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

356 Pages·1978·4.05 MB·English

by  J. A. Bondy (auth.), D. A. Holton, Jennifer Seberry (eds.)

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 con-
ference.
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.