Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 686 Combinatorial Mathematics Proceedings of the International Conference on Combinatorial Theory Canberra, August 16-27, 1977 Edited by D. A. Holton and Jennifer Seberry Springer-Verlag Berlin Heidelberg New York [ ~ Australian Academy of Science Canberra Editors D. A. Holton Department of Mathematics University of Melbourne Parkville, Victoria 3052/Australia Jennifer Seberry Applied Mathematics Department University of Sydney Sydney. N. S. W. 2006/Australia Distribution rights for Australia: Australian Academy of Science, Canberra ISBN 0-8584?-049-? Australian Academy of Science Canberra AMS Subject Classifications (1970): 05-04, 05A15, 05A17, 05A19, 05A99, 05B05, 05815, 05B20, 05B25, 05B30, 05B40, 05845, 05899, 05C10, 05C15, 05C20, 05C25, 05C30, 05C35, 05C99, 15 A 24, 20 B 25, 20 H 15, 20 M 05, 50 B30, 52 A45, 62-XX, 62 K10, 68 A 20, 94A10, 82A05 ISBN 3-540-08953-5 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38?-08953-5 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, re- printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210 PREFACE The International Conference on Combinatorial Theory was held at the Australian NationalUniversity from August 16-27 1977. The names of the eighty-nine participants are listed at the end of this volume. This Conference was sponsored jointly by the International Mathematical Union and the Australian Academy of Science and was organised under the auspices of the Academy. Grants from the IMU and the Australian Government enabled us to invite a number of overseas specialists to the conference. With the exception of Professor Tutte, whose paper will appear elsewhere, the texts of the talks of the invited speakers appear in these Proceedings. We wish to thank our sponsors and the Australian Government for their support. In addition to the invited addresses, three instructional series of talks were given. Professors Tutte and Bondy gave four lectures on the Reconstruction Conjecture, Professor Hughes gave four lectures on Designs and Professors Mullin and Vanstone gave four lectures on (r, X) Systems. The first two of these series will appear elsewhere. Professor Bondy's lectures will appear in the Journal of Graph Theory under the title "Graph reconstruction - a survey". This paper is coauthored by Professor R.L. Hemminger. The work by Professor Tutte is to appear in Graph Theory and Related Topics, the Proceedings of the Conference held in Waterloo in July 1977. Professor Hughes' material will appear in a book that he is currently writing. Only the material of Professor Mullin therefore, appears in these Proceedings. At the conference there was a large number of contributed talks. Of these, twenty-eight appear in this volume. Papers which are given by title only in the Table of Contents will appear elsewhere. I t t a k e s a g r e a t many p e o p l e t o make a c o n f e r e n c e t h e s i z e o f t he p r e s e n t one run s m o o t h l y . We thank a l l t h o s e p e o p l e who so w i l l i n g l y c h a i r e d s e s s i o n s and r e f e r e e d p a p e r s . Thanks t o o must go t o t h e A u s t r a l i a n N a t i o n a l U n i v e r s i t y , t he A u s t r a l i a n Academy o f S c i e n c e and t h e C a n b e r r a C o l l e g e o f Advanced E d u c a t i o n . The ANU provided us with a number of lecture theatres, as well as library and other facilities. Neville Smythe of the ANU was a great help to us in the preconfer- ence organisation and in arranging typing and photocopying during the conference. IV Considerable help was provided by the staff of the Academy. We particularly wish to express our thanks to Pat Tart and Beth Steward for their assistance which started many months before the conference. They were invaluable registering delegates, producing the daily newsletter, organising entertainment, and taking n + 1 jobs off our shoulders and executing them efficiently. Jack Deeble was a great help in the publication of these Proceedings. At the Canberra CAE we were greatly helped by Peter 0'Hallaron and Alan Brace. We thank Peter especially, for his liaison work between the College and the conference and his general assistance, particularly with regard to social events. We are grateful to the College for providing both lecture facilities for an afternoon session and transport for delegates during the conference. We cannot let this opportunity go by of thanking Bernhard Neumann and Cheryl Praeger for the part they played in the running of the conference. The original idea of holding the conference was Bernhard's and he consistently gave his support through- out. Cheryl also was invaluable, especially in the early days of planning when the conference was on a very flimsy financial footing. Finally we would like to thank Marjorie Funston, Helen Wort and Janet Midgley for their fine secretarial work in the periods of pressure before and after the conference. D .A.H. J.S. TABLE OF CONTENTS INVITED ADDRESSES J.A. Bondy: Reflections on the legitimate deck problem. 1 P. ErdSs: Some extremal problems on families of graphs. 13 M. Hall, Jr.: Integral properties of combinatorial matrices. 22 H. Hanani: A class of three-designs. 34 Frank Harary, Robert W. Robinson and Nicholas C. Wormald: Isomorphic Factorisations II~ Complete multipartite graphs. 47 Daniel Hughes: Biplanes and semi-biplanes. $5 R.C. Mullin and D. Stinson: Near-self-complementary designs and a method of mixed sums. 59 T.V. Narayana: Recent progress and unsolved problems in dominance theory. 68 J.N. Srivastava: On the linear independence of sets of 2 q columns of certain (1, -1) matrices withagroup structure, and its connection with finite geometries. 79 R.G. Stanton: The Doehlert-Klee problem. 89 M.E. Watkins: The Cayley index of a group, i01 INSTRUCTIONAL LECTURE R.C. Mullin: A survey of extrema! (r, I)-systems and certain applications. 106 VII CONTRIBUTED PAPERS C.C. Chen: f~l the enumeration of certain graceful graphs. III Keith Chidzey: Fixing subgraphs of K 116 m, n" Joan Cooper, James ~filas and W.D. Wallis: Hadamard equivalence. 126 R.B. Eggleton and A. Hartman: A note on equidistant permutation arrays. 136 lan Enting: The combinatorics of algebraic graph theory in theoretical physics. 148 C. Godsil: Graphs, groups and polytopes. 157 Frank Harary, W.D. Wallis and Katherine Heinrich: Decompositions of complete symmetric digraphs into the four oriented quadrilaterals. 165 D.A. Holton and J.A. Richard: Brick packing. 174 R. Hubbard: Colour symmetry in crystallographic space groups. 184 H.C. Kirton and Jennifer Seberry: Generation of a frequency square orthogonal to a i0 x i0 latin square. 193 J-L. Lassez and H.J. Shyr: Factorization in the monoid of languages. 199 Charles H.C. Little: On graphs as unions of Eulerian graphs. 206 Sheila Oates Hacdonald and Anne Penfold Street: The analysis of colour symmetry. 210 Brendan D. McKay: Computing automorphisms and canonical labellings of graphs. 223 Elizabeth J. Morgan: On a result of Bose and Shrikhande. 233 VIII M.J. Pelling and D.G. Rogers: Further results on a problem in the design of electrical circuits. 240 R. Razen: Transversals and finite topologies. 248 R.W. Robinson: Asymptotic number of self-converse oriented graphs. 255 D.G. Rogers and L.W. Shapiro: Some correspondences involving the SchrSder numbers and relations. 267 Jennifer Seberry: A computer listing of Hadamard matrices. 275 Jennifer Seberry and K. Wehrhahn: A class of codes generated by circulant weighing matrices. 282 G.J. Simmons: An application of maximum-minimum distance circuits on hypereubes. 290 T. Speed: Decompositions of graphs and hypergraphs. 300 E. Straus: Some extremal problems in combinatorial geometry. 308 D.E. Taylor and Richard Levingston: Distance-regular graphs. 313 H.N.V. Temperley and D.G. Rogers : A no t e on B a x t e r ' s g e n e r a l i z a t i o n o f t h e Temper ley -L ieb o p e r a t o r s . 324 Earl Glen ~nitehead, Jr,: Autocorrelation of (+1, -I) sequences. 329 N.C. Wormald: Triangles in labelled cubic graphs. 337 PROBLE~ 1. A problem on duality (Blanche Descartes) 346 2. Perfect matroid designs (M. Deza) 346 3. Permutation graphs (R.B. Eggleton and A. Hartman) 347 4. Tiling (P. Erd~s) 347 5. Groups (P. Erd~s and E.G. Straus) 347 6. Graphs (D.A. Holton) 347 05Cxx REFLECTIONS ON THE LEGITIMATE DECK PROBLEM J.A. Bondy University of Waterloo Waterloo, Ontario Canada ABSTRACT We study the following problem: given a collection H = (Hill N i N n) of n graphs, each on n-i vertices, when does there exist a graph G whose vertex- deleted subgraphs are the members of H? i. LEGITIMATE DECKS A deck of n cards is a collection (Hill ~ i ~ n) of n graphs, each having n-i vertices. If there exists a graph G with vertex set {l,2,...,n} such that G.1 M H.1 (i < i _< n) (where G.1 denotes the subgraph of G obtained on deleting vertex i) the deck (Hill ~ i ~ n) is said to be legitimate, and we call G a generator of the deck. Decks which are not legitimate are, of course, illegitimate. The deck shown in figure l(a) is legitimate: a generator is displayed in figure l(b); but the deck of figure 2 is illegitimate, because we see from H 1 that every generator is acyclic, and from H 2 that no generator can possibly be so. O O O I I 3 4 H I H 2 H 3 H 4 G (a) (b) Figure i O O O O © H I H 2 H 3 H 4 Figure 2 A less obvious example of an illegitimate deck is given in figure 3. O O i H I H 2 H 3 H 4 H 5 H 6 Figure 3 In the Reconstruction Conjecture 1 , the problem is to show that no deck has more than one generator, up to isomorphism. The Legitimate Deck Problem, by contrast, seeks a characterization of those decks having at least one generator (in other words, legitimate decks). It was first mentioned by Harary 7 in 1968, more as an aside to the Reconstruction Conjecture than as a problem of independent interest. However, it does appear to be quite a basic question, having links with much existing graph theory. This paper surveys the first few tentative steps which have been made towards an understanding of legitimate decks. Our notation and terminology is that of Bondy and Murty 2 . However, all graphs are assumed to be simple. Before proceeding, we make a couple of simple observations, based on a fundamental result in the theory of reconstruction known as Kelly's lemma 9 . Let H = (Hill -< i -< n) be a legitimate deck, G a generator, and F any graph with ~(F) < n. Then Kelly's lemma gives a formula for the number s(F,G) of subgraphs of G which are isomorphic to F in terms of the deck H: n s(F,H i) i=l s(F,G) n-v(F) (i) Since s(K2,G) = e(G) and g(G) - ~(G i) = dG(i), the number of edges and the degree sequence of any generator of a deck can be determined. The following proposition is now easily established. PROPOSITION. Let H be a legitimate deck, and let G be a generator of H. Then (i) if no two vertex degrees in G are consecutive integers, H has a unique generator (up to isomorphism) which can be obtained from any card H.i by adding a new vertex and joining it to the vertices of H°i whose degrees do not occur in the degree sequence of G; (ii) if all the cards in H are isomorphic, the unique generator of H is vertex- transitive. 2. THE KELLY CONDITIONS A variant of Kelly's lemma, this time involving induced subgraphs, can be invoked to yield a strong necessary condition for legitimacy. As before, let H = (Hill -< i -< n) be a legitimate deck, G a generator, and F any graph with ~(F) < n. Then the number s'(F,G) of induced subgraphs of G which are isomorphic to F is given by n I s' (F,Hi) i=l (2) s' (F,G) n-~ (F) Since the numbers s'(F,G) must clearly be integers, we obtain the following condition on H: n (KI) n-v(F) ] I s'(F,H.) i= I l for each graph F with ~(F) < n. This condition appears to detect the vast majority of illegitimate decks; for instance, our deck of figure 3 fails (KI) when F = KI, 3. However, it is not as discriminating as might initially be supposed. Let G be a vertex-transitive graph on a prime number p of vertices. Then, by (2) s'(F,G v) p.s'(F,Gv) vcV s'(F,G) p-~(F) p-~(F) Since s'(F,G) is an integer and (p,p-~(F)) = i, we see that p-~(F) I s'(F,G v) (3) for each graph F with 9(F) < p. Consider, now, a deck of p cards, each of which is a vertex-deleted subgraph of a vertex-transitive graph. By (3), this deck satisfies (KI). However, it follows from our proposition that the deck is legitimate only when all of its cards are isomorphic. An example of an illegitimate deck formed in this way is given in figure 4.

Combinatorial Mathematics: Proceedings of the International Conference on Combinatorial Theory Canberra, August 16–27, 1977

Similar Combinatorial Mathematics: Proceedings of the International Conference on Combinatorial Theory Canberra, August 16–27, 1977