Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

360 Pages·2007·12.688 MB·English

by  Geoffrey Grimmett

Title Pages
Combinatorics, Complexity, and Chance: A
Tribute to Dominic Welsh
Geoffrey Grimmett and Colin McDiarmid
Print publication date: 2007
Print ISBN-13: 9780198571278
Published to Oxford Scholarship Online: September 2007
DOI: 10.1093/acprof:oso/9780198571278.001.0001
Title Pages
(p.i) Oxford Lecture Series in Mathematics and its Applications
(p.ii) OXFORD LECTURE SERIES IN MATHEMATICS AND ITS APPLICATIONS
(p.iii) Combinatorics, Complexity, and Chance
Series Editors
John Ball Dominic Welsh
1. J. C. Baez (ed.): Knots and quantum gravity
2. I. Fonseca and W. Gangbo: Degree theory in analysis and applications
3. P. L. Lions: Mathematical topics in fluid mechanics, Vol. 1:
Incompressible models
4. J. E. Beasley (ed.): Advances in linear and integer programming
5. L. W. Beineke and R. J. Wilson (eds): Graph connections: Relationships
between graph theory and other areas of mathematics
6. I. Anderson: Combinatorial designs and tournaments
7. G. David and S. W. Semmes: Fractured fractals and broken dreams
8. Oliver Pretzel: Codes and algebraic curves
9. M. Karpinski and W. Rytter: Fast parallel algorithms for graph
matching problems
10. P. L. Lions: Mathematical topics in fluid mechanics, Vol. 2:
Compressible models
11. W. T. Tutte: Graph theory as I have known it
12. Andrea Braides and Anneliese Defranceschi: Homogenization of
multiple integrals
13. Thierry Cazenave and Alain Haraux: An introduction to semilinear
evolution equations
Page 1 of 4
Title Pages
14. J. Y. Chemin: Perfect incompressible fluids
15. Giuseppe Buttazzo, Mariano Giaquinta and Stefan Hildebrandt: One-
dimensional variational problems: an introduction
16. Alexander I. Bobenko and Ruedi Seiler: Discrete integrable geometry
and physics
17. Doina Cioranescu and Patrizia Donato: An introduction to
homogenization
18. E. J. Janse van Rensburg: The statistical mechanics of interacting
walks, polygons, animals and vesicles
19. S. Kuksin: Hamiltonian partial differential equations
20. Alberto Bressan: Hyperbolic systems of conservation laws: The one-
dimensional Cauchy problem
21. B. Perthame: Kinetic formulation of conservation laws
22. A. Braides: Gamma-convergence for beginners
23. Robert Leese and Stephen Hurley (eds): Methods and algorithms for
radio channel assignment
24. Charles Semple and Mike Steel: Phylogenetics
25. Luigi Ambrosio and Paolo Tilli: Topics on analysis in metric spaces
26. Eduard Feireisl: Dynamics of viscous compressible fluids
27 Antonín Novotný and Ivan Straškraba: Introduction to the
mathematical theory of compressible flow
28 Pavol Hell and Jaroslav Nešetřil: Graphs and homomorphisms
29 Pavel Etingof and Frederic Latour: The dynamical Yang–Baxter
equation, representation theory, and quantum integrable systems
30 Jorge Ramirez Alfonsin: The diophantine Frobenius problem
31 Rolf Niedermeier: Invitation to fixed parameter algorithms
32 Jean-Yves Chemin, Benoit Desjardins, Isabelle Gallagher and
Emmanuel Grenier: Mathematical geophysics: An introduction to rotating
fluids and the Navier-Stokes equations
33 Juan Luis Vázquez: Smoothing and decay estimates for nonlinear
diffusion equations
34 Geoffrey Grimmett and Colin McDiarmid: Combinatorics, complexity,
and chance
(p.iv)
Great Clarendon Street, Oxford OX2 6DP
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Page 2 of 4
Title Pages
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ISBN 0–19–857127–5 978–0–19–857127–8
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Page 3 of 4
Illustration
Combinatorics, Complexity, and Chance: A
Tribute to Dominic Welsh
Geoffrey Grimmett and Colin McDiarmid
Print publication date: 2007
Print ISBN-13: 9780198571278
Published to Oxford Scholarship Online: September 2007
DOI: 10.1093/acprof:oso/9780198571278.001.0001
Illustration
Dominic J. A. Welsh
Page 1 of 1
PREFACE
Combinatorics, Complexity, and Chance: A
Tribute to Dominic Welsh
Geoffrey Grimmett and Colin McDiarmid
Print publication date: 2007
Print ISBN-13: 9780198571278
Published to Oxford Scholarship Online: September 2007
DOI: 10.1093/acprof:oso/9780198571278.001.0001
(p.v) PREFACE
Dominic Welsh retired from Oxford University in 2005 having taught and
inspired generations of mathematicians. His formal retirement was marked by a
grand dinner in Merton College attended by many of his ex-undergraduate
students, and by a research workshop (and dinner in Merton) attended by many
of his ex-graduate students and some close colleagues. He has now resumed his
combinatorial researches following this flurry of celebration.
Dominic has been extremely influential in many aspects of Discrete
Mathematics, especially in the theories of graphs, matroids, algorithmic
complexity, cryptography, and knots, together with discrete physical models
(including percolation) and swathes of applied probability.
His doctoral thesis was written under the supervision of John Hammersley. It
was the basis for the important publication entitled ‘First-passage percolation,
subadditive processes, stochastic networks, and generalized renewal theory’,
written jointly by Hammersley and Welsh in 1965. This classic paper laid the
foundations for first-passage percolation and subadditive stochastic processes.
The decade of the 1960s witnessed the emergence of Combinatorial Theory as a
systematic topic of mathematics. In 1969, Dominic organized a ‘Conference on
Combinatorial Mathematics and its Applications’ in Oxford, inviting an
impressive list of diverse participants including Paul Erdő os and Roger Penrose.
The now biennial British Combinatorial Conference was thus born, and it has
remained one of the most important regular meetings in the subject worldwide.
Page 1 of 2
PREFACE
The publication in 1976 of Dominic's book ‘Matroid Theory’ was another
important moment for Combinatorics. In this first major text on the subject, he
made the case for this abstraction of linear independence as a powerful concept
in Combinatorics. Percolation Theory and Matroid Theory are now well
established and important topics in mathematics, and the authors (Grimmett and
Oxley, respectively) of the current standard texts on these two subjects were
each research students of Dominic.
Dominic has written many fine research papers. His influence is recognized
equally through his books, his questions, and his lectures and personal
interactions. The books have broken new ground, and have invariably presented
interesting material in an accessible and engaging way. The gift of asking the
right question at the right moment is much esteemed by mathematicians, and
Dominic has often hit the nail on the head, even if some of his conjectures have
not stood the test of time. Dominic continues to travel extensively for
mathematical visits and conferences. When looking for him at a research
meeting, you must head for (p.vi) where the action is, where the discussions
are most active and the excitementhighest.
The topics represented in this volume reflect Dominic's wide vision of
Combinatorics, and the contributors include some of his ex-graduate students
together with several eminent colleagues. There are articles on graphs,
matroids, polynomials, random structures, algorithms, and knots, under the
theme ‘Combinatorics, Complexity, and Chance’. We trust that this volume will
be a suitable tribute to an inspiring colleague, and we offer it to him with
affection and appreciation.
Bridget Welsh has kindly supplied the two photographs of Dominic.
Geoffrey Grimmett
Colin McDiarmid
Page 2 of 2
LIST OF CONTRIBUTORS
Combinatorics, Complexity, and Chance: A
Tribute to Dominic Welsh
Geoffrey Grimmett and Colin McDiarmid
Print publication date: 2007
Print ISBN-13: 9780198571278
Published to Oxford Scholarship Online: September 2007
DOI: 10.1093/acprof:oso/9780198571278.001.0001
(p.ix) LIST OF CONTRIBUTORS
(p.ix) LIST OF CONTRIBUTORS
Peter J. Cameron, School of Mathematical Sciences, Queen Mary,
University of London, Mile End Road, London E1 4NS, UK.
Laura E. Chávez Lomelí, Department of Mathematics, Simon Fraser
University, 8888 University Drive, Burnaby, British Columbia, V5A
1S6, Canada.
Graham E. Farr, Clayton School of Information Technology, Monash
University, Clayton, Victoria 3800, Australia.
Alan Frieze, Department of Mathematical Sciences, Carnegie Mellon
University, Pittsburgh PA 15213, USA.
Jim Geelen, Department of Combinatorics and Optimization,
University of Waterloo, 200 University Avenue West, Waterloo,
Ontario, N2L 3G1, Canada.
Bert Gerards, Centrum voor Wiskunde en Informatica, Kruislaan
413, 1098 SJ Amsterdam, The Netherlands.
Stefanie Gerke, ETH Zürich, Institute of Theoretical Computer
Science, CAB H 36.2, Universitätsstrasse 6, CH 8092 Zürich,
Switzerland.
Luis A. Goddyn, Department of Mathematics, Simon Fraser
University, 8888 University Drive, Burnaby, British Columbia, V5A
1S6, Canada.
Andrew Goodall, Department of Mathematics, University of Bristol,
Bristol BS8 1TW, UK.
Geoffrey Grimmett, Statistical Laboratory, Centre for Mathematical
Sciences, University of Cambridge, Wilberforce Road, Cambridge
CB3 0WB, UK.
Page 1 of 3
LIST OF CONTRIBUTORS
Mark Jerrum, School of Informatics, University of Edinburgh, The
King's Buildings, Edinburgh EH9 3JZ, UK; School of Mathematical
Sciences, Queen Mary, University of London, Mile End Road, London
E1 4NS, UK.
Bráulio Maia Junior, Departamento de Matemática e Estatística,
Universidade Federal de Campina Grande, Campina Grande, Paraíba,
58105–305, Brazil.
Koko Kalambay Kayibi, Design Research Group, Mathematics
Research Centre, School of Mathematical Sciences, Queen Mary,
University of London, Mile End Road, London E1 4NS, UK.
Manoel Lemos, Departamento de Matemática, Universidade Federal
de Pernambuco, Recife, Pernambuco, 50740–540, Brazil.
(p.x)
Lászlo Lovász, Microsoft Research, Redmond, WA 98052, USA;
Department of Computer Science, Eötvös University, Pázmány Péter
sétány 1/C, H-1117 Budapest, Hungary.
Colin McDiarmid, Department of Statistics, Oxford University, 1
South Parks Road, Oxford OX1 3TG, UK.
Tereza R. B. Melo, Departamento de Física e Matemática,
Universidade Federal Rural de Pernambuco, Recife, Pernambuco,
52171–900, Brazil.
Steven D. Noble, Department of Mathematical Sciences, Brunel
University, Kingston Lane, Uxbridge UB8 3PH, UK.
Marc Noy, Departament de Matemàtica Aplicada II, Universitat
Politècnica de Catalunya, Campus Nord, C. Jordi Girona 1–3, 08034
Barcelona, Spain.
James Oxley, Department of Mathematics, Louisiana State
University, Baton Rouge LA 70803–4918, USA.
Jorge Ramírez Alfonsín, Université Pierre et Marie Curie, Paris 6,
Equipe Combinatoire et Optimisation, 4 Place Jussieu, Paris 75252
Cedex 05, France.
David Romero, Instituto de Matemáticas, Universidad Nacional
Autónoma de México, 62210 Cuernavaca, Morelos, Mexico.
Abdón Sánchez-Arroyo, Instituto Nacional de Estadística, Geografía
e Informática, Héroe de Nacozari 2301, Puerta 3, Nivel 1,
Aguascalientes, Ags., Mexico C.P. 20270.
Angelika Steger, ETH Zürich, Institute of Theoretical Computer
Science, CAB H 19.2, Universitätsstrasse 6, CH 8092 Zürich,
Switzerland.
David Stirzaker, St John's College, Oxford OX1 3JP, UK.
Eric Vigoda, College of Computing, Georgia Institute of Technology,
Atlanta GA 30332, USA.
Page 2 of 3
LIST OF CONTRIBUTORS
Andreas Weißl, ETH Zürich, Institute of Theoretical Computer
Science, CAB H 24, Universitätsstrasse 6, CH 8092 Zürich,
Switzerland.
Geoff Whittle, School of Mathematics, Statistics and Computer
Science, Victoria University, PO Box 600, Wellington, New Zealand.
Page 3 of 3
ORBIT COUNTING AND THE TUTTE POLYNOMIAL
Combinatorics, Complexity, and Chance: A
Tribute to Dominic Welsh
Geoffrey Grimmett and Colin McDiarmid
Print publication date: 2007
Print ISBN-13: 9780198571278
Published to Oxford Scholarship Online: September 2007
DOI: 10.1093/acprof:oso/9780198571278.001.0001
ORBIT COUNTING AND THE TUTTE
POLYNOMIAL
Peter J. Cameron
DOI:10.1093/acprof:oso/9780198571278.003.0001
Abstract and Keywords
This chapter summarizes the various attempts to extend the Tutte polynomial of
a matroid to a polynomial which counts orbits of a group on various sets of
objects that the usual Tutte polynomial counts. In other words, the aim is to
produce a hybrid of the Tutte polynomial and the cycle index polynomial. There
have been various attempts at this, some of which are good for some aims but
not for others.
Keywords:   Tutte polynomial, matroid, orbit counting, cycle index polynomial, orbital chromatic
polynomial, flow and tension polynomials
This chapter is a summary (excluding some technical details), of various
attempts by me and others to extend the Tutte polynomial of a matroid to a
polynomial which counts orbits of a group on various sets of objects that the
usual Tutte polynomial counts. Said otherwise, the aim is to produce a hybrid of
the Tutte polynomial and the cycle index polynomial. There have been various
attempts at this, each of which is good for some jobs but does not work for
others.
1.1 Structure and symmetry
A question that we meet in an elementary course on combinatorics or probability
is: How many ways can we select n objects from a set of k objects? The answer
depends on whether we are allowed to repeat objects in the sample, and
whether we care about the order in which the selections are made. The well-
known answers are given in the table.
Page 1 of 12