Graph Colouring and the Probabilistic Method

by Michael Molloy, Bruce Reed

Overview

Over the past decade, many major advances have been made in the field of graph colouring via the probabilistic method. This monograph provides an accessible and unified treatment of these results, using tools such as the Lovasz Local Lemma and Talagrand's concentration inequality.

The topics covered include: Kahn's proofs that the Goldberg-Seymour and List Colouring Conjectures hold asymptotically; a proof that for some absolute constant C, every graph of maximum degree Delta has a Delta+C total colouring; Johansson's proof that a triangle free graph has a O(Delta over log Delta) colouring; algorithmic variants of the Local Lemma which permit the efficient construction of many optimal and near-optimal colourings.

This begins with a gentle introduction to the probabilistic method and will be useful to researchers and graduate students in graph theory, discrete mathematics, theoretical computer science and probability.

Product Details

ISBN-13:	9783540421399
Publisher:	Springer Berlin Heidelberg
Publication date:	12/06/2001
Series:	Algorithms and Combinatorics , #23
Edition description:	2002
Pages:	326
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1. Colouring Preliminaries.- 2. Probabilistic Preliminaries.- 3. The First Moment Method.- 4. The Lovász Local Lemma.- 5. The Chernoff Bound.- 6. Hadwiger’s Conjecture.- 7. A First Glimpse of Total Colouring.- 8. The Strong Chromatic Number.- 9. Total Colouring Revisited.- 10. Talagrand’s Inequality and Colouring Sparse Graphs.- 11. Azuma’s Inequality and a Strengthening of Brooks’ Theorem.- 12. Graphs with Girth at Least Five.- 13. Triangle-Free Graphs.- 14. The List Colouring Conjecture.- 15. The Structural Decomposition.- 16.—,— and—.- 17. Near Optimal Total Colouring I: Sparse Graphs.- 18. Near Optimal Total Colouring II: General Graphs.- 19. Generalizations of the Local Lemma.- 20. A Closer Look at Talagrand’s Inequality.- 21. Finding Fractional Colourings and Large Stable Sets.- 22. Hard-Core Distributions on Matchings.- 23. The Asymptotics of Edge Colouring Multigraphs.- 24. The Method of Conditional Expectations.- 25. Algorithmic Aspects of the Local Lemma.- References.

Editorial Reviews

From the reviews of the first edition:

"The presented book contains many … chapters, each of which presents a proof technique and apply that for a certain graph coloring problem. … The book ends with a vast bibliography. We think that this well-written monograph will serve as a main reference on the subject for years to come." (János Barát, Acta Scientiarum Mathematicarum, Vol. 69, 2003)

"The book is a pleasure to read; there is a clear, successful attempt to present the intuition behind the proofs, making even the difficult, recent proofs of important results accessible to potential readers. … The book is highly recommended to researchers and graduate students in graph theory, combinatorics, and theoretical computer science who wish to have this ability." (Noga Alon, SIAM Review, Vol. 45 (2), 2003)

"The probabilistic method in graph theory was initiated by Paul Erdös in 1947 … . This book is an introduction to this powerful method. … The book is well-written and brings the researcher to the frontiers of an exciting field." (M.R. Murty, Short Book Reviews, Vol. 23 (1), April, 2003)

"This monograph provides an accessible and unified treatment of major advances made in graph colouring via the probabilistic method. … Many exercises and excellent remarks are presented and discussed. Also very useful is the list of up-to-date references for current research. This monograph will be useful both to researchers and graduate students in graph theory, discrete mathematics, theoretical computer science and probability." (Jozef Fiamcik, Zentralblatt MATH, Vol. 987 (12), 2002)

From the Publisher