Table Of ContentTHEORY AND PRACTICE
OF COMB INA TORICS
Grrierol Editor
Peter L. HAMMER, University of Waterloo. Ontario, Canada
Advisory Editors
C. BERG.F,,Universited e Paris, France
M,;A;.fiARRISON, University of California, Berkeley, CA, U.S.A.
Y’. kcEE, Uhiiierslty of Washington, Seattle, WA, U.S.A.
S.G. VAN LINT, California Institute ofTechnology, Pasadena, CA, U.S.A.
‘Gr~C.’RO.~A;’h;l$ssach~sIentsttsi tute of Technology, Cambridge, MA, U.S. A.
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NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD
NORTH-HOLLAN D
MATHEMATICS STUDIES 60
Annals of Discrete Mathematics (12)
General Editor: Peter L. Hammer
University of Waterloo, Ontario, Canada
Theory and Practice
of Combinatorics
Acollection of articles honoring Anton Kotzig
on the occasion of his sixtieth birthday
Edited b y
Alexander ROSA
McMaster University, Hamilton, Ontario, Canada
Gert SABlDUSSl
Jean TURGEON
Universityof Montreal, Quebec, Canada
1982
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD
North-Holland Publishing Company, I982
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Library of Congress Cataloging in Publication Data
Main entry under title:
Theory and practice of combinatorics.
(Annals of discrete &hatics ; 12) (Morth-Holland
mathemtics studies ; 60)
English and French.
1. Combinstorial analysis--Addresses, essays, lectures.
2. Kotzig, Anton. I. Kotzig, Anton. 11. Rosa, Alexander.
111. Sabidussi, Gert. IY. Turgeon, Jean. V. Series.
VI. Series: North-Holland mathematics studies ; 60.
W64 T46 5111.6 81-22581
ISBN 6-444-86318-4 A4CE
PRINTED IN THE NETHERLANDS
Anton Kotzig
This Page Intentionally Left Blank
PREFACE
It was a surprise. “Sixty! Really? Why not publish a volume in his honor?” The
question was flattering, but Professor Kotzig didn’t know it was being asked about
him. He is familiar with the saying of Marcus Porcius Cat0 (234149 B.C.): “I would
rather have men ask why no statue has been erected in my honor, than why one has!”
But the question kept coming up again and again, and, during the summer of 1979,
the volume gradually became a firm project, in time to be announced to the surprised
hero, on the day of his sixtieth birthday.
Anton Kotzig was born October 22, 1919 in KoCovce, Czechoslovakia where his
father was director of the village school. As he came to Canada in 1969, he had already
had a long and productive career as an applied mathematician. He had obtained his
Ph.D. in mathematics in 1947 at the Komensky University of Bratislava. The thesis
dealt with non-parametric tests: correlations between two statistical variables,
measured by the number of inversions in the corresponding permutations. In 1969,
he was granted the degree of Doctor of Science, the highest possible scientific degree
in Czechoslovakia from the Charles University in Prague, for a thesis on connectivity
and regular connectivity of finite graphs.
In his native land, he had been entrusted with important responsibilities. From
1940 to 1948, he worked as a statistician for the Central Bureau of Social Insurance
for Slovakia, where he was chairman of the Department of Mathematics and Statistics,
from 1942 to 1948. In January, 1949, he was appointed director of the Department of
Economics, Classification and Control at the Slovak National Bureau for the pur-
chase and distribution of agricultural products. He left this occupation in 1951 to
become professor at the Bratislava University of Economics, where he stayed until
1959 and served as rector from 1952 to 1958. These two decades of his life were thus
full of constant contact with the applications of mathematics.
In 1959, he leaves the Bratislava University of Economics to found the Mathe-
matical Institute of the Slovak Academy of Sciences, of which he remains the director
until 1964. He is then appointed director of the Department of Applied Mathematics
of the Komensky University in Bratislava, until 1969. In 1965-1966, he also serves as
interim dean, for one year, of the Faculty of Sciences at the same University.
Professor Kotzig managed to maintain a high level of research in mathematics
throughout these long years of administrative duty-a remarkable achievement. In
1969, his list of publications included more than sixty papers in research journals
and four books: his Prague thesis, a book on Mathematical Methods in Economics
(The Academy of Sciences, Bratislava 1961), a “Mathematical outline of a dynamic
model” for a centrally planned economy and a Foundations of Higher Mathematics
handbook. He was also editor-contributor of a Mathematical Encyclopedia, in
Slovak. For his abundant research in mathematics and for certain applications that
proved to be very important in practice, he received the highest honors granted to
scientists in his country. In 1965, it was the Czechoslovak Order for “outstanding
viii Preface
contribution to the fatherland’. In 1966, he was laureate of Czechoslovak Academy
of Sciences Prize for his “excellent scientific and educational activities” and, in 1969,
laureate of the Czechoslovak State Prize “for the development of powerful mathe-
matical methods in macroeconomy”.
But, as impressive as this part of Professor Kotzig’s career can be, this is not yet the
motive of the present volume. The editors and contributors rather intend to celebrate
the extraordinary richness and diversity of his contribution to our mathematical
life since his arrival in Canada in 1969. We must first note that circumstances had
then deprived him of his personal notes, his collection of books and reprints; he had
to start again, “from square one”, so to speak, in an unfamiliar country, none of whose
two official languages he then mastered well. Above all, he was separated from his
team of collaborators that had gradually gathered around him over the years in
Czechoslovakia. He faced the challenge with such flexibility and vigor that, since
1969, he has now published more than fifty papers and successfully directed several
MSc. and Ph.D. theses. These publications deal with a wide variety of topics: com-
binatorial topology, optimization, Hamiltonian and strongly Hamiltonian graphs,
various valuations of graphs, magic squares and magic stars, latin squares, quasi-
groups defining special decompositions of complete graphs, perfect systems of differ-
ence sets, additive sequences of permutations, tournaments. This variety is reflected
in the present volume, whose themes are all somehow related to Professor Kotzig’s
own recent work, either published or presented orally at various conferences and
seminars. His numerous collaborators form a network spanning a large number of
universities in Canada and around the world: Montreal, McGill, UQAM, UQTR,
College Militaire Royal de Saint-Jean, McMaster, Toronto, Waterloo, Calgary,
Halifax, Simon Fraser, Auburn, Paris, Grenoble, Budapest, Haifa. The international
character of this network is also reflected in the list of authors of the present volume
in his honor, an honor he fully deserves !
Thanks are due to the contributors of the volume and to the anonymous team of
generous referees who upheld in their judgements the demanding mathematical
standards of Annals of Discrete Mathematics. Finally, we are grateful to the North-
Holland Publishing Company who welcomed this “Festschrift” in the series Annals
of Discrete Mathematics.
May 22, 1981 Alexander Rosa
Gert Sabidussi
Jean Turgeon
CONTENTS
Preface vii
1. J. ABRHAM,B ounds for the sizes of components in perfect systems of
difference sets 1
2. M. AJTAI, V. CHVATALM, . M. NEWBOREN., SZEMER~CDroIs, sing-free
subgraphs 9
3. B. ALSPACH, C. TABIBA, note on the number of 4-circuits in a tourna-
ment 13
4. L. BABAIF, .R.K. CHUNG,P . E R ~ sR,. L. GRAHAMJ.,H . SPENCER,
On graphs which contain all sparse graphs 21
5. L. BABAIP, .E R ~ SRe,p resentation of group elements as short products 27
6. C. BERGES, ome common properties for regularizable graphs, edge-
critical graphs and B-graphs 31
7. A. BERMANSk, ew Hadamard matrices of order 16 45
8. J.C. BERMONDG,. FARHSI,u r un probleme combinatoire d’antennes en
radioastronomie I1 49
9. P. CAMIONU, n algorithme de construction des idempotents primitifs
d’ideaux d’algebres sur IF, 55
10. C.J. COLBOURNE, . MENDELSOHNK,o tzig factorizations 1 existence
and computational results 65
11. K. CUL~KC,h aracter strings as directed Euler paths 79
12. M. DEZAS, .A. VANSTONOEn, maximal equidistant permutation arrays 87
13. J .A. EDWARDSG, .M. HAMILTONA,. J.W. HILTONB, . JACKSOND,o mino
squares 95
14. P. E R ~So,me problems on additive number theory 113
15. P. ERDUS, A. HAJNALE, . SZEMER~ODnI, almost bipartite large chro-
matic graphs 117
16. P. FRANKLC,o vering graphs by equivalence relations 125
17. B. GRUNBAUMG, .C. SHEPHARDA, nalogues for tilings of Kotzig’s
theorem on minimal weights of edges 129
18. R.K. GUY,S ets of integers whose subsets have distinct sums 141
19. P. HELL,F .S. ROBERTS,A nalogues of the Shannon capacity of a graph 155
20. C. HUANGE, . MENDELSOHNA., ROSA, On partially resolvable t-
partitions 169
21. D. KEEDWELLD,e compositions of complete graphs defined by quasi-
groups 185
22. &.J. LAUFER,R egular perfect systems of difference sets of size 4 and
extremal systems of size 3 193
23. C.C. LINDNERW, .D. WALLISA, note on one-factorizations having a
prescribed number of edges in common 203
24. T. LOBBOSS, ystemes de transition et coloriages euleriens 21 1
