Combinatorial Design Theory

462 Pages·1987·11.903 MB·English

by  Charles J. Colbourn and Rudolf Mathon (Eds.)

NORTH-HOLLAND MATHEMATICS STUDIES 149
Annals of Discrete Mathematics (34)
General Editor: Peter L. HAMMER
Rutgers University, New Brunswick, NJ U.S.A.
Advisory Editors
C. BERGE, Universite de Paris, France
M. A. HARRISON, University of California, Berkeley, CA, U.S.A.
V: KLEE, University of Washington, Seattle, WA, U.S .A.
J. -H. VAN LINT CaliforniaI nstitute of Technology, Pasadena, CA, U.S.A.
G. -C.R OTA, Massachusetts Institute of Technology, Cambridge, MA, U.S .A.
NORTH-HOLLAND -AMSTERDAM NEW YORK OXFORD .TOKYO
COMBINATORIAL DESIGN THEORY
Edited by
Charles J. COLBOURN
Department of Computer Science
University of Waterloo
Waterloo, Ontario
Canada
Rudolf MATHON
Department of Computer Science
University of Toronto
Toronto, Ontario
Canada
1987
NORTH-HOLLAND -AMSTERDAM NEW YORK OXFORD .TOKYO
OElsevier Science Publishers B.K, 1987
All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or
otherwise, without the prior permission of the copyright owner.
ISBN: 0444703284
Publishers:
ELSEVIER SCIENCE PUBLISHERS B.V.
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THE NETHERLANDS
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Library of Congress Cataloging-in-Publication Data
Combinatorlal design theory / edited by Charles J. Colbourn, Rudolf
Mathon.
p. cm. -- (North-Holland mathematics studies : 149) (Annals
of discrere mathematics ; 34)
Festschrift for Alex Rosa.
ISBN 0-444-70328-4 (U.S.)
1. Combinatorial designs and configurations. 2. Rosa, Alexander.
I. Colbourn. C. J. (Charles J.), 1953- . 11. Mathon. R. A.
111. Rosa. Alexander. IV. Series. V. Series Annals of discrete
mathematics ; 34.
0A166.25.C652 1987
511'.6--dc19 87-2231 1
CIP
PRINTED IN THE NETHERLANDS
V
PREFACE
Combinatorial design theory has experienced an explosive growth in recent
years, both in theory and in applications. This volume is dedicated to one indi-
vidual who has played a major role in fostering and developing combinatorial
design theory, Alex Rosa.
Although Alex is a young man at the middle of his career, he is truly an indi-
vidual worthy of such an honour. Thestrength anddiversity of his contributions
to the theory of Steiner systems over the past twenty years have directed
research into new and exciting avenues, and thereby contributed dramatically
to the vibrancy of research on designs found today. His research has en-
compassed difference methods, colourings, decomposition and partitioning,
recursive and direct construction methods, analysis of designs, and relation-
ships with graph theory and with geometry. But Alex’s contributions extend
beyond this impressive research. Virtually all design theorists have profited
from Alex’s astonishing ability to organize and disseminate research infor-
mation. For the past fifteen years, he has maintained a bibliography on Steiner
systems, which has proved a great boon to researchers; at the same time, he
has unselfishly taken much of his time to help researchers by providing infor-
mation about a vast range of queries on combinatorial designs. Alex’s will-
ingness and ability to help others make him a truly special individual.
In 1987, Alex Rosa is celebrating his fiftieth birthday. We take this opportunity
to express our thanks to Alex for his contributions thus-far, and to honour
him on the occasion of his fiftieth birthday.
This volume is a collection of forty-one research papers on combinatorial
design theory and related topics. They extend the current state of knowledge
on Steiner systems, Latin squares, one-factorizations, block designs, graph
designs, packings and coverings; they develop recursiveanddirect constructions,
analysis techniques, and computational methods. Hence the collection reflects
the current themes in combinatorial design theory, and captures the multi-
faceted nature of the field. It is not possible here to summarize all of the
contributions contained in the papers of this volume, so we simply remark
that collectively the papers represent advances on a number of important
problems.
vi Preface
We hope that readers of the volume find the individual papers as useful and
as enjoyable as we have.
Charles J. Colbourn Rudolf A. Mathon
Waterloo, Ontario Toronto, Ontario
July, 1987 July, 1987
vii
ACKNOWLEDGEMENTS
In a project such as the publication of this volume, there are always many
people without whom the project would never reach fruition. We want to thank
all of the people who provided assistance. Primarily we thank the authors for
such an enthusiastic response, and for helping us meet a very tight schedule.
We also thank the referees; we list their names here to acknowledge their
invaluable assistance: J. Abrham, B.A. Anderson, F.E. Bennett, A.E. Brouwer,
J.I. Brown, C.J. Colbourn, J.H. Dinitz, R. Fuji-Hara, M. Gionfriddo, T.S.Griggs,
J.J. Harms, A. Hartman, K. Heinrich, P. Hell, M. Jimbo, D. Jungnickel,
E.S. Kramer, D.L. Kreher, C.C. Lindner, S.S. Magliveras, R.A. Mathon,
W. McCuaig, B.D. McKay, E. Mendelsohn, D.M. Mesner, W.H. Mills,R.C.Mullin,
K.T. Phelps, V. Rodl, F.A. Sherk, D.R. Stinson, L. Teirlinck, S.A. Vanstone,
and W.D. Wallis.
We also thank all of those who helped with preparing the final manuscript,
including many of the authors, but primarily Karen Colbourn and Zoe Kaszas.
Finally, we thank Peter Hammer and the mathematics staff at North-Holland
for supporting the project and providing excellent technical assistance, as
always.
Annals of Discrete Mathematics 34 (1987) 1-26
0 Elsevier Science Publishers B.V. (North-Holland) 1
The Existence of Symmetric Latin Squares with
One Prescribed Symbol in each Row and Column
L.D. Andersen
Department of Mathematics
Institute of Electronic Syst.ems
Aalborg University
Strandvejen 19
DK-9000 Aalborg
Denmark
A.J .W . Hilt on
Department of Mathematics
University of Reading
Whiteknights
Reading RG6 2AX
United Kingdom
Visiting:
Division of Mathematics
Auburn University
Auburn
Alabama 36849
U.S.A.
TO ALfX ROSA ON MIS 3ITTIETU BIRTUDAY
ABSTRACT
Let P be a partial symmetric nXn Latin square in which up to
n +1 entries are specified such that there is at least one specified entry
in each row and column. We say exactly when P can be completed to an
nXn symmetric Latin square L. This is the first part of a proof of the
symmetric Latin square analogue of the Evans Conjecture.
1. Introduction
A partial nXn Latin square P is an nXn matrix in which some cells may be
empty and the non-empty cells will contain exactly one symbol, such that no symbol
occurs more than once in any row or in any column. P is called symmetric if whenever
a cell (i,j)c ontains some symbol, then cell (j,i)a lso contains the same symbol.
2 L.D .A ndersen and A.J . W.H ilton
Our concern in this paper is to characterize those partial symmetric nXn Latin
squares with up to n+l cells occupied, satisfying the restriction that there is at least
one occupied cell in each row and in each column, which can be completed to a sym-
metric nXn Latin square.
Before we state our main result we recall that in a symmetric Latin square of even
side, each symbol occurs an even number of times on the diagonal, and in a symmetric
Latin square of odd side, each symbol occurs exactly once on the diagonal. Both these
facts are easy to see. We shall call the diagonal of a partial symmetric Latin square of
side n admissible if the number of symbols occurring a number of times not congruent
to n modulo 2 is less than or equal to the number of empty cells (so that the parity
can be made right for every symbol with the ‘wrong’ parity). Let t(i) denote the
number of times that a symbol i occurs on the diagonal. Then, if the symbol set is
{I ,..., n}, the condition for admissibility can be written.
I{i:t (i) # n (mod 2)) Is n - g t(i).
i-1
For odd n, this simply means that a diagonal is admissible if and only if no sym-
bol occurs more than once on it. Obviously, a partial symmetric Latin square cannot be
completed to a symmetric Latin square if its diagonal is not admissible.
Our main result is the following:
Theorem 1. Let n23 and let P be a partial symmetric Latin square of side n with
admissible diagonal; suppose also that there is at least one occupied cell in each row
and in each column. Let c be the number of non-empty cells of P.
If c=n, then P can be completed to a symmetric Latin square of side n if and
only if P is not of the form of any of the Types El, 01 or 02 of Figure 1.
If c=n+l, then P can be completed to a symmetric Latin square of side n if and
only if P is neither of the form of any of the types El, 01 or 02 with a further diagonal
cell filled, nor of the form of the types E2 or L5 of Figure 1.
Remark. No doubt the meaning of the phrase ‘of the form of’ is self-evident. But,
to be formal, a partial Latin square is of the form of one of these Types if it can be
transformed to a partial Latin square of one of these Types by permuting rows, per-
muting columns the same way, and relabelling the symbols.
It is easy to see that the partial squares in Figure 1 cannot be completed sym-
metrically. In types El and 01 the symbol 1 needs an extra diagonal occurrence but
cannot get it; in Types E2 and 02 it is impossible to make the symbol 1 occur in the
first row; finally we leave it to the reader to check that Type 155 cannot be completed.
The analogous problem for Latin squares with no symmetry requirement to the
problem considered here was settled by Chang 161; he characterized the possible diago-
nals of a Latin square. A completely different proof of Chang’s result was given by Hil-
ton and Rodger [lo].
The result in this paper is a necessary preliminary to a more general result [5] in
which we characterize the partial symmetric nXn Latin squares with up to n+l cells
occupied which can be completed. For our proof of this more general result, we need to
have the result in this paper proved separately, for the general method of that paper
Symmetric Latin Squares 3
n even:
Type E l Type E2
even
odd:
n
1 does
not occur
Type 01 Type 02
Figure 1
Type L5
4 L.D. Andersen and A.J. W. Hilton
fails when every row and column has an occupied cell. This general question has been
known and unsolved for a number of years.
The analogous more general problem for Latin squares with no symmetry require-
ment was settled by Smetaniuk [12]a nd, independently, Andersen and Hilton 131. Sme-
taniuk only considered the case when n-1 cells were occupied; Andersen and Hilton
considered the case when n cells were occupied. Damerell 181 extended Smetaniuk's
method to cover the case when n cells were occupied. Finally, Andersen [2] dealt with
the case when n +1 cells were occupied.
2. Edge-Colourings of Graphs
In this section we discuss briefly edge-colourings of graphs, and the relationship
between symmetric Latin squares and edge-colourings of complete graphs with loops.
The reason for this is that the proof of Theorem 1 is easier to present in terms of
edge-colourings.
An edge-colouring of a graph G is an assignment of a colour to each edge and loop
of G in such a way that all edges and loops incident with the same vertex have distinct
colours. The chromatic indez CHI'(G2 of G is the least integer k for which G has an
edge-colouring with Ic colours. Let K, denote the graph obtained from the complete
graph K, by adding a loop at each vertex. It is well-known that CHI'(K,')=n.
A symmetric Latin square S of side n gives rise to an edge-colouring of K," as fol-
lows. Let the vertices of Kf be vl, * * ,v,. Define the colour of the edge viwi to be the
entry of clelsl i(si,n$ o,f S, and the colour on the loop on vi to be the entry of cell (i,i)of
S, i#j, lsjsn. This clearly defines an edge-colouring of K,' with n
colours, and it obviously works the other way round as well; it also works for partial
.
symmetric Latin squares and edge-coloured subgraphs of K,' In this connection, we
note that if an edge-coloured subgraph of Kf corresponds to a partial symmetric Latin
square P, then the number of filled cells of P is equal to the degree sum
5
d(wi)=2 IE(G)I+ IL(G)I, where E(G)i s the edge-set of G and L(G)t he loop-set
i-1
of G (a loop is not an edge; a loop on vi contributes one to the degree d(vi) of (vi).W e
shall say that G has admissible loop-colouring if the diagonal of P is admissible.
In this terminology, Theorem 1 becomes:
Theorem 2. Let nz3 and let G be an edge-coloured subgraph of Ki with an
admissible loop-colouring; suppose also that each vertex of G has at least one edge or
loop on it, and that G spans Ki. Let t =2 lE(G)I + IL(G) I.
If t =n, then the edge-colouring of G can be extended to an edge-colouring of Ki
with n colours if and only if G is not any of the Types el, 01 or 02 of Figure 2.
If t =n +I, then the edge-colouring of G can be extended to an edge-colouring of
Ki with n colours if and only if G is not any of the Types el, 01 or 02 with a further
loop added, nor of the form of Types e2 or 95 of Figure 2.
When G is one of the edge-coloured graphs of Figure 2, possibly with a loop
added in case el, 01 or 02 we refer to G as being a bad case.