, U S,U * - U S,-,). > 6(1- S(i - 1)/2) IS[. Hence 17J 3 ISi[ - IS, n S,l - IS, n &I - . - - IS;n IS1 5 I& U GU

- - - U T,l=

+

a contradiction, establishing the lemma.

- - - + 11;.1> Sj(1-

6 ( j - 1)/4) IS1 > 16,31,

0

Lemma 4. Let G be an on-line graph with at most n vertices. Then when , sets have size t. Partition(n, d ) terminates, at most 2 / ~residual Proof. For each j E (1, . . . , r } let Aj be the intersection of the preneighborhoods of all of the vertices in Ci. By the definition of the algorithm, if Cj has size f , A, has size at least E,n. Furthermore, we claim that if two sets C, and Cj both have size t , lAinAjl < E,+,n = n ~ : / 2 ,which by Lemma 3, with 6 = E , finishes the proof. To prove the claim, let v be the last vertex added to either Ci or Cj and suppose it was added to Ci. Before it was added, Cihad size t and Cj had size t - 1. Since v was added to a set having smaller cardinality than C,, C, U IJ is not legal. This implies that the intersection of the pre-neighborhood of u with Ai has fewer than elements. Since Ai is contained in the preneighborhood of v , the claim is established. 0

Proof o f Lemma 2. For any integer k 3 2, the number of residual sets that have size at least k at most n / k . By Lemma 4, the number of residual sets of size less than k is at most 2 ( 1 / ~+ , l / ~+,- - * + l/ck-,) which is bounded above by 1 / E k . Hence the number of residual sets is at most n& + 1/Ek. Taking k = loglogn/2 yields n / k + l / e kC 2n/log log n + 2(log log n)''Ogn =z4n/log log n. 0 Next, Partition is used to construct a second partitioning algorithm called Partition*. For k 3 2, let n,, be the largest integer such that nk/log,log, nk s 2k (e.g. n 2 = 4 and n 3 = 16). The on-line algorithm Partition* takes as input any graph and produces a union of disjoint partitions as follows: Place incoming vertices in a single class until the first vertex is received that has a neighbor in that class. Starting with that vertex, apply Partition(n2, 4) to the first n2 vertices. Apply Partition(n3, 8) to the next n 3 - n , vertices, and in general, apply Partition(nk,2 9 to vertices { n k - , + 1, . . . ,n k } .

Lemma 5. Suppose Partition* is applied to an on-line graph G on n vertices. If G rC. an independent set then Partition* produces a single class. Otherwise it produces


323

at most 20nllog log n sets, and each has chromatic number strictly less than that of G. Proof. The behavior of Partition* on an independent set is apparent from the definition. On a general graph, every set produced is either a first-fit set (which is independent) or a residual set, which lies in the neighborhood of some vertex of G, and thus has chromatic number strictly less than that of G. It remains to bound the number of sets created. Let k be the least index such that 2"-' < n/log log n Q 2". For any i, at most ni - ni-l vertices are partitioned by Partition(ni,2). This results in at most 2' first-fit sets and, by Lemma 2, at most 4ni/loglogni residual sets, for a total of at most 5(2') sets. Thus, the total number of sets created by Partition* is at most (1 5(22) 5(23) 5(24) 5(2")) s 10(2k)Q 2On/log log n sets. 0

+

+

+

+- - - +

Finally the algorithm Color is defined recursively from Partition*: Run Partition* on G. For each class besides the first (independent) class produced, color it by a recursive call to Color. It is easily shown by induction on the number of vertices of G, that Color partitions any input graph into independent sets. Define c(n, k ) to be the maximum number of colors used by Color to color an input graph on n fewer vertices and chromatic number at most k. Obviously c(n, 1) = 1 and c(n, k ) G n. Define h(')(n) = n, h ( n )= h(')(n)= max(1, log log n/20}, and for k b 2, h(k)(n)= h(k-')(h(n)).Note that for all k and n positive, h(')(n) is a concave function of n.

Theorem 6. c(n, k ) 6 n/h("-')(n)for k b 1 and n

1.

Proof. We prove the result by induction on k; the case k = 1 is trivial. Suppose Color is applied to a graph on n vertices having chromatic number k. By Lemma 5, the main call Partition* produces at most n / h ( n ) classes each having chromatic number at most k - 1. Each of these is recursively colored using Color. Thus the number of color classes created can be bounded above by: t

c(n, k ) s

c(ni, k

max tsn/h(n)

- 1)

i=i

by the induction hypothesis. Since n/k@-*)(n)is a convex function of n, the right hand side is bounded above by taking all of the nits to be equal and t to be as large as possible, yielding an upper bound of n/h(n){h(n)/h(k-2)(h(n))} = n/h@-')(n). 0

324

L. Louhz et al.

Corellrvy 7 . The algorithm Color has a worst case performance ratio of at most (2n/log* n)(l + o(1)).

Proof. By Theorem 6, the performance ratio of color on a graph on n vertices with chromatic number k is most n/kh(k-')(n),which is maximized for fixed n when k is the least index with hCk-l)(n) = 1 (or, possibly 1 less than that) and this index is asymptotically equal to log* n / 2 . 0 Note that the performance ratio of O(n/log(2k-2)(n)) for graphs of chromatic number bounded by k can be improved(!) to O(n logCzke3) ( n ) / l ~ g ( * ~(n)) - ~ ) for k 3 3 by modifying Partition' as follows. Initially, instead of constructing a single independent set, apply a bipartite graph coloring algorithm using at most O(log IVl) colors, switching to Partition(n2,4) only when a vertex is received that cannot be processed by that algorithm. Of course this does not improve the worst case performance ratio over general graphs. Thus for graphs of chromatic number at most 3, there is an algorithm that achieves performance ratio O(n/log log n). On the other hand, the only known lower bound is the O(log n) lower bound for trees mentioned in the introduction (note that Szegedy's lower bound is not useful for graphs of bounded chromatic number). It would be interesting to close the gap.

References [l] D. Bean, Effective coloration, J. Symbolic Logic 41 (1976) 469-480. [2] J.L. Bentley and C.C. McGeoch, Worst-case analyses of self-organizing sequential search heuristics, Communications of ACM, to appear. [3] J.R. Bitner, Heuristics that dynamically organize data structures, SIAM J. Comp. 8 (1979) 82-110. [4) A. Borodin, N. Linial and M. Saks, An online algorithm for metrical task systems, Proc. 19th Annual ACM Symp. on Theory of Computing (1987) 373-382. [S] R.R.K. Chung, R.L. Graham and M. Saks, Dynamic search in graphs, in Discrete Algorithms and Complexity (Academic Press, 1987). 161 F.R.K. Chung, R.L. Graham and M. Saks, A dynamic location problem for graphs, Preprint. [7] (a) GyM& and J. Lehel, On-line and first-fit colorings of graphs, J . Graph Theory, to appear. [8] C. Gonnet, J.I. Munro and H. Suwanda, Toward self-organizing search heuristics, Proc. 20th IEEE Symp. Foundations of a m p . Sci. (1979) 169-174. [9] D.S. Johnson, A. Demers, J.D. UUman, M.R. Garey and R.L. Graham, Worst case performance bounds for simple one-dimensional bin packing algorithms, SIAM J. Computing 3 (1974) 299-325. [lo] A.R. Karlin, M.S. Manasse, L. Rudolph and D.D. Sleator, Competitive snoopy catching, Proc. 27th IEEE Symp. Foundations of Comp. Sci. (1986) 244-254. [11] H.A. Kientead, An effective version of Dilworth's theorem, Trans. Amer. Math. SOC.268 (1981) 63-77. [12] H.A. Kierstead, The linearity of first-fit colorings of interval graphs, SIAM J. on Discrete Math., to appear. (13) H.A. Kierstead, G.F. McNuIty and W.T.Trotter, A theory of recursive dimension for ordered sets, Order 1, 67-82.


325

[14] H.A. Kierstead and W.T. Trotter, An extremal problem in recursive combinatorics, Congressus Numerantium 33 (1981) 143-153. [15] M. Manasse, L. McGeoch and D. Sleator, Competitive algorithms for on-line problems, Proc. 20th Annual Symp. on Theory of Computing (1988). [16] R. Rivest, On self-organizing sequential search heuristics, CACM 19 (1976) 63-67. [17] J.H. Schmerl, Recursion theoretic aspects of graphs and orders, in Graphs and Order, I. Rival, ed. (D. Reidel, 1984). 467-484. [18] D. Sleator and R. Tarjan, Amortized efficiency of list update and paging rules, CACM 23 (1985) 202-208. [19] M. Szegedy, personal communication. [20] R.E. Tarjan, Amortized computational complexity, SIAM J. Alg. Disc. Methods 6 (1985) 306-318. [21] A. Wigderson, Improving the performance guarantee for approximate graph coloring, J. ACM 30 (1983) 729-735. [22] D.R. Woodall, Problem no. 4, Combinatorics (Proc. British Combinatorial Conference), London Math. Soc. Lecture Note Series 13, T.P. McDonough and V.C. Marvon, Eds. (Cambridge University Press, 1974) 202.



327

THE PARTITE CONSTRUCTION AND RAMSEY SET SYSTEMS Jaroslav NESETkIL Department of Applied Mathematics, Charles University, Prague, Czechoslovakia

and Vojt6ch RODL Department of Mathemah, Czech Technical University, Prague, Czechoslovakia

This paper deals with Ramsey properties of finite set systems of a given type. We present new and simple proofs of some of the most general results in Ramsey theory for set systems. The proofs rely on a new proof of the Partite Lemma which is combined with an amalgamation technique known as Partite Construction.

Introduction The following result [18] is one of the most famous and fundamental of combinatorial statements. Finite Ramsey Theorem. For every choice of positive integers t, a, b there exists a positive integer c such that c -+ (b);.

Here c+ (b); is a short hand notation (due to Erdos and Rado) for the following statement: For every partition of the collection of all a-element subsets of a set X of size c, there exists a b-element subset B of C such that all a-element subsets of B belong to one class of the partition. This theorem has been generalized many times and several of these generalizations are both profound and difficult to prove. Motivated by general results due to Rado [19] and Graham, Leeb and Rothschild [3], one of the main streams of the research was formed by efforts to prove a very general result which would imply all the known (usually difficult) instances. Thus development culminated with the proof of the Ramsey theorem for systems, which we shall state after introducing a few standard notions. A type A = (&; 6 E A) is an indexed collection of positive integers. Throughout this paper we will fix an index set A and a type A. A system A of type A is a pair (X, A) where X is a finite linearly ordered set, A = (&; u E A), and E (As customary, here ( f )denotes the set of all for 6 f 6 ’ . k-element subsets of X.) We shall suppose that & n.Ua,=8 Elements of the sets & are called edges of A.

(c).

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

J. NeSetfil, V. Rod1

328

A is a subsystem of B = (Y,K) if X is a subset of Y with the induced order, and A& = J\rS r l P(X)for every 6 E A. Two systems (X, A) and (Y, N)are isomorphic if there is a monotone bijection f .X+ Y taking A onto K. A subsystem of B isomorphic to A is called a copy of A in B. Denote by (A") the set of all copies of A in B. A system F is called irreducible if every pair of points of F is contained in an edge of F. The arrow C-+(B):' is defined by analogy with the classical Erdiis-Rado case: C+(B)f if for every partition ( ~ ) = d l U . - - U dthere t exists B ' E ( ~ ) and an i , exists such that ( y ) ) c d i .The following result was proved by the authors in [6] and [7].

The Ramsey Theorem for Systems. Let t be a positive integer and let A , B be systems. There exists a system C such that

c-, (B)f. Moreover, if A , B do not contain an irreducible system F then C may be chosen with the same property. The original proofs of this result were difficult and complex, see [7, 121. Let us remark that related resuits were obtained by Abramson and Harrington and by Prommel [l, 171. However, for some special cases (such as partitions of edges of graphs and hypergraphs) several simple proofs were found, see "13, 141. These proofs are variations on a common theme - the systematic use of amalgamation of partite systems. This Amalgamation technique has been known to authors since 1976 and was effectively used in several papers [lo, 13,14,15]. This technique did not imply the Ramsey theorem for systems. A breakthrough was achieved in 1980 [16], with a proof of an old conjecture of Erdos related to the Ramsey property of rectangle free graphs. (This seemingly esoteric question is a cornerstone of the area and yields e.g. the existence of complex designs; see the discussion below in Section 3). The purpose of this note is to further extend the methods of [14, 161 and to present a new proof of the Ramsey theorem for systems. (H.J. Prommel and B. Voigt recently found a different simple proof of this result. Their method is also a variant of the Amalgamation technique.) It appears that our approach is strong and flexible enough to yield virtually all the known Ramsey theorems for special classes of set systems. The paper is divided in three parts: In the first part we derive the Partite Lemma which is the starting point of our amalgamation technique. This part uses one new trick. In the second part we apply the Partite Construction. This part is routine for anyone familiar with the Partite Construction and we closely follow the ideas of [14]. For the convenience of the reader we outline the proof of the basic properties of the Partite Construction. In the final part we state several strengthenings of the above results which follow from our method.

Partite construction and R a m e y set system

329

1. The Partite lemma An a-partitie system a is a pair ((Xi):=l, A) where (i) X = UfXlXi is an ordered set satisfying X, <X,< - < X, (ii) A = (Aa;6 E A), Aa E (n”,) (iii) J M n X i J e l f o r e v e r y M E A a i, = l , . .. , a , ~ E A .

-

Sets Xi are called parts of A, elements of M edges of A. Property (iii) implies that edges are transversals with respect to the family X,< - * - < X,. Given a subset Y G X we denote by tr(Y) the trace of Y , i.e. the set ( i ; Y n Xi # 0 ) . A is called tramversal if /Xi/ = 1 for every i = 1, . . . , a. A is a subsystem of B= X ) if there exists a monotone injection i : ( 1 , . . . ,a } + (1, . . . . , b } such that XiE for i = 1, . . . , a and

((x)f=l, xli,

.cl,= Xan

(3

for (IE A.

As before ((Xi)tl, A) is isomorphic to ((K):=,, X ) if there is an order preserving bijection from X = Xi to Y = K sending each part X,onto the corresponding and taking A to X . A subsystem of B isomorphic to A is called a copy of A in B; the set of such copies is denoted again by (A”). The arrow notation carries over in the obvious way.

ui

x

ui

The Partite lemma. Let t be a positive integer and let A and B be a-partite systems. Moreover, let A be transversal. Then there exists an a-partite system C such that

C-, (B):. Proof. Put A = ((Xi);=,, A), B = ((K):=l, X). As A is transversal we may suppose without loss of generality that UaEd A, is the set of all subsets of X (this may be achieved by adding a set of “dummy” edges to A and X). Without loss of generality we may also suppose that every vertex y E Y is contained in a copy of A. This is a general comment (see e.g. [9]): if B* is the subsystem of B induced by (2) and C*+(B*):’ then we can easily construct a system C such that C-, (B): by enlarging every B* E (2:) to a system B. Now take N to be a sufficiently large (indeed very large) number; the actual value of N will be estimate later. Define an a-partite system C = ((Zi)bl, O), 6 = (aa;6 E A) as follows. Set Zi= Y;: X - - X % (N times); i.e. each element of Zihas the form (xi; xi E &, j = 1, . . . ,N).Put Z = Uf=,Ziand for j = 1 , . . . , N define a projection arj:Z+ Y by arj(xk;k = 1, . . . , N)= x i . Clearly nj maps 2, to Yk. We define B = (Oa; 6 E A) as follows. First put Xa= ./\rb U N,where XL is the set of all edges of N which belongs to a copy of A in B, N,= X’ - X; (note that we cannot assume that = 0).

-

J, NesetpI1, V. Rod1

330

The set {(xf; j = 1, . . . , N); k = 1, . . . , na} belongs to 0, if tr({xf; k = 1, . . . , R ~ } )= tr({xf; k = 1, . . . , na}) for all j , j ‘ s N and one of the following possibilities occurs: (1) {xi”; k = 1, . . . , na} E X i for every j = 1, . . . , N ; (2) there exists a non-empty set o G (1, . . . , N} and a set { x k ; k = 1, . . . , R ~ E} NL such that {xf; k = 1, . . . , R ~ = } { x k ; k = 1, . . . , na} for all j E o and for all j @ o there is some q with

{ ~ fk;= 1, . . . , Q } E N ; . (Note that, in general, q # 6, however q is determined by tr({xf; k = 1, . . . , ns}). We shall prove that C - , ( B ) f provided N is large enough. This follows easily from the following two facts. Fact 1. A‘ E (2) iff nj(A’)E

(A”)

for every j = 1, . . . , N .

Proof. Obvious from the definition of 0. 0 In order to state Fact 2, we introduce some notation. Put (2) = { A l , . . . ,A r } . Put R = (1, . . . ,r}. Think of the set R N endowed with Hales-Jewett (combinatorial) lines. A line is a set L of the following form: Fix w c (1,. . . , N} and a’= (at,. . . , & ) e R N and put ~ = { ( ,a. ,. . , aN); ai=a4 f o r i e o ,

aj=aj fori,jEo.

Clearly ILJ= r. Given a = (a1, . . . , aN)E R N denote by V ( a ) the set of all V(a). By virtue of vertices x of Z which satisfy nj(x)E A , . Put V ( L )= UaEL Fact 1 the set (2) is in 1-1 correspondence with the set R N . Fact 2. Let L be a line of RN. Then V ( L ) induces a copy of B in C.

Proof. Check the definition of V ( L ) . 0 Now we invoke the classical Hales-Jewett theorem [5] and choose N sufficiently large such that for every partition of R N into t classes one of the classes contains a monochromatic line. This implies C + (B):. Indeed, let (2) = .dlU * U d,be a partition. By Fact 1 this induces a partition R N = .d; U U Sg: by (Y E d,! iff V ( a )induces a copy belonging to di.By the Hales-Jewett theorem there exists a monochromatic line I in RN. This in turn, using Fact 2, B ’ E (g) such that ( 5 )is monochromatic. 0

-

--

2. The Partite construction

In this part we prove the Ramsey theorem for systems by means of the Partite Construction. We follow closely the construction given in [14, 131.

Partite construction and Ramsey set systems

331

Let k, A, B be fixed. Consider A as a transversal a-partite system and B as a transversal b-partite system. Explicitly

B = ({Yl, * * * ,Y b } , 4. Put p = r(a, t, b) = min{n; n+ (b):} (the classical Ramsey number). Put q = (z), ( { ' y p ) ) = {MI, . . . ,M q } . We shall construct "pictures" Po, . . . ,P k , . . . , Pq by induction on k. Picture Pq will be the desired system C. Let P o = ((Xp)ip,l, 0") be a p-partite system where for each choice of b parts . . . , the subsystem of Po induced by them contains a copy of B. Such a "picture" P" may be formed as a disjoint union of copies of B. Suppose pictures P k = ((Xf)f=',Ok), k < q, be given. Consider Mk+l and the a-partite system Dk+l induced in P k by parts Xf where yi belongs to Mk+'. By the Partite lemma there exists an a-partite system Ek+' such that

x,

x,

Ek+'_* (D"")f. Extend each copy of Dk+' in Ek+' to a copy of Pk in such a way that the distinct copies of Pk intersect in vertices of Ek+l only. In this amalgamation the parts of distinct copies of P k are preserved. Denote the resulting amalgamation by ($:)*Pk. (If a more explicit definition is needed, see [15]). Put Pk+'= ($;) * Pk. Finally, put C = Pq. We claim that C has the desired properties.

Claim 1. Every irreducible subsystem in C is a subsystem of B. Proof. Induction on k. This being trivial for k = 0, in the inductive step the amalgamation does not create any new irreducible system. 0 Claim 2. C+ ( B ) f . Proof. We apply backward induction on k = q, q - 1, . . . , 1. In the inductive step (k 1+ k) we apply the Partite lemma and find a copy of Pk such that all copies of A with trace Ak are monochromatic. this leaves us, for k = 0, with a copy P' of Po in C where the color of a copy of A in P depends only on its trace. However, such a copy of Po contains a monochromatic copy of B by the construction of P" and the fact that p _* (b):. 0

+

3. Applications The Partite construction yields, in the spirit of [9] and [ll], several results stronger than the Ramsey theorem for systems. We list some of them. A. Horn-connectedgraphs

First we give an auxiliary definition. Let B = (X, 4)be a set system. A set Y c X is called a cut of B if there is a partition of X - Y into two disjoint sets

J. Ndetfil, V. Rod1

332

Y,, Y, such that no pair {yl, y2}, yl E K, y2 E Y2,is covered by an edge of B. We shall also consider the cut Y as a subsystem of B determined by Y. A system B is called homA-connected if no cut Y of B has a homomorphism into A. Here a homomorphism is an edge-preserving mapping. (The notion of a Horn Kkconnected graph coincides with the notion of chromatically k-connected graph, see [15].) It is also convenient to recall the following notion from [9]: Given a (possibly infinite) set 9 of systems, denote by Forb(9) the set of all those system A which do not contain any system F E 9 as a weak subsystem of B. (A is a weak subsystem of B if every vertex (edge, respectively) of A is a vertex (edge, respectively). Now we have the following result. Theorem 3.1. Let 9 be a set of homA-connected systems. Then for every positive t and every B E Forb(.@ there exim C E Forb( 9) such that C - (B):.

Proof. First fur P = ( T , 9)such that P - + ( B ) f . Put T = (1,. . . ,p}, (;)= { B , , . . . , B 4 } . Define picture P = ((X)iP,,, A') by:

x;= {i} x (1, . . . , q } and iff {(i,, m j ) ; j = 1, . . . , n b } E k = mj = mjvforj, j ' = 1, . . . , q, {ij;j= 1, . . . , nb} is an edge of B and belongs to Nb. (Thus Po is a disjoint union of copies B1, . . . , B, with traces induced by (i)). Clearly Po E Forb(9) and the amalgamation does not create any new homA connected subsystem of Pk+'.(Note that every a-partite system D', . . . , 0 4 may be homomorphically mapped into A.) 0 The Partite Construction is very convenient for constructing sparse Ramsey graphs. This is not surprising as one of the byproducts of the partite construction is a new easy construction of highly chromatic graphs without short cycles [lo]. There are various ways of defining sparseness and we list them in the order of increasing difficulty. 0. Sparse Ramsey theorems - Ramsey families

We say that 3 c (2) is a Ramsey family if for every partition

(;)=dlu..

.Ud,

there exists B' E 3 such that (%)c di for some i. We denote this by C 4 (B):

Partite construction and Ramsey set systems

333

We associate with 8 a uniform hypergraph H i = (X, E) where X = (2) and

E=

c)

[

; B E 8).

We have the following result for sparse Ramsey families.

Theorem 3.2. For A, B and positive integers t, 1 there exist C and a system 48 c (2)such that (1) c 3 (B):' (2) The hypergraph H i has no cycles of length s l . Proof. We proceed by induction on 1 and construct Q by means of the Partite Construction (see Section 2). Put 8 '= In the inductive step we assume that there exists a system Bk+'5 (:"*'.) such that H'&+I has no cycles of length s l - 1. We form the picture Pk+'= a'+'* Pk.Assuming that in Pk we have a system sek then in Pk+'we may define a system Bk+' as gk+'* Qk consisting of copies of B which have no cycles of length SZ (it appears that all copies in $23' and thus sek+' are transversal). See [15] where this argument is covered in details. 0

(z).

C. Sparse Ramsey theorems - cycles in copies We prove the following result concerning sparse copies.

Theorem 3.3. Let Be be a Hom A-connected system. Let t, 1 be positive integers. Then there exists a set system C with the following properties: (1) C+(B);4 (2) The hypergraph H a ) has no cycles of length Sl. Proof. We apply the Partite Construction. We start with picture Po introduced in the above proof of Theorem 3.1 We proceed by induction on k. In the inductive step we use Theorem 3.2 to obtain a Ramsey system Bk+' c ($:) without cycles of length , 1, 0 < r =s00 and multiple points are allowed. Let

with mE-')(O) = 1. Then

where k = [ 2 2 / M 1 .

To see that Theorem B is implied by Theorem 1, we observe that when n = 2 and r = m, m L - ' ( e . ) can be estimated by 4 or 1 according as is at least, or less than, min H , and it is 0 if %.E H. Theroem 1 actually holds for all k 2 1for which krn - 2 > 0; the stated value of k gives the best upper bound (see Lemma 4 in Section 2 ) while Theorem B uses k = 1. In the following section we review the known results related to ours. Section 3 contains some probabilistic and measure-theoretic lemmas necessary for the proof of Theorem 1. We note here that combining Lemmas 3 and 4 gives the following apparently new result.

e.

meorem 2. Let A l , . . . , AM be events of the same probability p and let

2 = p - l Ci,j P(A, nAi). Then p

;

(k l )

6-

kM-X

where k = [ 2 X / M ] . Theorem 1 is proved in Section 4. Our main use of Theorem 1 is to investigate the geometric graph G, and to improve a number of upper bounds for m y ) and lower bounds for xm. This is done by iterating the application of Theorem 1 to a number of appropriate configurations of increasing dimension. Our method is easier to handle in practical computations than the use of Theorem A. This is because the best use of the latter requires knowledge of the maximum size of an independent set in the subgraph of GH induced by the M points, which requires the solution of an NP-complete problem in general. In [23], for some big configurations D could only be estimated by the Davenport-Haj6s Lemma. The configurations we found useful are described in Section 5 . The algorithm we used in applying Theorem 1 to these cofigurations was complicated by the fact that many configurations were involved. We describe our computer implementation of this algorithm in Section 6. The results are presented in Section 7, together with a complete proof of our bounds on m y ) and xm(R") for 3 =sn s 9. Related unsolved problems are given in Section 8.

Bow& on the measureable chromatic number of R"

347

2. History

Investigation in this direction was initiated by Hadwiger [18], who proved in 1944 that if R" is covered by n + 1 closed sets (n 3 l), then one of the sets realises all positive reals as the distance between two of its points. In 1970, Raiskii [30, 311 dropped the restriction "closed" for n 3 2, and Woodall [28] independently did the same in 1973. Generalising Raiskii's idea, Larman and Rogers [23] proved Theorem A, and the following result, in 1972.

Theorem C . Suppose there are M points in R" (multiple points allowed) such that every ( D 1)-subset of these points has at least one pair of points at unit distance apart. Then any covering of R" using less than MID sets has at least one set realising all the positive reals as the distance between two of its points.

+

The existence of a configuration as in Theorem C clearly imples x(Rn) 3 MID, whereas Raiskii's theorem implies x(R") 3 n + 2. The best known bounds in dimensions 2 and 3 are . 4 s x(R2)

6 xm(R2)

7

(Hadwiger [19], L. and W. Moser [27]),

5dxm(R2), 64x,,,(R3)

(Falconer [14]),

5 6 x(R3) c %,,,(R3) s 21

(Raiskii [30], [31]; the colouring is an exercise for the reader).

Frank1 and Wilson [14] demonstrated the exponential nature of %(BY),thereby solving a longstanding problem of P. Erdos, by proving that x(R") 3 (1.2 + o(1))" and m y ' s (1.2 + u(l))-". On the other hand, x,,,(Rn) d (3 + o(1))" and hence m p ) a (3 + o(l))-", as proved by Larman and Rogers [23], using previous results of Butler [2] and Erdos and Rogers [ll]. Falconer [14] recently proved, without using my),that xm(R") 3 n 3, which is much worse than the Frankl-Wilson bound in high dimensions but notable if n = 2 or 3. On the other hand, the Frankl-Wilson bound is weak even in the 25th dimension. The Moser spindle (Fig. 1) and Theorem A prove that mf)s 3, and a natural generalisation, the Moser-Raiskii spindle, proves mi2)d [23]. These estimates

+

Fig. 1. The Moser spindle.

L.A. Szlkely, N.C. W o d

348

were improved by the first author to mf) < [34] (and further [33]) and mi3)s 5 [351. The latter inequality is an alternative proof for xm(R3)2 6. On the other hand, an easy lower bound is mi2)2 n / ( 8 f i )= 0.2267, given by a configuration of open disks of diameter 1 with centres at the vertices of a lattice of equilateral triangles of diameter 2. L. Moser obtained mi2)9 0.2293 by a slight but tricky modification of this set (see Croft [8] and W. Moser [B]). In Table 1 we quote the results of Larman and Rogers [23] for dimensions n up

Table 1. The best known applications of Theorem A to lower bounds MID for the reciprocal of m y ) , n S 25. By Theorem C, M I D is a lower bound for x(R"). n

M

D

2

7

2

Moser spindle

3

14

3

Moser-Raiskii spindle

4

23

4

Moser-Raiskii spindle

5

16

2

Half-cube

6

316

32

7

8

56 64

4

7-dimensional Gosset polytope Special Gosset spindle

9

64

4

Special Gosset spindle

10 11

165 220

9 12

18: 18

12

286 364

12 12

23

13 14

455

13

35

15

560 680

16 16

35 42;

17

816

16

51

18 19

969 1140

17

20

57 57

20 21

1330

20

22

1540 1771

20 21

23 24

4600 4692

46 46

25

2600

24

16

Configuration

Half-cube spindle

4

'

30

66; 77 84;

>

Erdos-T-S6s configurations

, Leech-Conway configuration Special Leech-Conway spindle

J

Erdos-T-S6s configuration "

Boundr on the measureable chromatic number of R"

349

to 25. Of these, only when n = 2 and 3 were better bounds for m y ) obtained prior to our present results.

3. Prelimianrylemmas The first two lemmas, due to RCnyi, are equivalent. We will use the second explicitly, and we include the first for comparison with a step in the proof of Theorem 1.

Lemma 1 (Rdnyi). Let f be any function from (0, l}"to R , and Q a probability space with measure o.Then I f (xAI,. . . ,xAn)d o 3 o for all events A ~. ., . ,A , c 8

iff

fkA,, . . . ,xA,)(w)2 0

for all sequences A l , . . . ,A ,

of events of probability 0 or 1 in Q and all w E R Lemma 2 (Rdnyi [26, 321). For i = 1,2, . . . ,k let Bi be a Boolean polynomial (with operations U,

f l

and complementation) and let ci be a constant. Then

2 ciP(Bi(Al,. . . ,A,)) 2 0 i

for all sequences A l , . . . ,A , of events in a probability space 8 if and only if the inequality holdr for all sequences A l , . . . ,A , of events in 8 of probabiZity 0 or 1. RCnyi actually proved only Lemma 2, but the equivalence with Lemma 1 is easy to see.

Lemma 3 (Kai Lai Chung [22]). If A l , . . . , A , are any events in a probability space and k is a natural number, then

Proof. By Lemma 2, it is enough to check that the inequality is true whenever exactly t events have probability 1 and the rest have probability 0 (t = 0, . . . , n). In this case, the inequality is equivalent to (k - t)2+ (k - t ) 3 0. 0 Lemma 3 can be strengthened, as is plain from its proof, but as we shall see, it is optimal for our present purposes. Its bound is optimized in turn by the following lemma, which is straightforward to check.

350

L.A. Sz&dy,

N.C. Wormald

Lemma4. LetM>O, z T > O , p > O . If

pSMk-Z

for every natural number k > . Z / M , then the sharpest of these bounds is for k = [2.Z/M]. It may be surprising that, although the method proof of Lemma 3 can give a more general result, we only require the stated version. This is because the way we shall use Lemmas 3 and 4 is the following. For a certain natural number M, and a fixed 2, reals a and b satisfying a 3 bt - (i) for all t = 0, . . . , M and 6 > Z / M will lead to the inequality m g ) ( r )s a / ( b m- 2).Thus, the sharpest inequality occurs when (b/a)M - (l/a).Z is maximised, subject to 1 3 ( b / a ) t( l / a ) ( ; ) for t = 0 , . . . ,M . In this linear programming problem, the maximum occurs when at least two of the constraints hold simultaneously, and it is easily seen that these must be consecutive integers t. It follows that a = ( " t ' )and b is integral. The next result apparently follows immediately from results in the theory of relative invariant measures in homogeneous spaces (see Helgason [21]). We prove it here since its extraction from that theory is difficult, and the proof, while using techniques reminiscent of that theory, is quite simple.

Lemma 5 . Every finite isometry-invariant measure on the Lebesgue-measurable sets of S" is confor some nonnegative constant c. Proof. The area a,, is isometry-invariant. Suppose p is an isometry-invariant measure on S" such that a,(S")=p(S") and there exists a measurable set X c S" with

Since both of

a,

and p are isometry-invariant, we have for every g E G

and hence

On the other hand, the function z E S"+

JG

k x ( g ( z ) )dh is constant and by

Boundr on the measureable chromatic number of W"

p(S") =

351

we have

We now have a contradiction by (3.1), (3.2) and Fubini's theorem. 0 It is also possible to give a rather elementary proof of Lemma 5 for n = 2 by the following sequence of steps. If p is isometry-invariant then p = can on: the digons determined by great circles (lunes) having an angle ( p / q ) n ,all lunes, all spheric triangles (by Girard's area formula using the areas of lunes), all open sets. For n = 1 and 3, the lemma also follows immediately from the unicity of the Haar measure, as these spheres are topological groups. For z E S"(r) and p > 0, let T ( z ,p) denote S"(r) n S"(Z, p ) .

Proof. Let G' denote the subgroup of G fixing P, with corresponding measure h', and let g' denote an element of G'. We prove the lemma by showing Zl(P, Q) = JG, Zl(P, g ' ( Q ) ) dh' = J G ~Z2(P, g'(Q)) dh' = Z2(P, Q). The third of these is immediate, and the first follows from (g')-' E G' c G, so we examine the second in detail. We have

(by Fubini's theorem),

(by Lemma 5, regarding the second integral as the measure of X t l T ( g ( P ) ,

(by the invariance of the Haar measure on G), = Z2(P,

Q)

(after reversing the order of the first two integrations). 0

m)),

352

L.A. SzAely, N.C. Wonnald

4. Roof of Tbeorem 1

Let X be an H-independent measurable set in S"(r). We assume without loss of generality that X is closed, since in any case X contains a closed set whose measure (or upper Lebesgue density, if r = = ) differs from that of X by a negligible amount. Suppose firstly that S"(r) is itself a topological group; i.e. r = or r < m and n = 1 or 3. Let C be a big cube if r = TX: and let C = S"(r) if r < w. Let * denote the group operation; i.e. addition in R", multiplication of complex numbers of unit absolute value in S'(r) (with r scaled to 1) and multiplication of unit quaternions in S3(r). Applying Lemma 3 to A, = P, * (Xn C) in the probability spaces

and P ( A i )= an ( A ' ) in

S" (for arbitrary n )

(4.1)'

Un(S")

(4.2') In (4.2), the error term o(1) is for A(C)+=. After slight modification, we get

(4.3') We next integrate the inequalities (4.3) and (4.3') over g E G where the positions of PI, . . . , PM have been moved by g. The only non-constant terms are the numerators in the terms in the summations. They become

and

Bounds on the measureable chromatic number of R"

353

where we have used for brevity the notations

Edz) = and

E2(Z) =

I I

X-g(Pi)+g(4)+x"c(z)

G

Xg(P,)-l.g(P,)*X(Z)

G

dh

dh.

e.,

For fixed 4, z E R" (or S") and C, and an arbitrary set X,the values of E&) and E2(z) depend only on the intersection of X with the spheres S"(Z, and S" n S"(Z, respectively, and it is easy to check that they are normalised isometry-invariant measures on these spheres. Hence, by Lemnia 5,

q)

2.)

and

z) is identically equal We also have that xxnc(z)El(z) (and ~ ~ ( z ) E , (respectively) to 0 if E H. Hence integration of (3.3) and (3.3') over g E G yields

( ')

- Z)A(X n C)lA(C) + o(1)

3 (Mk

(4.4)

and

(k

') (Mk- Zbn(X)/an(S"). 3

(4.4')

The theorem now follows for Iw" and S' and S3,and the stated value of k is optimal by Lemma 4. To complete the proof, we consider arbitrary Sn(r) with r c c o . Let b l , . . . , bM E Y ( r ) and let k be a natural number. Then

(In fact, integration of this inequality gives an alternative proof for Lemma 3, and integration of a more general one gives Lemma 2.) Setting bi = g(e.) in (3.5) and integrating over g E G, we get

By Lemma 5 , the first integral is a,(X)/a,(S"). The second integral is clearly 0 if -

ce E H,and otherwise, by Lemma 6, it has the upper bound mg-')(%Vl

-(qq/2i)2~"(x)/fz,(S").

The theorem now follows. 0

L.A. SzCkely, N.C. Wormald

354

We remark that the proof given for S"(r) (sifting points) works for R" as well if the integration of (4.5) to (4.6) is on an appropriate big part of the isometry group of R" of type G x C rather than on G,but the formulae become more complicated. Although we could not apply the argument given for groups (sifting sets) to non-group spheres, the dual approaches of sifting points and sifting sets are in accordance with the duality of Lemmas 1 and 2. 5. List of configurations

Here we list and briefly describe the configurations that we need. The dimension of the configuration is denoted by n, M is the number of points, r is the radius and x +-k denotes that the distance fi occurs k times. S ( k , rn, n) denotes a Steiner system with these parameters. Existence and properties of the Steiner systems used can be found in [3] and [20]. Some of these configurations are spherical designs or subsets thereof, in the sense of [9]. nZcube Take n coordinates, put f 1 into two and 0 into n - 2 of them in every possible way. Here 3 S n c 24, M = 2(n2 - n), r2 = 2, 4+-48(:)+4(;), 6+-8n( n - 1 8tn(n-1). 2+8n( n - 1

),

),

n3-cube Take n coordinates, put f l into three and 0 into n - 2 of them in every possible way. Here 3 C n s 24,

M

= 8(

6+-640(:)

;)

, r2 = 3, 2 +96(:),

+ 192(:),

8+96(n

4 +%(n

4

')n

+ 12(;),

4

l>n

+ 12(4),

10+-96(:),

12+4(;).

Simplex Take a regular simplex of unit edge length. Here 3 s n ~ 2 4 M , =n Let t2 = f , t, = 1 - ((n - l)/n)2fn-,; then r: = (n/(n + 1))2tn. 1+-(" l').

+ 1,

Cross polytopes See [5]. Here 3 s n S 24, M = 2n, r2 = $, 1+- (7)- n, 2 e n .

Erd6s- T. Sbs configuration See [23]. Take n + 1 coordinates, put 1 into three and 0 into n - 2 of them in every possible way. Here 10s n S 24, M = (" 3 I ) , r2 = 3(n2 - n - 2)/(n + 1)2,

')(" 2

;l>o 52)/2.

'), 4-(;)(" i 2 ) ( n + 1)/2, 6 2+('; (In some dimensions these configurations are the best for the Larman-Rogers principle.)

355


Steiner-3

Take a Steiner system S(2, 3, n) [20] and let the vertices of the conguration be the columns of the incidence matrix of S(2, 3, n) (i.e. the characteristic functions of the edges). Here n - 1= 6, 8, 12, 14, 18, 20, 24; M = 4(2), r2 = 3(n - 3)/n,

The dimension of the configuration is n - 1, since the sum of the coordinates of vertices is constant. Steiner-3-cube

Sign the nonzero entries in the vertices of Steiner-3 in all possible ways. Here n =7, 9, 13, 15, 19, 21; M =4(n - l)n/3, r 2 = 3, 4 t (4n - 10)n(n - l), 6 t 8 +(4n - lO)n(n - l),

-y-

$;)(:(I)

1)

12 +2(n - l)n/3.

We remark that the Steiner-3-cube for n = 7 is the 7-dimensional Gosset polytope; see [36]. Steiner-4

Take a Steiner system S(3, 4, n) [20] and let the vertices of the configuration be the columns of the incidence matrix of S(3,4, n). Here n - 1= 7, 9, 13, 15, 19, 21; M = i(;), r2 = 2(n - 4)/n, 4+i(;)(n-4),

6+6(1)(4( n - 1 )-6n+20),

8+:(1)(!(1)-$("2')+3n-9).

Steiner-4-cube

Sign the non zero entries in the vertices of Steiner-4 in all possible ways. Here n = 8, 10, 14, 16, 20, 22; M = 4(3, r2 = 4, 4+(24n-88)(1), 8+32(9)(:(1)

6+64(1)(

(n - l)(n - 2)

- $(n - l)(n -2) +3n -9) - - +25 ) ,

3n

- - +25 )

+ 48(1)(n

'

- 4) + 12(;),

12+-(24n -88)(J),

L.A. Sz&kcly, N.C. Wormald

356

Steiner 4-318 cube Keep only those vertices of the Steiner-4-cube whose sum of coordinates is zero. Here n - 1 = 7, 9, 13, 15, 19, 21; M = $(:), r2= 4,

4+9(;)(n

- 4),

6-:(;)(24("

8 ~ ; ( 4 ) ( 9 ( ; ) -48( n - 1

2') -

108n + 360),

) + 156n -492),

.lOtftjJ)(24( n - 1 )-108n+360),

1 2 ~ (15 ~ ) (n ~ ) ( n - 4 ) ,

Sreiner 3-618 cube Keep only those vertices of the Steiner-3-cubewhose sum of coordinates is f l . Here n = 7 , 9, 13, 15, 19, 21; M =2(;), r 2 = 3 ,

4+(:)(4.5n

- 11.5), 6 t (I)(.(.

8 t n ( n - l), 12

-(; )

- 1) - 4.5n

+ 7.5),

.

Halfcube Consider those 0 - 1 sequences of length n whose sum is even. Here 3 ~ n S 1 3 M=2n-1, , r2 =n / 4 ,

Gosset 6 See 2171, IS]. Here n = 6, M = 27, r2= y , 8 t 2 1 6 , 1 6 ~ 1 3 5 See . [6] for the analogous Hessian polytope. Gosset 8

See [17], IS]. Here n = 8 , M=240, r 2 = 3 , 3 ~ 6 7 2 0 ,6 ~ 1 5 1 2 0 ,9 ~ 6 7 2 0 , 1 2 t 1 2 0 . See [6] for the analogous Witting polytope. The coordinates we used for Gosset 8 are given in [7].

Bounds on the measureable chromatic number of R“

357

Gosset 7 See Steiner-3-cube in 7 dimensions. Pentagonal configuration Represent R 4 as ( x , y, u, v). Put a regular pentagon of unit edgelength centred at the origin into the plane (x, y, 0,O) and put a regular star-pentagon of unit edgelength centred at the origin into the plane (0, 0, u, v). Here n = 4, M = 10, the configuration is not spherical, 16 3 5 , (l6 1)2/4t 5 , (l6- 1)2/4t 5 .

+

Dodecahedron, icosahedron Turned out to be useless. “Other” polytope See [6]. Here n = 6, M = 72, r2= 3, 3 t 7 2 0 , 6 +1080, 9 t 7 2 0 , 12t36. S(5, 8924)

See [3]. Here n = 23, M = 759, r2= 16/3, 8t170016, 12t106260, 1 6 t 11385. S(5, 6 2 4 )

See [3]. Here n =23, M = 7084, r2=4.5, 2 ~ 4 7 8 1 7 0 , 6-2975280,

8 t

8607060, 10 t9563400, 12+3464076.

$66 , W See [3]. Here n = 11, M = 132, r2= 5, 1 ~ 2 9 7 04, t2640, 2 t2970, 3 t66. S(4,5,11) See[3].Heren=10, M=66,rZ=15/22, 1 + 9 9 0 , $ t 6 6 0 , 2 t 4 9 5 . Perm (+ + - - -) Take the permutations of 2 “1” and 3 “-1”. Here n = 4, M = 10, rz = 24/5, 8 ~ 3 0 1, 6 ~ 1 5 . Perm (-11OOO) Permute -110o0 every possible way. Here n = 4, M = 20, 3 = 2, 2 6 6 0 , 4t60, 6 4 4 0 , 8 ~ 1 2 .


358

Quartercube

-

See [23]. Here n = 10, M = 28, r2 = 5 , 4-2560, 4 3200.

8+14080,

124-12800,

S(4,5,23)

See [3]. Here n =22, M = 1771, r 2 = 90/23, 4679965, 64-371910, 8 t 717255, 104-398475. S ( 4,7,W

See [3]. Here n = 22, M = 253, r2 = 112/23, 6 t 17710, 12t 14168. The following few configurations are direct products. The vertices of the product configuration are ( u l , .. . , v,, ul,.. . ,urn), where ( u l , . . . , u,) is a vertex of the first and (ul, . . . ,u,) is a vertex of the second'configuration. Si denotes the regular unit simplex of R'.

Gosset 8 x S1 n = 9 , M=480,r2=3.75, 3-13680, 240.

6+43680, 9-43680,

12-13680,

15t

n = 10, M = 720, r2 = 4, 3 ~ 2 0 8 8 0 64-85680, , 9t110880, 12-40680,

15t

Cosset 8 x S2 720.

Gosset 8 x S3

n = 11, M =960, ? =4.125, 3t28320, 64-141120, 9t208320, 124-81120, 15+- 1440. Cosset 6 x Cosset 6 n = 12,

M =729, ? = 4 , 3+-11664, 6+-100602, 9+-116640,

12-36450.

Gosset 6 x Cosset 8 n = 14, M = 6480, r2= 5, 3 4-233280, 6~3343680,9 4-8527680, 12 4-6988680, 15t 1866240, 18+- 32400.

Cosset 6 x Cosset 7 n = 13, M = 1512, r2 = 4.25, 3 t32508, 6-354564, 15c 7560.

9 4 3 1 4 6 8 , 12 -216216,


359

Gosset 7 X Gosset 7

n = 14, M = 3136, r2= 4.5, 3 ~ 8 4 6 7 2 , 6 + 1227744, 9 ~ 2 2 8 9 2 8 0 , 12 c 1227744, 15 t84672, 18 t 1568. Moser configuration

See Fig. 1. This configuration is realizable on S2(r) if 0.5862 < r < 0.6277 or 0.8195 < r. Morning star

Take a regular unit simplex centred at the origin in R3. Reflect all the vertices in the opposite faces and keep two copies of the vertices of the original simplex. Here n = 3, M = 12, it is non-spherical, O t 4 , 1 c48,1 2 t 8 , v ' c 6 , where 1 is the distance of a vertex from its mirror image in the opposite face, and v is the distance between two outer vertices. The morning star can be made into a spherical configuration in R4by translating the outer vertices by a distance i along the fourth dimension axis, keeping the unit distances. If x is the distance from one of the outer vertices to the vertex of the opposite face of the original tehahedron in R3, r is the radius of the S3containing the configuration and j is the fourth coordinate of its centre, then the following relations hold: x2+i2=1

rZ = ( j - i)2 +

(m + id)'

r'=j'+:

P = i2 + v2 = !

(a+ V x i Q '

( V m + id)'.

We often use a subconfiguration W of a spherical configuration T. A useful tactic is to specify one point in T as the pole, and then let W = T f l S where S is a sphere of radius q centered at the pole. If q is the minimum distance occurring in T then W is the arctic. If q is lh times the radius of T then W is the equator. G2

The equator of Gosset 8 choosing any of its points as a pole. Here n = 7, M = 126, ?=8, 8 t 2 0 1 6 , 16+3780, 2 4 ~ 2 0 1 6 3, 2 t 6 3 . G21

The intersection of the equator of Gosset 8 with pole (1, 1, 1, 1, 1, 1, 1 , l ) and the arctic of Gosset 8 with pole (0, 0, 0, 0 , 2 , 0 , 0 , 2 ) . Here n = 6, M = 32, r2 = 6, 84-240, 1 6 t 2 4 0 , 2 4 t 1 6 .

360

L A . Sz&kely, N.C. Womrold

G22 The intersection of the equators of Gosset 8 for (1,1,1,1,1,1,1,1) and (0,0,0,0,2,0,0,2). Here n = 6 , M=60, ?=8, 8 ~ 4 8 0 ,1 6 t 7 8 0 , 24t480, 3 2 ~ 3 0(We . note that the arctic of the Gosset 8 is the Gosset 7.)

Limpace The Steiner system S(5,8,24) generates a set algebra. We suppose the underlying set to be {1,2, . . . ,24}. The vertices of linspace are the characteristic functions of the elements of the set algebra. The blocks of S(5,8,24) (octads) were produced as the orbit of {1,2,3,4,5,8,11,13} under the permutation group Mathiu 24 (see [3]) generated by the three permutations (1,2, . . . ,24); (2,16,9,6,8)(3,12,13,18,4)(7,17,10,11,22)(14,19,21,20,15);

(1, 22)(2,11)(3,15)(4,17)(5, 9)(6,19)(7,13) (8,20)(10,16)(12,21)(14,18)(23,24).

A n y arctic of linspace is isometric to S(5,8,24) and any equator is isometric to the set of 12-element sets in the algebra (dodecads). We kept the dodecads as increasing sequences of 12 elements of { 1,2, . . . ,24}, lexicographically ordered.

a, a.

Only four different distances occur in linspace; fi, 4 and The notation doili2- ik means the subconfiguration of doili2 - - - ik-l lying at distance fi from the first point of ilj2 * * * i k V l , while dodecs = do is the whole set of dodecads. A similar notation is used for octads. All these configurations are kept in lexicographical order. The statistics relating to these configurations are given in Table 2. Here, the column headed by YIN answers the question of whether this configuration was actually used in proving the bounds on rnp) and rn$)(r) given

--

Table 2. Subconfigurationsof the dodecads and octads. Y/N

N Y N Y N N N Y

N Y N N N

Configuration do do2 do3 do22 do23 do32 do33 do332 do333 do3332 do3333 oct2 oct3

n 23 22 22 21 21 21 21 20 20 19 19 22 22

r 46 4/43 46 45 421/2 4/43 46 4/43 d6 4/43 46 45 421/2

n 2576 495 1584 184 288 315 952

195 560 117 324 280 448

d2

d3

d4

637560 45540 249480 8408 16704 18540 93060 7080 32760 2520 11250 19600 40320

2040192 71280 153904 8208 23040 28440 266080 10800 90720 3852 29664 19040 56448

637560 5445 249480 220 1584 2475 93060 1035 32760 414 11250 420 3360

d6 1288 0 792 0 0

0 476 0 280 0 162 0 0


361

in Table 6. (Y = Yes.) The column headed by di refers to the frequency of the distance fi.Note also that the table gives r2, which is rational.

L The Leech lattice was discovered by Leech [24,25] and many of its remarkable properties were found by Conway [4]. We use Conway's description of it, as a set of lattice points on the sphere of radius 4~ centred at the origin. We call this configuration L. The squares of distances occumng in L are 16i (i = 2,3, 4, 5,6, 8). Specifying a linear order of the vertices of L, we define subconfigurations Lil - is - . ik (is E {2,3,4, 5, 6, 8}) in the same way as we did for dodecs, with the one difference that the pole of L used in generating L2, L3 and L4 was (-4, - 4 , 0 , . . . ,0) rather than the first point of L in the linear ordering. For all other configurations, the pole was the first point stored, and points in a subconfiguration are stored in the same order as in L. To specify the linear order of the vertices of L (we did not use lexicographic), we give the following pseudo-pascal program for generating them. As before, each octad is an 8-subset of (1, . . . ,24}, stored as an increasing 8-tuple. Each dodec is a 12-subset of (1, . . . ,24}. Both are read lexicographically. Also, append denotes appending (x(l), . . . ,x(24)) to the output file.

- -

procedure subsl(k, I ) ;

fort:= 1 to 24 tox(t):=O; x ( i ) := k and x ( j ) := 1.

procedure subs2(i, A);

f o r j : = l to 2 4 s e t x ( j ) : = l i f j ~ AanLx(j):=-l otherwise. x(i):= - 3 * x ( i ) . program generate L;

begin for i : = l to 23 do for j : = i

+ 1 to 24 do

begin subsl(4,4); append; subsl(-4, -4); append; subsl(4, -4); append; subsl(-4, -4); append;

end; repeat A:=next octad=a(l),

for k : = O to 127 do

. . . , a(8);


362

besin (c(l), c(2), . . . 47)) :=the binary representation of k; c(8) : = 0 or 1 so that 2 ( i ) is even; foci:= t o 8 d o x ( a ( i ) ):= 2 if c ( i ) = 0 and -2 otherwise; set x ( i ) : = O for all i $ A ; append; end; for i : = l to 24 do be&

subs2(il A); append ; for j := 1 to 24 do x ( j ) := - x ( j ) ; append; end; until A = last octad;

reperrt A := next dodec; for i:= 1 to 24 do begin subs2(i, A);

append; end; ~ t iAl = last dodec;

for i:= 1 to 24 do begin subs2(i, { 1, 2, . . . , 24));

append; subs2(i10); append; end;

end.

The configurations L, L2, L3, L4, L22, L23, L24, L232 and L233 all have a nice distance-invariance property: the frequencies of the distances from a point x

Table 3. Subconfigurations of L. ~

Y/N Y

N N N Y Y

N

N Y

N Y

N Y Y Y

N Y Y

N N

N N

N N

N N Y

N Y Y

N N

N

N Y

N N Y Y Y

N N N

Configuration

n

L2 23 L4 L22 L23 L3 2 L33 L34 L35 L43 L232 L233 L323 L332 L333 L334 L335 L342 L343 L344 L433 L434 L442 L443 L444 L2333 L3443 L3444 L3445 L4442 L4444 L23333 134434 L34444 L44443 L44444 L3 44443 L344444 L444444 L4444443 L444 44 44 L4444444 4 L4444444444

23 23 23 22 22 22 22 22 22 22 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 20 20 20 20 20 20 19 19 19 19 19 18 18 18 17 17 16 14

r2

24 30 32 64/3 24 352/15 144/5 448/15 80/3 30 64/3 24 258/11 208/9 28 256/9 220/9 164/7 201/7 208/7 144/5 448/15 24 30 32 24 372/13 384/13 340/13 24 32 24 2624/93 88/3 30 32 312/11 320/11 32 30 32 32 32

M 4600 47104 93150 891 2816 2025 15400 22275 7128 22528 567 1680 1232 972 5892 6885 1620 1092 7392 10290 7336 10431 1360 10240 19962 976 3360 4745 1504 1280 17400 552 1447 2180 4224 7590 684 1003 6372 1536 2658 2144 748

d2 2049300 47692800 114760800 149688 798336 467775 7484400 12162150 2673000 12063744 61236 294840 179256 120285 143256 6 1766610 245430 141330 1917216 2962050 1852200 2942100 195480 2810880 6870240 103032 450624 737532 154528 178560 5795040 34044 98963 186792 581184 1262976 25265 47678 998784 94976 203136 150912 21936

d3

d4

2049300 6476800 524620800 362700800 1049241600 2010363300 18711 228096 798336 2365440 334125 1247400 53014500 45368400 114604875 82328400 9355500 12474000 117494784 82632704 8505 90720 294840 819840 449680 129360 282366 69255 7275420 7231770 10213290 9371700 306180 752490 100380 353976 11946480 10444000 23868705 17821440 11812500 10177440 24584805 17947440 195480 532480 16916480 23731200 88159320 48660480 103032 269248 2410896 2143968 4956372 3821760 386096 555360 178560 460800 65307240 37232640 34044 83712 430848 409622 1029894 800112 3872448 2823168 12163884 7053312 96761 84816 211199 168488 8343972 4976640 493824 358400 1448288 837632 929440 532480 124828 49152

d5 0 167878656 1049241600 0 0 0 12474000 37422000 898128 39739392 0 0 0 0 1400490 2308500 7290 0 2946720 7929600 2993760 8502480 0 8478720 48660480 0 622176 1661760 34272 0 37232640 0 105100 343440 1520640 7053312 26240 71352 4976640 206848 837632 532480 49152

d6 2300 6476800 114760800 0 1408 0 231000 1559250 0 1813504 0 840 0 0 14640 38070 0 0 62720 355110 68880 420840 680 486400 6870240 488 15456 77716 0 640 5795040 276 1648 14872 121536 1262976 504 3786 998784 24832 203136 150912 27936

d8 0 0 46575 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9981 0 0 0 0 0 8700 0 0 0 0 3795 0 0 3186 0 1329 1072 374

5

a

2a. 0

a

H

Q-

3

W

8

L.A. Szkkxly, N.C. Wormald

364

to all other points is independent of x (this follows from the results of [9]). This made it easier to compute their distance distribution (see also [16, p. 1291). The others were done by computing the distances between all pairs of points, using a computer. The results are in Table 3, where Y / N is as for Table 2, and di refers to the frequency of the distance Although we almost always used the first point in a configuration as the pole in generating subconfigurations, other subconfigurations were sometimes found using different poles. However, a little investigation made it appear that no particular advantage was to be gained by using these other configurations.

a.

6. Method of cornputation

The basic items of data computed were ordered pairs, called jump points, of two types: singular and nonsingular. These are arranged into sets Sing(n) and Nonsing(n), 2 =sn S 24. The only requirement is that if a jump point (r, b) is in Sing(n) then m$")(r)d b, and if (r, b) is in Nonsing(n) then mp)(r') s b for all r' r , including r' = 30. Given Sing(k - 1) and Nonsing(k - l), we obtain sets Sing(k) and Nonsing(k) by applying Theorem 1 for each of the configurations which can be embedded in an Sk, or in R k . Theorem A was also applied, but turned out to be useless for k 2 3. Almost all the configurations used are spherical. Such a configuration of dimension n and radius ro gives rise to a jump point (ro,b) in Sing@) if n = k 1, and to a number of jump points (r, b)~Nonsing(k), with r Z r o , if risk. Non-spherical configurations (such as the pentagonal configuration) give rise to jump points with r = m. A convenient place to begin the computation is with dimension 2. Firstly, and trivially, we can put (0, 1) in Nonsing(2). W e also obtain the following elements of Nonsing(2) by applying the Larman-Rogers principle (Theorem A) to the given configurations:

+

(4,

&two

(&,$)-the (0.819417678, +)-the

(G, a)

points at distance 1 apart; vertices of a unit equilateral triangle; Moser configuration.

Finally, put into Sing(2) by applying Theorem A to the vertices of a unit 3-simp1ex in s2(fl). The Moser configuration can also be embedded in S2(r) for 0 . 5 8 6 2 s r s 0.6277. This does not produce any more elements of Nonsing(2), but can be viewed as giving an infinite subset of Sing(2) if required. However, our implementation of the computations made such information difficult to deal with, and a little experimentation showed that it made very little difference to jump

Bounds on the measureable chromatic number of W"

365

points obtained in higher dimensions, and no observable difference to our bound on my)(..) for small n. Hence, this extra interval of existence of an embedding of the Moser configuration was not made use of. (In fact, our vast experience at repeated computations showed that omitting a jump point, or even any single configuration, from the computations, rarely made much difference to the bounds obtained, especially in dimensions much higher than that of the omitted configuration.) Given a list of elements of Sing(k - 1) and Nonsing(k - l), we proceed to calculate elements of Sing(k) and Nonsing(k) as folows, via Theorem 1. In these d lead to a sphere of radius calculations, two points & and 4 with %.= r , = d d m . We call r, the radius induced by d in Sk(r). To estimate mik-')(r1),we use the best upper bound implied by Sing(k - 1) and Nonsing(k 1). The configurations of the correct dimensions are taken one at a time. Except for the morning star, which we postpone discussing, each distance present in the configuration is taken in turn, and the configuration is expanded or contracted so that the chosen distance becomes 1. Its adjusted radius and distances are computed. For a non-spherical configuration, its dimension must be at least k. Theorem 1 is then applied with r = m. The result is an element (00, b) of Sing@). On the other hand, for a spherical configuration, two things are done. Firstly, if its dimension is k + 1, then a singular jump point (ro, b) is obtained by Theorem 1, where ro is the adjusted radius of the configuration. Secondly, for each (adjusted) distance d in the configuration, other than 1, and for each jump point (r, b) in Sing(k - 1) or Nonsing(k - l), do the following. Set rl = d 2 / 2 1 / m (if it is real or infinite). Then embedding the configuration in Sk(rl) and applying Theorem 1 causes mtk-')(r)to be called for when d. Hence, we obtain a jump point (rl, b,). We call this the jump point induced by d and (r, 6 ) . It is singular either if (r, 6) was singular or if any elements of Sing(k - 1) were used for obtaining bounds, and is then added to Sing(k). If (r, b) is in Nonsing(k - l), the computation is repeated using only bounds obtained from Nonsing(k - l), to obtain an element of Nonsing(k). The treatment of the morning star is slightly different, since the distances of the configuration, when embedded in a sphere of given radius, mu$t be computed by a different method. We did this using the equations relating r, 1 and v given in the description in Section 5 . As when treating the other configurations, for each jump point (ro, b) in Sing(k- 1) or Nonsing(k-l), the values of r, I and u were computed so that mik-')(ro) was called for when apply Theorem 1, first with &P, = I, then with = v. If no singular jump points were used for bounds, this gave an element of Nonsing(k), because of the monotonicity of the appropriate functions (we omit details). Otherwise, it gave an element of Sing(k), as long as r, 1 and v were all positive reals. We did not expand or contract the morning star. Finally, all the elements of Sing@ - 1) and Nonsing(k - 1) were directly copied

sq=

=.

366

L.A. Sz&kely, N.C. Wormald

into Sing(k) and Nonsing(k), since they were all obtained by Theorem 1 or Theorem A, and both of these give the same bound when the dimension of the space is increased by any positive integer. This method was implemented in a Pascal program. For a jump point (r, b), the bound b was represented as a real, and we may expect very high accuracy in the results here: probably to 14 significant figures as far as our present computations are concerned. In preliminary computations, we stored r as a real for all jump points, and assumed that b was an upper bound on mik)(r')whenever Ir' - rl < whether or not the jump point was singular. This possibly gave invalid results, erring on the side of providing stronger bounds, and was done because of the inaccuracies in representing reals on a computer. In all later computations, valid bounds were obtained by effectively storing r2 as a rational whenever it had been computed that Nor2 is an integer, where No is any suitable integer. Such a jump point is called precise. With No = 64 x 81 x 5 X 7 x 11, we obtained virtually the same results as with the preliminary computations. For imprecise jump points, r was stored as a real. If a jump point (r, b , ) is induced by a distance d and an imprecise jump point (I, b), then (r, b) is always applicable in the computation of b l , but r, will not be precise. In all other cases that an imprecise jump point (r, b) in Nonsing(k) was used in a computation, it was only applied to mp)(r') for r' > r E, where we used E = One can show that the errors in our computations of the values of r and r' could not exceed ~ / 2 Hence, . the actual value of r is indeed less than r', and so this jump point really does provide a bound on mik)(r').At some other places in the computation, it was also necessary to err on the side of safety. Steps had to be taken to reduce the proliferation of jump points. Obviously, if (rl, b,) is in Nonsing(k) and (r,, b,) is in Sing@) or Nonsing(k) with r, s r2 and b , 6 b Z ,then (r,, 6 2 ) can be deleted. Similarly, if these jump points are both in Sing(k) and r, = r2 and b , zs b2, then delete (r2,b,). This policy still left too many jump points: over lo00 when k - 9 , with the number doubling to tripling with each extra dimension. The extra information carried along with jump points, given necessary details of how they were obtained, limited the total number of jump points to 20,000 or so, when practical considerations were made. To circumvent this problem, note that deleting jump points still gives valid results. A constant 6 was chosen, and at the end of the computation of Sing(k) and Nonsing(k) described above, it was ensured that for every nonsingular jump point (r, b ) retained, all jump points (r', b') with r' > r and b ' a b ( 1 - 6) were deleted. With 8 = 0.001, several thousand jump points were required for all ks24. Reducing 6 below this value did not seem to change any results much. For example, 6 = O.OOO1 gives exactly the same bound on

+

7. Results

The method of computation described in Section 6 was implemented in a Pascal program and run on an IBM 4341 at the University of Auckland. The results

Table 4. Jump points (r, b ) used in the proof of lower bounds on rnp)(r) for n 6 8 and large r. r

jump

sing

r2

O 1/4 3/10 1/3 3/8

3-1 3-2 3-3 3-4 3-5 3-6

0.0 0.500000 0.547722 0.577350 0.612372 0.918417

N N Y N

4-1 4-2 4-3 4-4 4-5 4-6

0.0

N Y

5-1 5-2 5-3 5-4 5-5

0.559016 0.577350 0.784464 0.929876

Y

N

Y N

?

O 5/16 1/3 ?

N

?

Y

-

0.0

N

0.577350 0.612372 0.849836 0.938510

Y

? 1/3 3/8

N

7

6-1 0.0 6-2 0.612372 6-3 .0.884651 6-4 0.945249

N Y N N

O 3/8

7-1 7-2 7-3

0.707106 1.007326 1.033499

Y N N

1/2

8-1

1.079198

N

7

Y

-

o

Y N

?

?

?

b

configuration

dim

unit

1.000000 0.500000 0.400000 0.333333 0.250000 0.187500

copied copied perm2+3 copied simplex morning star

4

2

3 3

1

1.000000 0.312500 0.316216 0.142857 0.137931 0.128000

copied halfcube n2cuba cross polytopa cross polytopa pentagonal

5 5 4 4 4

1.000000 0.222222 0.209773 0.097222 0.095394

copied Gosset G21 halfcube halfcube

1.000000 0.142857 0.072072 0.070812

induced radii and jump points used

I

0.540061

0.5

2 3 1 1 1

0-547722 0.500000 0.612372 0.918417 1.618033

3-3 3-2 3-5 3-6 3-6

6 6 5 5

2 2 1 1

0.559016 0.577350 0.784464 0.929876

4-2 4-3 4-4 4-5

copied Steiner3cube Gosset Gosset

7 6 6

2 1 1

0.577350 0.849836 0.938510

5-2 5-4 5-5

0.075000 0.053146 0.053113

Gosset Steiner3cube Steiner3cube

8 7 7

2 1 1

0-612372 6-2 1.007220 6-4 1.031395 6-4

0.612372 0.884651 0.945249

6-2 6-3 6-4

0.0

0.972446 0.986597

6-1 6-4 6-4

0.042638 0.034197

Steiner3cube Gosset

7 8

1 2

1.068357 0.707106

1.033499 1.224744

7-3 7-3

1.008479 1.414213

7-2 7-3

0.577350

3-4

0.618033

3-5

0.0

4-1

0.0

5-1

0.0

3-1

0.612372

5-3

f

0-

3

B ? i

8-2

o

? ?

7-3 7-1

L.A. Sz&kly. N.C. W o d

368

quoted in this section took an hour or so of computation time. (The time required for Table 3 was much greater.) After computing Sing(k) and Nonsing(k) for 3 c k s n, the program made a list of all the jump points required to establish the upper bounds obtained for rn‘:’(m) and for rnik)(r) for large r (i.e. the element (r, b) of Nonsing(k) for which r was greatest). The results for n = 8, with 6 = 0, are shown in Table 4. The column headed “jump” gives a name to each jump point (r, b), “sing” shows Y for singular and N for non-singular and “r2” gives r2 if the jump point is precise and “?” if not. The name of the configuration used to establish the jump point is given, “copied” appears in the “configuration” column if the jump point was copied from the previous dimension, and “dim” gives the dimension of the configuration. Given a configuration, number the different distances appearing in it in increasing order. The one expandedlcontracted to equal 1 is given as “unit”, and the radii induced Table 5. Jump points ( r , b ) used in the proof of lower bounds on m y ) ( r ) for n s 9 and large r. r

n 3

0.0

3 3 3 3 3

0.500000 0.547722 0.577350 0.612372 0.918417

4 4 4

0.0 0 .!A7722 0.559016 0.577350 0 .?a4464 0.329876

4 4 4 4

-

Sing

r2

copied gosset G2 1 Steiner3cube gosset gosset

6 6 7 6 6

2 2 2 1 1

1 .oooooo 0.142857 0.075000 0.053146 0.053113

copied Steiner3cube gosset Steiner3cube Steiner3cube

7 8 7 7

2 2 1 1

-

0.069444 0.042638 0.034197

n2cube Steiner3cube gosset

9 7 8

2 1 2

N

3

Y

ce

0.034604 0.028820

Steiner3cube n2cube

9 9

1 2

?

5

6 6 6 6 6 6

0.0 0.577350 O.blZ342 0.612372 0.884651 0.945249

N N

7

0.0

7

0.612372 0.707106 1.007326 1.033499

N N

ca

2 2 3 1 1 1

1.000000 0.222222 0.20?920 0.142857 0.072072 0.070812

N

3

0

1.295163

4 5

2 2 2 1

N

0

3/10 5/16 1/3

5/16 1/3 3/8

9

1

5 6 6 5 5

N Y Y

N

9

3

copied halfcube gosset C21 half cube halfcube

N

7

N Y

ce

2

1.000000 0.312500 0.222222 0.209773 0.097222 0.095394

N

Y

0.707106 1.079198

4

copied perm2 t 3 halfcube n2cube cross polytope cross polytope pentagonal

N

0.0

8 8

unit

1.000000 0.400000 0.312500 0.316216 0.142857 0.137931 0.128000

Y N

0.559016 0.577350 0.612372 0.849836 0.938510

8

dim

copied copied perm;! t 3 copied simplex morning star

0

1/4 3/10 1/3 3/8

5

7 7

configuration

1.000000 0.500000 0.400000 0.333333 0.250000 0.187500

N N

5 5 5 5

7

bound

Y

N

7

N

1

N Y

N N

Y

0 1/3 3/8 3/8 3

7

0

3/8 1/2

N N

7

Y

1/2

N Y

?

7

5 4 4 4

1


369

by the others are shown separately (preserving the order), together with the jump point used to estimate mik-')(r1)for each induced radius r,. Briefer results, omitting the induced radii, are given in Table 5 for n = 9, again with 6 = 0. In Table 6, we give for 6 = O.OOO1 the nonsingular jump points with greatest radii, for all dimensions up to 24, and also the singular jump points with r = to, where these give an improved bound (i.e. in all dimensions but 2, 3, 5 , 6 and 7). The implied lower bounds on %,,,(S(")(r))are also listed, as well as the configurations which gave these jump points. We do not give a proof as in Table 5 because thousands of jump points were involved. The configurations listed in Section 5 which were found by the computer to be useless in establishing these bounds were: Steiner-4-cube, Steiner 4-2 cube, "other" polytope, S(5, 8, 24), S(5, 6, 24), S(4, 5, l l ), perm(-1, 1,0,0,0), S(5, 4,23), S(4, 7, 23), Gosset 8 X Si (i = 1, 2, 3), Gosset 6 x Gosset i (i = 7, 8), Table 6. Our best upper bound b on rnp'(r) for large r. Its reciprocal gives a lower bound 1.b. on X,,,(S(")(r)). n

r

b

1.b.

2 3 4 4 5 6 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24

0.81941 0.91841

0.28571428 0.18750000 0.12800000 0.13793103 0.09539473 0.07081295 0.05311365 0.03419769 0.04264237 0.02882153 0.03460966 0.02234835 0.02850742 0.01789325 0.02404499 0.01437590 0.01902089 0.01203324 0.01490547 0.00981770 0.01301242 0.00841374 0.00990294 0.00677838 0.00847505 0.00577854 0.00767212 0.00518111 0.00680895 0.00380311 0.00515955 0.00318213 0.00436088 0.00267706 0.00329095 0.00190205 0.00228112 0.00132755 0,00160037 0.00107286 0.00129086

4 6 8 8

Do

0.92987 0.93851 0.94524 1.03349 ca

1.05654 ca

1.12915 ca

1.44913 ca

1.85404 ca

2.61007 ca

3.93276 Do

2.59807 Do

8.04155 Do

5.88784 Do

6.45497 Do

5.88784 on

5.88784 Do

7.22649 ca

6.03845 ca

8.59909 Do

8.34770 ca

10.9457

11 15 19 30 24 35 29 45 36 56 42 70 53 84 68 102 77 119 101 148 118 174 131 194 141 263 194 315 230 374 304 526 439 754 625 933 775

configuration

morning star pentagonal cross polytope halfcube Gosset Steiner3cube Gosset Steiner3cube n2cube Steiner3cube n2cube quartercube n2cube n2cube n2cube G6xG6 nlcube Steiner3,6/8cube n2cube G7xG7 n2cube Steiner3,6/8cube n2cube Steiner3,6/8cube n2cube L44 44443 n2cube L444444 n2cube Steiner3,6/8cube nlcube L3445 nlcube L333 L23 L35 L2 L2 linspace linspace

dim

3 4 4 5 6 7 8 7 9 9

10 10 11 11 12 12 13 13 14 14 15 15 16 15 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24

3m

L.A. SzCkely, N.C. Wormold

G22 and L, as well as those indicated in Tables 2 and 3. Not all of the remaining general configurations were useful in all dimensions. For, example, the n2-cube was used of dimensions 4, 5 and 9 to 21, but the n3-cube was only used of dimensions 10 and 11.

8. Open problems 1. Is miz)s !? The method of the present paper does not seem strong enough to solve this. 2. What is inf,,,,,mg)? This is closely related to the following question of Erdos: what is maxlH,,lX(GH) in the plane? It is known that these quantities can be about l / r G ( r G ) , but it is not known if they have polynomial bound. For example, the best known results for r = 2 are

(See Proposition 1.2(a), (b) and Corollary 2.5 in [34].) 3. Is it possible to use our method to prove an exponentially small upper bound for my)? If the answer is yes, the configurations involved must have strong design-like properties. Is an improvement of (1.2 + o(l))-” possibly this way? Is there an E > 0 and a configuration in BB“ of at least (1 B)” vertices, such that the shortest distance occurs at least in B% of the distances? If the answer is yes, mathematical induction may work. 4. What is the connection begween the structure of H and mg)? There are some results for n = 2. The first author proved [34] mg!b)=s if b / a 2 1.401 and conjectured [351 that

+

r n Z ) = O ifsupH=m,

(5.1)

Using a weaker concept of upper density (measured on concentric cubes around the origin) Weiss [37] proved (5.1) by ergodic theory. An alternative proof was given by Falconer and Marstrand, who also proved (5.2) (see [12, 10, 131). Furstenberg conjectured the generalization of Weiss’s theorem: “If a set of the plane has positive upper density measured in concentric cubes centred in the origin, then it contains vertices of a regular triangle with every long enough edge length. Further, it contains every big enlargement of every finite planar configuration”. The same conjecture will be formulated for the upper density concept used by us as well. (5.1) is a special case of the conjecture for a two point configuration.

Bow& on the measureable chromatic number of Iw"

371

5. How can one construct large 1-independent sets in R"? This seems to be related to the sphere-packing problem in R" (see [23], Assertion 2). On the other hand, even in R2, the densest sphere-packing does not give the densest 1-independent set (see Croft [8]). In R3 (from a table of Leech [25]), putting spheres of radius f at the centres of spheres in a best-known packing yields a set of density 10.80-'. Using the same table and a similar construction yields the following densities in higher dimensions n: n = 3 :10.80-' n = 4 :25.83-' n = 5 : 82.54-' n = 6: 171.62-' n = 7 :1634.lo-'. The growth here is faster than exponential. Another approach to proving rnP)al/k is to prove the existence of a measurable k-colouring of Gl in R". The bound rnP) 3 (3 o(1))-" can be obtained by this means, but the function o(l), coming from Butler [2], is hard to compute for small values of n. 6. What happens to xm and if the distances in R" are defined by another norm? 7. In the definition of measurable chromatic number we may replace the Lebesgue measurable sets by any other a-algebra and ask for the corresponding restricted chromatic number. The question looks the most meaningful for sets having the property of Baire (see Oxtoby [29]) because of the deep analogy between them and the Lebesgue measurable sets. Is it true that the chromatic number x b , in which the colour classes must have the property of Baire, is always equal to xm?

+

ME)

Acknowledgement The blocks of S(5,8,24) were obtained with the kind help of Marston Conder using Cayley.

References [l] N.G.de Bruijn and P. Erdiis, A colour problem for infinite graphs and a problem in theory of relations! Nederl. Mad. Wetensch. Proc. Ser. A. 54 (1951) 369-373. [2] G.J. Butler, Simultaneous packing and covering in Euclidean space, Proc. London Math. Soc. 25 (1972) 721-735. [3] P.J. Cameron and J.H. van Lint, Graphs, codes and designs, London Math. SOC. Lecture Notes Ser. 43 (Cambridge University Press, 1980). [4] J.H. Conway, A group of order 8,315,553,613,086,720,000, Bull. London Math. Soc. 1 (1969) 79-88. [S] H.S.M.Coxeter, Regular Polytopes (Dover Publ. Inc., New York, 1973). [6] H.S.M. Coxeter, Regular Complex Polytopes (Cambridge University Press, London, 1974).

372

L.A. Sz&ely, N.C. Wormald

171 H.S.M. Coxeter, The pure Archimedean polytopes in six and seven dimensions, Proc. Camb. Phil.SOC.24 (1927) 1-9. [S] H.T. Croft, Incidence incidents, Eureka (Cambridge) 30 (1%7) 22-26. [9] P. Delsarte, J.M. Geothals and J.J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977) 363-388. [lo] P. Erdiis, Some combinatorial, geometric and set theoretic problems in measure theory, in: Measure Theory, Oberwolfach 1983, Lecture Notes in Mathematics 1089 (Springer-Verlag. 1984) 321-327. 111) P. Erd6s and C.A. Rogers, Covering space with mnvex bodies, Acta Arithmetica 7 (1962) 281-285. 1121 K.J. Falconer and J.M. Marstrand, Plane sets with positive density at infinity contain all large distances, Bull. London Math. SOC.18 (1986) 471-474. [13] K.J. Falconer, The realization of small distances in plane sets of positive measure, Bull. London Math. Soc. 18 (1986) 475-477. [I41 K. J. Falconer, The realization of distances in measurable subsets covering W", J. Combinat. Theory A 31 (1981) 187-189. [ 151 P. Frank1 and R.M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (4) (1981) 357-368. [I61 D. Gorenstein, Finite Simple Groups: An Introduction to Their Classifications (Plenum Press, New York-London, 1981). 1171 T. Gosset, On the regular and semi-regular figures in space of n dimensions, Messenger of Mathematics 29 (1900) 43-48. 1181 H. Hadwiger, Ein Uberdeckungssatz fur den Euklideschen Raum, Portugaliae Math. 4 (1944) 140-144. 1191 H.Hadwiger, Ungelhte Probleme No. 40, Elemente der Math. 16 (1%1) 103-104. [20] M. Hall, Jr.. Combinatorial Theory (Blaisdell Co., Mass., 1%7). [21] S. Helgason, Groups and Geometric Analysis (Integral geometry, Invariant differential operators and Spherical functions) (Academic Press, 1984). [22] Kai Lai Chung, On the probability of the Occurrence of at least rn events among n arbitrary events, Ann. Math. Stat. 12 (1941) 328-338. [23) D.G. Larman and C.A. Rogers, The realization of distances within sets in Euclidean space, Mathematika 19 (1972) 1-24. [24]J. Leech, Some sphere packings in higher space, Canadian J. Math. 16 (1964) 657-682. [25j J. Leech, Notes on sphere packings, Canadian J . Math. 19 (1%7) 251-267. [26] L. bv&, Combinatorial Problems and Exercises (North-Holland, Amsterdam-New YorkOxford, 1979). (271 L. Moser and W. Moser, Solution to Problem 10, Canad. Math. Bull. 4 (1%1) 187-189. I281 W.Moser, Research problems in discrete geometry (mimeographed) (1981). [29] J.C. Oxtoby, Measure and Category; A Survey of the Analogies Between Topological and Measure Spaces (Springer-Verlag, New York, 1971). [N]D.E. Raiskii, The realisation of all distances in a decomposition of the space R" into n + 1 parts (Russian), Mat. Zametki 7 (1970) 319-323. 1311 D.E. Rasiskii, the realisition of all distances in a decomposition of the space R" into n + 1 parts, Math. Notes 7 (1970) 194-1%. [32] A. Renyi, Foundations of Probability (Holden-Day, San Francisco-Cambridge-LondonAmsterdam, 1970). [33] L.A. Szekely, Inclusion-exclusion formulae without higher terms, Ars Combinatoria 23B (1987) 7-20. [34] L.A. Szdkely, Measurable chromatic number of geometric graphs and sets without some distances in Euclidean space, Combinatorica 4 (1984) 213-218. [35] L.A. SzCkely, Remarks on the chromatic number of geometric graphs, Graphs and Other Topics, Teubner-Texte zur Mathematik, Band 59 (Leipzig, 1983) 312-315. [36] P. du Val, On the directrices of a set of points in a plane, Roc. London Math. Soc. 35 (1933) 23-74. 1371 B. Weiss, Personal communication. I381 D.R. Woodall, Distances realized by sets covering the plane. J. Combinat. Theory A 14 (1973) 187-20.


373

A SIMPLE LINEAR EXPECTED TIME ALGORITHM FOR FINDING A HAMILTON PATH Andrew THOMASON Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 ISD, England We give a simple algorithm which either finds a hamilton path between two specified vertices of a graph G of order n, or shows that no such path exists. If the probability of an edge in G is p L 12n-f the algorithm runs in expected time cnp-’ and uses storage cn, where c is an absolute constant.

It is well known that the problem of finding a hamilton cycle in a graph is NP-complete, and there is some value in having algorithms which are fast for most input graphs. Let %(n,p) be the probability space of graphs with n labelled vertices, where edges appear with probability p. Johnson [9] asked whether there exists an algorithm with linear expected running time, when the input is a randomly chosen element of % ( n , p ) . (The algorithm is allowed to enquire of an oracle which tells whether two specified vertices are adjacent; this commonly employed oracle models many situations, in which such an enquiry is cheap.) Gurevich and Shelah [7] answered the question by displaying an algorithm which runs in expected time cnlp, c being an absolute constant, and requires O(n’) storage, provided p is constant. In fact some of their argument works if p >> n-4. In this note we give another such algorithm, which has the advantages of being simpler, requiring only O(n) storage and working for p 3 12n-4. The order n / p for the expected time is best possible, as observed in [7],since in order to find the n edges of the cycle we expect to have to ask at least n / p questions of the oracle. Angluin and Valiant [l] gave an algorithm which almost surely finds a hamilton cycle in polynomial time if p 2 c(log n ) / n , and an improvement appears in Shamir [ll]. Finally BollobPs, Fenner and Frieze [3] gave an algorithm which is about best possible. One of the best known algorithms for finding a hamilton path [8] requires time n22”, so the problem is one of finding a fast algorithm which works on all graphs in % ( n , p ) except for a proportion somewhat smaller than 2-“. This will be our algorithm A2. A very fast algorithm, which works on most graphs but not on as many as A2 does, is used to bring the running time down to O ( n / p ) . This will be algorithm Al. The algorithm A2 was originally based on a study of hamilton cycles in “concrete random graphs” in [12]. 0012-365X/89/$3.500 1989, Elsevier Science Publishers B.V. (North-Holland)

374

A. Thomason

We shall suppose that the algorithm has to find a hamilton path in an input graph G between two specified vertices s and t. (An easy modification gives the hamilton cycle version.) The graph G will be randomly chosen from %(n, p), with p 3 12n-i and n large. If Q is an s - I path and u and u are vertices on Q , we write UQUto denote the section of Q between u and u . We say u precedes u on Q if when traversing Q from s to t the vertex u is encountered before the vertex v. We say an algorithm is successfur if it finds a hamilton s - c path or proves that G does not have one. It fuih if it gives up without succeeding. Our algorithm is as follows.

Step 1. Apply algorithm A l . Step 2. If algorithm A1 fails, apply algorithm A2. Step 3. If algorithm A2 fails, apply algorithm A3.

If at any stage algorithm A1 has made more than 13n/p probes on edges, it fails. Step 1. Set Q = s. Step 2. Repeat whilst n - lQl k 14nil: replace the path Q = sQw by sQwx where x is a neighbour of w in G - Q - t. If no such neighbour is found, algorithm A1 fails. Step 3. Let Q =sQv. Find a set Y of 13.41 neighbours of u in G - Q - t and a set Z of [3n'] neighbours of t in G - Q - Y . Find an edge y z , y E Y , z E 2. Replace Q by sQuyzr. If the sets Y, 2 or the vertices y, z cannot be found, algorithm A1 fails. Step 4. Repeat until Q is a hamilton s - t path: Choose w E G - Q . Find two neighbours u and u of w which are adjacent on Q. If no such pair exists, algorithm A1 fails. Replace Q by sQuwuQt. In order to present the main algorithm we need to define a s2-mutching in a bipartite graph with vertex classes X and Y.This is a set of edges F such that each vertex of X is incident with at most two edges of F and each vertex of Y is incident with at most one edge of F. A maximum SZmatching is a ~Zmatching with the maximum possible number of edges. Algorithm A2

Step 1. Find V,, the set of all vertices in G of degree at most pn/4. If IV,l 3 24p-' algorithm A2 fails. If V, = B go to step 5.

Simple linear expected time algorithm

375

Step 2. Construct a maximum s2-matching from V, to G - V,- {s, t } . Let V, be a set of 2 IV,l vertices of G - V, - {s, t } containing the vertices of degree one in this s2-matching. Step 3. Let H be the subgraph of G induced by V, U V, U {s, t } . Form H* from H by joining every pair of vertices in V ( H * )- V,. Step 4. Find a hamilton s - t path Q* in H*, using any algorithm such as A3. If no such path exists, then G has no hamilton s - t path and algorithm A2 succeeds. Let k = [ 2 m 1 . Given vertices u, v in G - V, and a subset X c V ( G ) , a 3-path in G - X is a path uulvlu,found by choosing k neighbours of u in G - X - {v}, a disjoint set of k neighbours of v in G - X - { u } , and finding some edge between these sets. Step 5. If V, # 8, construct from Q* an s - t path Q in G with V, c V ( Q ) as follows: if uQ*v is a maximal subpath of Q* contained in E ( Q * ) E ( G ) , replace uQ*v by a 3-path uulvlvin G - V ( Q * ) , and call the new path Q * . Repeat whilst E ( Q * ) - E ( G ) Z0, then put Q = Q * . If V, = 0, let Q be a 3-path ssltlt in G . If at some stage a required 3-path is not found, algorithm A2 fails. Step 6. Repeat whilst lQl S 3k: select adjacent vertices uv on Q with u, v E G - V,, and replace uv in Q be a 3-path in G - V ( Q ) . If no such 3-path is found, algorithm A2 fails. Step 7. Repeat until Q is hamilton s - t path: choose w E G - Q . Construct 2k vertex disjoint paths Pl, . . . ,Pu, of length at most two from w to Q - s as follows. Choose 5k neighbours of w . If 2k of these lie on Q - s, these give P,, . . . , P2k. Otherwise select 3k neighbours of w not on Q , and find a maximal independent set of edges between these neighbours and Q - s. These give P,, . . . ,Pu, unless there are less than 2k independent edges, in which case algorithm A2 fails. Let { x l , . . . ,X 2 k ) be the end vertices of these paths and let { y , , . . . ,y2k} be their predecessors on Q . If there is an edge yiyj, replace Q by sQyiyjQxiewP+jQt. If there is no edge yiyj, algorithm A2 fails.

Algorithm A3 Any algorithm which always succeeds in time 2"+"(") and in space O ( n ) may be used here. It is shown, e.g. in [7],that the dynamic programming algorithm of [8] and the proof of Savitch's theorem [lo] can be blended to this end. Thorem. Algorithm A is successful. The expected running time is at most cnlp, provided p 3 12n-i, and the storage is at most cn, where c is an absolute constant. Proof. The proof follows directly from Lemmas 1 and 2 below.

A. Thomason

376

Lemma 1. The algorithm A 1 succeedr with probability at least 1 - 2-75'ppn-2. The time used and storage required are bounded by cnlp and cn respectively, where c is an absolute constant.

Lemma 2. The algorithm A2 succeeds with probability at least 1 - 2-2n+o(n).The time used and storage required are bounded by ~ ( 2 ' ~+ 'n'/p) ~ and cn respectively, where c is an absolute constant. Before proving these lemmas, we define P(n, p , k) to be the probability of scoring at most k successes in a sequence of n independent Bernoulli trials, each with probability p of success. If k = ( 1 - p ) p n with p > 0 then

P(n, p , k) < e-p2pnn

(-F)

(see f l ] ;this follows from ChernoB's bound [4],or see [2, chapter 11).

Proof of Lemma 1. A graph G in %(n, p ) can be thought of as being generated by the following process. For each pair u,u E V ( G ) , conduct b independent Bernoulli trials, each with probability p ' of success, where ( 1 -p')* = 1 - p . If any trial is successful, the edge u v is inserted; if all are unsuccessful, the edge uv is let out. Now algorithm A 1 never attempts to establish the non-existence of a hamilton path, so the probability of its failure on a random input graph is simply the probability that it fails to find a hamilton path. Since the algorithm questions the oracle at most three times about any edge, an upper bound for the probability of failure of the algorithm can be obtained under the assumption that the answers to each question are independent of previous answers, and each answer is "yes" with probability p' > p / 3 , where ( 1 - P ' ) ~= 1 - p. The constraints on time and storage are manifest, so we just check the first claim of the lemma. Let I = 14nfI - 1. The probability that some vertex w in step 2 fails to have a neighbour x is at most (1 -p')'. Hence the probability of step 2 failing to work after 4np-I probes is at most

p(4np-',p', n

-

[4n51)+n(l -p')'<e-"/24+ne-p"30 it follows likewise since

6 > x n + y >xn

3 (1- l / r ) n ,

(c) is similar to (b). 0 If v e V ( G ) and ScVCG), N ( v ) denotes the set of vertices adjacent to (“neighbouring”) v, N ( S ) : = U , , , N ( v ) and the binding number [6] of G is defined to be bind(G):=min{IN(S)I/ISI :Sc V(G), SZB, N ( S ) # V(G)}.

Theorem 2.4. Zf r 3 2 and bind(G) >y = t ( r - 1+ V

m),

then K,,, E G.

Proof. We may suppose inductively that K, E G. Suppose that no vertex of G \ K, is adjacent to all the vertices of K,. Suppose that exactly s vertices of G\K, are non-adjacent to exactly one vertex of K, and n - r - s vertices of G\ K, are non-adjacent to two or more vertices of K,. For some v in K,, at least s/r vertices of G\K, are non-adjacent to v but are adjacent to all other vertices of K,, so (if K,,, & G) together with v they form an independent set X of cardinality at least

D.R. Woodall

392

1

+ s f r . Thus n - 1 - s / r a IN(X)I > y

whence

1x1 a y ( l + s / r ) ,

r(n - 1 ) > y ( r + s ) + s.

Also, for some w in K,, at least (s + 2(n - r - s ) ) / r vertices of G\K, are non-adjacent to w. Let Y consist of these together with w itself, so that

whence

r(n - 1 ) >y(2n

- r - s).

Averaging ( 1 ) and ( 2 ) and using ( 2 ) again gives

r(n-I)>ny+$s>ny+

y ( 2 n - r ) - r ( n - 1) 2Y

,

whence

2ny2 - 2n(r - 1)y + ry

- r ( n - 1 ) < 0,

so certainly 2y2-2(r-1)y-r ( r + l ) x + y .

(5)

Thus Kr+l is the smallest graph that does not satisfy the condition. (Compare Characterization 2.3(b).) And if G is any graph that does not satisfy the condition, with (necessarily) n > r vertices, then

G>xn+ya

r-1-y n r

r-2 +y k n + 1 =n + 1 r

~

2n/t

(6)

since y 6 1. If r k 2 then K,+l S G by Theorem 3.1, while if r = 1 then 6 > 0 implies that K,+, = K , S G . 0

Characterization 3.3. Let x and y be real numbers such that x < 1, y > 0 and + y B 0. Let P be the property

x

{ N ( S ) l < xn

Then qP') property.

+ y IS1for some S c_ V ( G )such that S # 0 and N ( S ) # V ( G ) .

= YP")= @(K,+,), where

K, is the largest complete graph with the

Forbidden graphs

395

Proof. If T := V(G)\N(S), so that T f 0, N ( T ) # V ( G ) and S E V ( G ) \ N ( T ) , it is easy to see that (N(S)IGxn+y I S l ~ $ I N ( T ) ( c y - ' ( x + y - l ) n + y - ' ( T I

and I N ( S )~~y - ' ( x +y - 1)n +y-' IS1 j JN(T)J b x n +y IT]. Thus (replacing x and y by y-'(x + y - 1) and y-' if necessary) there is no loss of generality in supposing that y S 1. The integer r satisfies (5) (and therefore (4)). We must prove that if IN(S))>xn + y (SI for every S E V ( G ) such that S # 0 and N ( S ) # V ( G ) , then K,,, c G . But this hypothesis implies (6), and K,,, G G by Theorem 3.1 if r 3 2. If r = 1 we get a contradiction to G being edgeless by taking S:= V ( G ) . 0

4. Further results involving the minimum degree

In this section and the next we shall investigate %(P'), where P = P(x, y) is the property 6 c x n + y. In this section our main aim is to justify the so-far unexplained regions in Fig. 1 . I f x + y < O t h e n K 1 $ $ ( P ) a n d s o %(P')=8. I f x + y a O a n d x b l thenall graphs satisfy P. If x < 0, then qP') and %(P) are both finite sets. If n = 0, then q P ' ) depends only on Ly] :if Ly] < 0 then %(P') is empty, if Ly] = 0 then %(P') consists of the edgeless graphs, if ly] = 1 then %(I")comprises the circuit-free graphs (forests), and so on. From Characterization 2.3(b) we know what happens if y 6 0 (see Fig. l), so from now on we assume 0 <x < 1 and y > 0. The regular trivalent graphs with n = 10 and girth 5 (the Petersen graph), and n = 14 and girth 6, show that there exist values of n and t for which the following result is best possible.

Theorem 4.1. Let t 3 4 be an integer and G a graph with n vertices and minimum degree

4:

- t)/(5t- 14) + 2 - r)/(7t - 26) + 2

(t even), (t odd).

Then G has girth Gt.

proof. The condition ensures that 6 2 2 if n < t and 6 3 3 if n a t . We may therefore suppose that 6 3 3. If G has girth >t and r is even, then all the vertices at distance 0, 1,2, . . . ,at from a given vertex are distinct, and so

n 3 1+ 6 + 6(6 - I ) + S(6 - I)*+ -

+ 6 + 6(6 - 1 ) [ 1 + ( i t - 2)2] = 1+ 6 + S(6 - l ) ( t - 3 )

31

+ 6(6 - I)+'

D.R. Woodall

3%

since 6 - 1 2 2. And if t is odd then a similar argument starting from a pair of adjacent vertices gives

n > 2 { 1 + (6 - 1) + ( 6 -I)’+ * . + ( 6 - I)*(~-’) 1 3 2 { 1 + ( 6 - 1) + ( 6 - 1)*[1+ f ( t - 5 ) 2 ] }

=2{6 + (6 - l)’(t -4)}. Suppose first that t is even. Then, since 6 3 3 and so 6’ - 6 3 56 - 9,

n - f a 1 + 6 + 6 ( 6 - l)(t - 3 ) - t 31

+ 6 + (56 - 9)(t - 3) - t

= ( 6 - 2)(5t - 14).

And if t is odd then, since 6’ - 26

n - t 3 26

+ 1 3 46 - 8 = 4( 6 - 2),

+ 2(6 - l)’(t - 4) - t

3 2(6

- 2) + 8(6 - 2)(t - 4) - t + 4

L (6 - 2)(7t - 26).

In either case we have the required contradiction.

0

Characterization4.2. Let t 3 4 be an integer and let x and y be real numbers such that (a) 2x + y 3 1, (b) tx + y C 2 , (c) ( t + 1)x + y 3 2 and

(4

- 14)x + y a 3 ((6t (8t 26)x + y 3 -

Let P be the property 6 6 xn

qP‘)

( t even), ( t odd).

+ y . Then

= qP“) = F(C3,

. . . , C,) = F(C3, . . . , C,),

the class of graphs with girth at least t

+ 1.

< 4 and y > 0. (b) ensures that C3, . . . , C,q q P ) . We must prove that if b >xn + y then G contains one of C,, . . . , C,. This is obvious if n = 1 (vacuously, since 6 2 1 ) or 2 S n G t (when 6 3 2 ) . If t + 1 s n < 6t - 14 and t is even, then (c) implies 6 2 3 and Theorem 4.1 implies Ppoof. (a), (b) and (c) imply O < x

the result, while if n 3 6t - 14 and t is even then (b) and (d) imply (5t - 14)6 > (51 - 14)(xn

+ y ) = [(6t - 14)x + y ] ( n - t ) - (tx + y ) [ n - (6t - 14)] 3 3(n

- t ) - 2[n - (6t - 14)]

=n -t

+ 2(5t - 14)

and the result again follows from Theorem 4.1. The result for odd t follows similarly. 0

Forbidden graphs

397

Corollary 4.2.1. Zf t a 4 and P is any of the properties n n-2 n-1 +1, 6 s + 1 and 6 0. If n d 6 and G 4 q P ) , then (c) implies that

6 >xn

n + y Z x n +Y-36 2'

so that K3 G G. And if n 3 6 and G $ %(P) then (a) and (c) imply that 36 > 3(xn + y) = (6.x + y)(n - 3) - (3x + y)(n - 6) 3 3 ( n - 3) - 2(n - 6) = n + 3, so the result follows from Theorem 4.5.

0

Theorem 4.7. If G h a graph such that 6 > $(n + 4) then G has a K 3 or a K3,3as an induced subgraph.

Proof. This follows from Theorem 4.3 if n d 10, so suppose n 3 11, 6 2 5. Suppose G does not contain K3 or K3,3.It must contain K2,3, since C , E G by Theorem 4.1, and if no vertex of G \ C4 is adjacent to more than one vertex of C4 then G \ C4satisfies the hypothesis and so K2,3 c G \ C4 by induction. Define u, v, U,V , W as in Theorem 4.3 and let

x:={ X E u u v: I N ( X n) ( u u v)l32}; since K2,31~G we can choose u, v, U, V so that X # 0. Choose a E X such that s := (N(a)f l WJ is maximal. Note that n d 346 - 44, so IWl= n - 26 d 1$6 - 44. Suppose w.1.o.g. a E U.There are two cases to consider.

Case 1. N ( a ) n (V\X) # 0. Let b belong to this set. Since N ( a ) nN ( b ) = 0 and IN(b) n Wl s 6 - 2, it follows that s + S - 2 s IWI d 146 - 44 and s d fS - 2i. Thus JN(a)n VI 3 6 - 1- s 3 $6 + 14. For each b' in N ( a ) rl X , IN(b') n U l 2 $6 + 14 similarly (by the maximality of s), and so IN(a) nX(d 1 or else K2,2 (UU V) and K3,3c G. So IN(a) n (V\X)l 3 46 + 4 3 3, and any three of these vertices form a K3,3with W since they are all adjacent to at least 6 - 2 verticesof Wand 3[lWl-(6-2)]dIWl-3. Case 2. N ( a ) n (V\X) = 0. Then there are at least max(2, 6 - 1- s) vertices of V that are adjacent to a, and each of them is adjacent to at most

D.R. Woodall

400

+

min(s, 1 W J- s) vertices of W and hence to at least max(6 - 1- s, s - 46 34) vertices of U. So some vertex in U\{a} is adjacent to two vertices of N ( o ) n V (giving a K3,3in C ) unless max(2, 6 - 1-s)max(b

- 2 -s,

s - 46 +24)

s IU\{a}l= 6 -2;

this implies

(6 - 1 -s)(6

- 2 - s ) =s6

and rnax(2, 6 - 1 - s)(s

-2

- 46 + 2 t ) c 6 - 2.

(11) is obviously false if s 2 S - 3, and so if (10) and (11) hold then they must both do so when 6 - 2 - s = s - 46 +2$, s = $6 - $. This gives

+ 5)+(S + 1) s 6 - 2 or h2 - 106 + 37 S 0, which is impossible. i(6

Characterization 4.8. Let x and y be real number such that (a) 3x + y < 2, (b) 6 x + y < 3 , (c) 4 x + y 2 2 a n d ( d ) 1 0 x + y 3 4 . LetPbetheproperty 6 a x n + y . Then q P ' ) = q P " ) = W ( K 3K3,3). , (Note that graphs such as 2oC5 and K5,s minus a 1-factor, which have n = 10, 6 = 4 and no K 3 or K3,3, show that (d) correctly delimits the region labelled K 3 , K3,3in Fig. 1. Here 2oG is the graph obtained from G by doubling each vertex and replacing each edge by a K2,*.)

Proof. (a) and (b) ensure that K 3 and K3,3$ q P ) . (b) and (c) imply that y > 0. If n c 4 and G $ q P ) , then (c) implies that Yn n 6>xn+y~xn+-2-, 4 2 so that K 3 c G. If 4 G n C 10 and G @ q P ) then (c) and (d) imply that 66>6(xn + y ) = ( 4 x + y ) ( l O - n ) + ( l O x + y ) ( n - 4 ) 3 2(10 - n )

+ 4(n - 4) = 2(n + 2),

so that G contains a K 3 or a K3,3 by Theorem 4.3. And if n 3 10 and G $ %(P) then (a) and (d) imply that

76 > 7(xn + y) = (31 + y)(10 - n ) + (lox + y)(n - 3) 2 2(10 - n ) + 4(n - 3) = 2(n + 4).

so the result follows from Theorem 4.7. I7 We write K,(r) for the Turan graph T,(rt), the complete r-partite graph on r sets of t vertices each. The following theorem makes no pretence of being best possible-indeed, if r = 2 it is worse than Theorem 4.3 whenever n > 2r - 2-but it is simple to prove and enables us to fill in the depth of the notches between the K,,, regions in Fig. 1.

Forbidden graphs

401

Theorem 4.9. Let r 3 2 and t 3 1 be integers and G a graph with n vertices and minimum degree 6 > n - (tn - t + l ) / ( r t- 1). Then K,,, or K,(t) E G. Proof. We prove the result by induction on t, noting that it holds if t = 1 (when the conclusion is that K , E G) by Theorem 2.l(c). So suppose that t 3 2 and neither K,,, nor K,(t) E G. We may suppose inductively that K,(t - 1)E G. Let G, = T,(nl) be the largest r-partite Turfin subgraph of G (necessarily induced), where n l = r ( t - l ) + s and O S s < r . Note that G1#G, since if n = n l a n d s = O then d 1 : = 6 ( G 1 ) = n l - t + 1 whereas 6 > n - t + l , and if l S s < r then a,= n1- t whereas 6>n-t

+ l-- rt ts- 1 > n - t .

There is no loss of generality in supposing that G \ G1 does not satisfy the degree condition, so that at least one vertex of G\Gl is non-adjacent to fewer than t n , / ( r t - l ) S t , hence to at most t - 1, vertices of G,. But this means that G contains a K,,, or a T,(n, l ) ,contrary to hypothesis. 0

+

Characteriza~on4.10. Let r > 2 and t 3 1 be integers and x and y real numbers such that (a) rtx + y < (r - l)t, (b) r(t - 1)x + y 2 (r - l)(t - 1) and (c) (rt - 1)x > rt - t - 1. Let P be the property 6 S xn + y. Then %(P') = %(P") = v ( K r + 1 , Kr(t)).

Proof. (a) and (b) imply that rx < r - 1 and y > 0. (a) and (c) imply x + y < 1 and so (with the previous sentence) ( r + 1)x + y < r, whence K,,, @ %(P). (a) ensures that K,(t) @ %(P). If n S r(t - 1) and G @ %(P) then (b) implies 6>xn+yaxn+-

yn n ( r - 1) 2 , r ( t - 1) r

so that K,,, G G by Theorem 2.l(c). And if n >r(t - 1) and G @ %(P) then (b) and (c) imply

(rt - 1)6 > (rt - l ) ( x n + y ) = (rt - l)[r(t- l ) x + y ] + (rt - l ) x [ n- r(t - l ) ] 2 (rt - l ) ( r - l ) ( t - 1) + [rt - t - l ] [ n- r(t - l ) ] = (rt - 1)n - (tn - t

+ l),

so that G contains K,,, or K,(t) as a subgraph by Theorem 4.9.

0

5. The shift theorems In this section we attempt to relate the forbidden subgraphs for the property 6 ~ x + ny for different values of x and y. If P = P(x, y ) denotes this property, let

D.R. Wmdall

402

-P denote the property 6 > x n + y . Let S c ( x , y ) be the set of graphs in q - P ) that have no proper subgraph in Y-P), and let P ( x , y ) be the set of graphs in q - P ) that have no proper induced subgraph in Y - P ) , so that S c ( x , y ) c P ( x , y ) and

%(F) = %P")

= W(SC(X,y ) ) = W(9"(x,

y ) )= W ( P ( x , y)).

In all the examples we have seen so far, it has been the case that F ( x , y ) = F ( x , y ) , and all the graphs in this set are regular. Lest the reader presume that this is true in general, we should immediately point out a consequence of Corollary 4.2.1 and Theorem 5.2 (below), that (for example) C5+ K2E F(:2). , (Here denotes "join": every vertex of C5 is adjacent to every vertex of E2.) Also C5+ K 2 E . P ( a , 2), since this graph has the same minimum degree and number of vertices as C5+ K2,and it is easy to check that every induced subgraph of it satisfies 6 ss $n + 2. Thus the irregular graph C5+ R2 is in both 2 ) and Sc(a,2 ) , while C5+ K2is in the second set only. If x f 2 , define the ship of ( x , y ) to be

"+"

$-(a,

S fixes the line x = 1, and on any other line through (1,O) the segment between 0 and 4 is mapped onto the segment between $ and 3, which is mapped onto the segment between 3 and f , and so on. Thus in Fig. 1 the region labelled K2 is mapped by S onto the region labelled Kf, which is mapped onto the region labelled K.,,and so on. If G is a graph with n vertices and minimum degree 6, define the shift S(G) of G to be G K,,-b; it has 2n - 6 vertices and minimum degree n. From Fig. 1 we see that the following conjecture is true if x + 2y s 1. The dotted lines in that figure between x = 1 and x = $ are the shifts of the continuous lines between x = 0 and x = 4, and so represent conjectures whose truth is implied by the following:

+

Conjecture (The Shift Conjecture). If 0 =sx s 1 and y b -1, then Sc(S(x, y ) ) = {S(G):G E F ( x , y)} and S"(S(x, y ) ) = { S ( G ) :G

E

Sc(x, y ) } .

w r n 5.1. l f O s x 6 l a n d y a - 1 , then { S ( G ) : G E F ( X , ~ ) } E ~ ~ ( S ( X , ~ ) ) and { S ( G ) : GE 9"(x, Y ) ) c 9 c ( S ( x , Y ) ) .

Proof. Note first that

so that G satisfies P ( x , y) if and only if S ( G ) satisfies P ( S ( x , y ) ) . Suppose now that G is a graph in 9"(x, y ) , so that G does not satisfy P ( x , y )

Forbidden graphs

403

but every proper induced subgraph of it does; in particular,

6 - 1 d x ( n - 1) + y .

(12)

By the previous paragraph, S(G) does not satisfy P(S(x, y)). We must prove that every proper induced subgraph of it does. Such an induced subgraph is either G1+ Kr,where Gl is a proper induced subgraph of G and r d n - 6, or G + Kr where r d n - 6 - 1. The latter graph has n + r vertices and minimum degree 6 + r and satisfies P(S(x, y)) since (12) is equivalent to (2 - x ) ( n

- 1 ) c 2 n - 6 - 1 +y,

(13)

which implies (2 - x ) ( 6

+r)

(n + r ) + y

because 6 + r d n - 1 and 2 - x degree S1, then

2 1.

And if Gl has n1 vertices and minimum

d1c x n l + y and (2 - x ) n l c2n1- + y (14) since G1 satisfies P, and Gl+Kr has n l + r vertices and minimum degree

min(nl, S1+ r). In either case the required inequality follows immediately from (14)This proves the second inclusion in the statement of the theorem. To prove the first, suppose that G is a graph in S c ( x , y) E 9 " ( x , y). We must prove that every proper subgraph of S(G) satisfies P(S(x, y)). It suffices to consider subgraphs of the same form as before, where now G1 is a proper subgraph of G (not necessarily induced). The only difference in the proof arises if r = n - 6 and n1= n. But then 61C 6, since every proper subgraph of G satisfies P(x, y) and G does not, and since Gl + Kr has minimum degree at most 6 - 1 + r, the result follows from (13). 0 Finally, there is an upwards shift theorem, although it is less interesting as equality frequently does not hold. Define U(x, y) := (x, y + 1- x ) and let U ( G ) be obtained from G by adding a single vertex adjacent to all vertices of minimum degree in G. Theorem 5.2. { U(G):G E F ( x , y ) } c Sc( U(x, y ) ) .

Proof. Let G be a graph in S c ( x , y ) with n vertices and minimum degree 6. Since G does not satisfy P(x, y) and every proper subgraph of it does, deleting any edge of G must lower its minimum degree. Thus no edge of G joins two vertices with degree greater than 6. It follows immediately that G has at least S + 1 vertices of degree 6, so that U(G) has minimum degree 6 + 1. U ( G )clearly has n + 1 vertices. Since 6 > x n + y it follows that 6 + 1 > x ( n + 1) + ( y + 1 - x ) , so that U(G)does not satisfy P ( U ( x , y ) ) . The deletion of any edge from U ( G )

404

D.R. WoodaU

results in a graph with minimum degree 6, which satisfies P ( U ( x , y)) since 6 - 1s xn + y. So it suffices to consider induced subgraphs of U ( G ) obtained by deleting vertices of G , which will have n , + 1 vertices and minimum degree at most d1 + 1 where 6, <xnl +y, and which therefore clearly satisfy P ( W Y ))*

References [l] L.W.Beineke, Characterizations of derived graphs, J. Combinat. Theory 9 (1970) 129-135. f2] L.Lovisz, A characterization of perfect graphs, J. Combinat. Theory Ser. B 13 (1972) 95-98. 131 W.Mader, Homomorphiesatze fiir Graphen, Math. Ann. 178 (1968) 154-168. [4] H. Meyniel, On the perfect graph conjecture, Discrete Math. 16 (1976) 339-342. [5] P. Turin, On an extremal problem in graph theory (in Hungarian), Mat. Fiz. Lapok 48 (1941) 436-452. [6] D.R. Woodall, The binding number of a graph and its Anderson number, J. Combinat. Theory Ser. B 15 (1973) 225-255. (71 D.R. Woodall, Subcontraction-equivalence and Hadwiger’s conjecture, J. Graph Theory 11 (1987) 197-204.


COMBINATORICS 1988 LIST OF CONTRIBUTORS J. Akiyama

I. Algor

Noga Alon

J.-C. Bermond

K. Berrada

Norman Biggs

BCla Bollobhs

J. Bond

Adrian Bondy

Endre Boros Graham Brightwell

Peter J. Cameron Amanda G. Chetwynd

Department of Mathematics, Tokai University, Hiratsuka, Kanagawa 259-12, Japan Department of Mathematics, Sackler Faculty, Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel Department of Mathematics, Sackler Faculty, Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel Laboratoire de Recherche en Informatique, UniversitC Paris-Sud, Bit. 490, 91405 Orsay Cedex, France Laboratoire de Recherche en Informatique, UniversitC Paris-Sud, Bit. 490, 91405 Orsay Cedex, France Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, England Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 lSB, England Laboratoire de Recherche en Informatique, UniversitC Pas-Sud, Bit. 490, 91405 Orsay Cedex, France Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 RUTCOR, Department af Mathematics, Rutgers University, New Brunswick, NJ 08903, USA Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 lSB, EngZand School of Mathematical Sciences, Queen Mary College, Mile End Road, London El 4NS, England Department of Mathematics, University of Lancaster, Cartmel College, Bailrigg, Lancaster LA1 4YL, England

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

c o m b ~ o r i c s1988

406

F.R.K. Chung Michel Deza Reinhard Diestel

Paul ErdEis

Phillippe Flajolet Zoltb Fiiredi

H. Furstenberg Roland Haggkvist AndrL Hajnal

Peter L. Hammer Julie Haviland

A.J.W. Hilton

Y.Katznelson Donald E. Knuth Monique Laurent

Imre Leader

Bell Communications Research, Morristown, NJ 07%9, USA CNRS, UA 212, UniversitC Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 lSB, England Mathematical Institute of the Hungarian Academy of Sciences, ReAitanoda utca 13-15, 1053 Budapest V., Hungary INRIA, Rocquencourt, 78150 Le Chesnay, France Mathematical Institute of the Hungarian Academy of Sciences, 1364, Budapest, P.O. Box 127, Hungary Department of Mathematics, The Hebrew University, Jerusalem, Israel Matematiska Institutionen, Stockholms Universitet, Box 6701, 113 85 Stockholm, Sweden Mathematical Institute of the Hungarian Academy of Sciences, Rehltanoda utca 13-15, 1053 Budapest V., Hungary RUTCOR, Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 lSB, England Department of Mathematics, University of Reading, Reading, P.O. Box 220, Whiteknights, Reading RG6 2AX, England Department of Mathematics, Stanford University, Stanford, CA 94305, USA Computer Science Department, Stanford University, Stanford, CA 94305, USA CNRS, Lamsade, Universitd Paris Dauphine, Place du MarCchal de Lattre de Tassigny, 75775 Paris Cedex 16, France Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 lSB, England Department of Computer Science, Eotvos University, Budapest, Hungary and Department of Computer Science, Princeton University, Princeton, NJ 08544,

USA

List of Contributors

Fumi Nakada Jaroslav NeSetfil

J.L. Nicolas

Boris Pittel Steen Rasmussen

Vojtgch Rod1 P. Rosenstiehl Michael Saks

407

Department of Mathematics, Tokai University, Hiratsuka, Kanaga 259-12, Japan Department of Applied Mathematics, Charles University, Malostranskk n6m. 25, 118 00 Prague 1, Czechoslovakia Groupe Logiques, Mathkmatiques Discr&tes,Informatique Institute des Sciences et de la Matihe, Universitk Lyon 1,/bd. du 11 Novembre 1918, 69622 Villeubanne Cedex, France Mathematics Department, Ohio State University, Columbus, OH 43210, USA Center for Nonlinear Studies and Theoretical Division, MS B258, LANL, Los Alamos, NM 87545, USA and Physics Laboratory 111, The Technical University of Denmark, 2800 Lyngby, Denmark Department of Mathematics, Czech Technical University, Prague, Czechoslovakia Centre d'Analyse et de Mathkmatique Sociale, 54, Boulevard Raspail, 75006 Paris, France RUTCOR, Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA and Bell Communications Research, Morristown, NJ 07960,

USA A. Skkozy

Paul D. Seymour L.A. Szkkely Andrew Thomason

Sinichi Tokunaga W.T. Trotter D. R. Woodall

N .C. Wormald

A Magyar Tudomhyos Akadkmia, Matematikai Kutato Intkzete, Realtanoda u. 13-15, Pf. 127, H1364 Budapest, Hungary Bell Communications Research, Morristown, NJ 07960, USA ELTE Matematikai Intkzet, Mdzeum krt 6-8, Budapest 1088, Hungary Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 lSB, England Tokyo University, Hongo, Bunkyo-ku, Tokyo, 113, Japan Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA Department of Mathematics, University of Nottingham, Nottingham NG7 2RD, England Department of Mathematics and Statistics, University of Auckland, Private Bag, Auckland, New Zealand



409

AUTHOR INDEX Volume 75 (1989) Akiyama, J., F. Nakada and S. Tokunaga, Packing smaller graphs into a graph Algor, I. and N. Alon, The star arboricity of graphs Alon, N., see Algor, I. Alon, N. and B. BollobBs, Graphs with a small number of distinct induced subgraphs Bermond, J.-C., K. Berrada and J. Bond, Extensions of networks with given diameter Berrada, K., see Bermond, J.-C. Biggs, N., Confluence of some presentations associated with graphs Bollobtk, B., Preface Bollobi, B., Paul Erdiis at Seventy-Five BollobBs, B., see Alon, N. BollobBs, B. and G. Brightwell, Long cycles in graphs with no subgraphs of minimal degree 3 Bollobiis, B. and S. Rasmussen, First cycles in random directed graph processes Bond, J., see Bermond, J.-C. Bondy, A., Trigraphs Boros, E. and P.L. Hammer, On clustering problems with connected optima in Euclidean spaces Brightwell, G., see Bollobiis, B.

(1-3) 7-9 (1-3) 11-22 (1-3) 11-22 (1-3) 23-30

(1-3) 31-40 (1-3) 31-40 (1-3) 41-46 (1-3) 1 (1-3) 3-5 (1-3) 23-30 (1-3) (1-3) (1-3) (1-3)

47-53 55-68 31-40 69-79

(1-3) 81-88 (1-3) 47-53

Cameron, P.J., Some sequences of integers Chetwynd, A.G. and A.J.W. Hilton, 1-Factorizingregular graphs of high degreean improved bound Chung, F.R.K. and P.D. Seymour, Graphs with small bandwidth and cutwidth

(1-3) 103-112 (1-3) 113-119

Deza, M., see Laurent, M. Diestel, R., Simplicia1 decompositions of graphs: A survey of applications

(1-3) 279-313 (1-3) 121-144

Erdiis, P. and A. Hajnal, On the number of distinct induced subgraphs of a graph Erdiis, P., J.L. Nicolas and A. S6rkiizy, On the number of partitions of n without a given subsum (I)

(1-3) 145-154

Flajolet, P., D.E. Knuth aand B. Pittel, The first cycles in an evolving graph Fiiredi, Z.,Covering the complete graph by partitions Furstenberg, H. and Y. Katznelson, A density version of the Hales-Jewett theorem fork=3

(1-3) 167-215 (1-3) 217-226

Haggkvist, R., On the path-complete bipartite Ramsey number Haggkvist, R., Towards a solution of the Dinitz problem? Haggkvist, R., A note on Latin squares with restricted support Hajnal, A., see Erdiis, P. Hammer, P.L.,see Boros, E. Haviland. J. and A. Thomason, Pseudo-random hyDergraDhs -_ Hilton, A.J. W., see Chetwynd, A.G.

(1-3) 243-245 (1-3) 247-251 (1-3) 253-254 (1-3) 145-154 (1-3) 81-88 (1-3) 255-278 (i-3j 103-112

I

-

(1-3) 89-102

(1-3) 155-166

(1-3) 227-241

410

A h r index

Katznetson, Y.,see Furstenberg. H. Knuth, D.E., see Flajolet, P.

(1-3) 227-241 (1-3) 167-215

Laurent. M. and M. Deza, Bouquets of geometric lattices: Some algebraic and topologicai aspects Leader, I., A short proof of a theorem of V h o s on matroid representations LovBsz, L., M.Saks and W.T. Trotter, An on-line graph coloring algorithm with sublinear performance ratio

(1-3) 279-313 (1-3) 315-317

Nakada, F., see Akiyama, J. Nektfil, J. and V. R M , The Partite construction and Ramsey set systems Nicdas, J.L.,see Erdiis, P.

(1-3) 7-9 (1-3) 327-334 (1-3) 155-166

Pittel, B.,see Flajolet, P

(1-3) 167-215

Rasmussen, S., see BoUob& B.

(1-3) 55-68 (1-3) 327-334 (1-3) 335-342

Riidl, V.,see NeSetKl, J. Roscnstiehl, P., Scaffold permutations

Saks, M.,see h v k , L. Sdrkazy, A., see Erdiis. P. Seymour. P.D., see Chung, F.R.K. SzCkeIy, L.A. and N.C. Wormald. Bounds on the measurable chromatic number of R"

Tbomason, A., A simple linear expected time algorithm for finding a hamilton path Thomason, A., Dense expanders and pseudo-random bipartite graphs Thomason, A., see Haviland, J. Tokmaga, S., see Akiyama, J. Trotter, W.T., see L o v k L.

W&,

D.R., Forbidden graphs for degree and neighbourhood conditions Wormaid, N.C., see Sdkely, L.A.

(1-3) 319-325

(1-3) 319-325 (1-3) 155-166 (1-3) 113-119 (1-3) 343-372 (1-3) 373-379 (1-3) 381-386 (1-3) 255-278 (1-3) 7-9 (1-3) 319-325 (1-3) 387-404 (1-3) 343-372

Graph Theory and Combinatorics, 1988: Proceedings: Cambridge Combinatorial Proceedings in Honour of Paul Erdos

Graph Theory and Combinatorics, 1988: Proceedings

Graph theory and combinatorics, Proc. 1988

Graph Theory: Conference Proceedings

Combinatorics and graph theory

Combinatorics and Graph Theory

Graph theory: proceedings of the Conference on Graph Theory

Combinatorics, Paul Erdos is eighty, Vol.1

Combinatorics, Paul Erdos is Eighty Volume 1

Combinatorics, Paul Erdos is Eighty Volume 1

Graph Theory, Combinatorics and Algorithms

Graph theory and combinatorial optimization

Recent trends in combinatorics: The legacy of Paul Erdos

Problems in Combinatorics and Graph Theory

Matrices in combinatorics and graph theory

The Mathematics of Paul Erdos II (Algorithms and Combinatorics 14)

Problems in Combinatorics and Graph Theory

Graph Theory: Conference Proceedings (Mathematics Studies)

The mathematics of Paul Erdos

Matrices in Combinatorics and Graph Theory (Network Theory and Applications)

The mathematics of Paul Erdos

Paul Erdos and his mathematics

Paul Erdos and his mathematics

Graph Theory, Combinatorics and Algorithms: Interdisciplinary Applications

Graph Theory, Combinatorics and Algorithms: Interdisciplinary Applications

Graph Theory, Combinatorics, and Algorithms: Interdisciplinary Applications

Combinatorics 1981: Combinatorial Geometrics and Their Applications: Colloquium Proceedings: Combinatorial Geometrics and Their Applications

Combinatorial Mathematics: International Colloquium Proceedings

Combinatorics 1984: Finite Geometries and Combinatorial Structures: Colloquium Proceedings: Finite Geometries and Combinatorial Structures

A Tribute to Paul Erdos

Combinatorics and Graph Theory (Undergraduate Texts in Mathematics)

Graph Theory and Combinatorics, 1988: Proceedings: Cambridge Combinatorial Proceedings in Honour of Paul Erdos