Combinatorics: Being the Proceedings of the Conference on Combinatorial Mathematics held at the Mathematical Institute, Oxford, 1972

373 Pages·1972·16.214 MB·English

by  D. J. A. Welsh, D.R. Woodall (eds.)

COMBINATORICS

COMBINATORICS
Being the Proceedings of the Conference on
Combinatorial Mathematics
held at the Mathematical Institute, Oxford
edited by
D.J.A. Welsh
University of Oxfora
and
D.R. Woodall
University of Nottingham
1972
The Institute of Mathematics and its Applications
The Institute of Mathematics and its Applications,
Maitland House, Warrior Square,
Southend—on—Sea, Essex. SSl ZJY
Copyright ® 1972 by
THE INSTITUTE OF MATHEMATICS AND ITS APPLICATIONS
All Rights Reserved
No part of this book may be reproduced in any form by
photostat, microfilm, or any other means, without written
permission from the publishers.
CONTENTS
Participants.. .. .. .. .. .. .. .. .. .. .. vii
Preface .. .. .. .. .. .. .. .. .. .. .. .. x
Pictures.. .. .. .. .. .. .. .. .. .. ..' .. 1
N. BIGGS
Sperner Type Theorems for Finite Sets .. .. .. .. 18
A. BRACE and D.E. DAYKIN
Questions about Symmetric Designs .. .. .. .. .. 38
P.J. CAMERON
Some Combinatorial Problems in Number Theory.. .. .. 47
S.L.G. CHOI
Computation of Steiner Minimal Trees.. .. .. .. .. 53
E.J. COCKAYNE and D.G. SCHILLER
Supermatroids .. .. .. .. .. .. .. .. .. .. 72
F.D.J. DUNSTAN, A.W. INGLETON and D.J.A. WELSH
0n Ramsey Like Theorems. Problems and Results .. .. 123
P. ERD6S and A. HAJNAL
Characters of Symmetric Groups Induced by Characters
of Cyclic Subgroups .. .. .. .. .. .. .. .. .. 141
H.O. FOULKES
The Number of Trees in a Wheel .. .. .. .. .. .. 155
F. HARARY, P. O'NEIL, R.C. READ and A.J. SCHWENK
Matroid Polynomials .. .. .. .. .. .. .. .. .. 164
A.P. HERON
The Number of Spanning Trees of Labelled Wheels, Fans
andBaskets.. .. .. .. .. .. .. .. .. .. .. 203
A.J.w. HILTON
Matroids: Unimodal Conjectures and Motzkin's Theorem.. 207
J.H. MASON
Some Families of Non-Hamiltonian Graphs .. 221
G.H.J. MEREDITH
The Hamiltonian Graphs on to O7 .. .. .. 229
G.H.J. MEREDITH and E.K. LLOYD
Which Infinite Set-Systems have Transversals - A
Possible Approach .. .. .. .. .. .. 237
C. St-J. A. NASH-WILLIAMS
Smoothly Embeddable Subgraphs - A Survey.. 254
J. SHEEHAN
Electric Currents in Regular Matroids .. 262
C.A.B. SMITH
The Enumeration of Graphs on Large Periodic Lattices.. 285
H.N.V. TEMPERLEY
On the Adjacency Matrix of a Graph .. .. 295
R.J. WILSON
Property B and the Four-Colour Problem .. 322
D.R. WOODALL
Two Results on Infinite Transversals.. .. 341
D.R. WOODALL
Unsolved Problems .. .. .. .. .. .. 351
vi
PARTICIPANTS
M. Beynon; The Open University, Bletchley, Buckinghamshire,
England.
N.L. Biggs; Royal Holloway College, London, England.
D.W. Bolton; The Mathematical Institute, Oxford, England.
P.q. Cameron; University of Oxford, Oxford, England.
S.L.G. Choi; University of British Columbia, Vancouver,
Canada.
E.J. Cockayne; University of Victoria, British Columbia,
Canada.
H.T. Croft; University of Cambridge, Cambridge, England.
R.M. Damerell; Royal Holloway College, London, England.
J. Davies; The Mathematical Institute, Oxford, England.
D.E. Daykin; University of Reading, Reading, Berkshire,
England.
F.D.J. Dunstan; The Mathematical Institute, Oxford, England.
P. ErdSs; Mathematical Institute of the Hungarian Academy of
Sciences, Budapest, Hungary.
H.0. Foulkes; University College of Swansea, Swansea, Wales.
J. Godfrey; University of Reading, Reading, Berkshire, England.
J.E. Graver; University of Syracuse, New York, USA.
G.R. Grimmett; The Mathematical Institute, Oxford, England.
R.K. Guy; University of Calgary, Calgary, Alberta, Canada.
F. Harary; University of Michigan, Michigan, USA.
vii
A.P. Heron; Government Communications Headquarters,
Cheltenham, Gloucestershire, England.
A.J.W. Hilton; University of Reading, Reading, Berkshire,
England.
A.W. Ingleton; The Mathematical Institute, Oxford, England.
E.K. Lloyd; University of Southampton, Southampton, England-
C.J.H. McDiarmid; University of Oxford, Oxford, England.
T.P. McDonough; University College of Wales, Aberystwyth,
Wales.
J.H. Mason; The Open University, Bletchley, Buckinghamshire,
England. >
V.C. Mavron; University College of Wales, Aberystwyth, Wales.
G.H.J. Meredith; University of Southampton, Southampton,
England.
R.A. Muse; University College of Swansea, Swansea, Wales.
C. St. J.A. Nash-Williams; University of Aberdeen, Aberdeen,
Scotland.
H. Perfect; University of Sheffield, Sheffield, England.
P. Rado; Royal Holloway College, London, England.
P.D. Seymour; The Mathematical Institute, Oxford, England.
J. Sheehan; University of Aberdeen, Aberdeen, Scotland.
C.A.B. Smith; University College, London, England.
H.N.V. Temperley; University College of Swansea, Swansea,
Wales.
D.J.A. Welsh; The Mathematical Institute, Oxford, England.
R.J. Wilson; The Open University, Bletchley, Buckinghamshire,
England. '
"viii'
D.R. Woodall; University of Nottingham, Nottingham, England.
I.M. Zuzarte; The Mathematical Institute, Oxford, England.
ix
PREFACE
This book is the result of an attempt to produce,
quickly and cheaply, the proceedings of the informal
conference on combinatorics held at Merton College and the
Mathematical Institute, Oxford, from 3rd to 7th July 1972.
The conference was organized by a committee consisting of
Norman Biggs, Hazel Perfect and ourselves, and we hope that
it will be the forerunner of regular combinatorial meetings
in Britain.
We should like to take this opportunity to thank all
the participants for their ready co-operation, and in
particular to thank P. Ers, who took the chair at the
unsolved problems session.
We should also like to thank Miss Rosemary Schwerdt of
the Mathematical Institute, for her calm efficiency in handling
the arrangements for the conference, and Miss Laurel Lucas of
the Institute of Mathematics and its Applications, who has
typed the whole of the book and drawn all the diagrams ready
for reproduction.
D.J.A. Welsh
D.R. Woodall