Graph colouring and the probabilistic method图着色与概率方法

2001-10  

湖南文艺出版社  

Molloy, Michael S.; Molloy, M. S.; Reed, B.  

326  

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.

Part I Preliminaries 　1 Colouring Preliminaries 　2 Probabilistic Preliminaries Part II Basic Probabilistic Tools 　3 The First Moment Method 　4 The Lovasz Local Lemma 　5 The Chernoff Bound Part III Vertex Partitions 　6 Hadwiger's Conjecture 　7 A First Glimpse of Total Colouring 　8 The Strong Chromatic Number 　9 Total Colouring Revisited Part IV A Naive Colouring Procedure 　10 Talagrand's Inequality and Colouring Sparse Graphs 　11 Azuma's Inequality and a Strengthening of Brooks' Theorem Part V An Iterative Approach 　12 Graphs with Girth at Least Five 　13 Triangle-Free Graphs 　14 The List Colouring Conjecture Part VI A Structural Decomposition 　15 The Structural Decomposition 　16 [omega], [Delta] and [chi] 　17 Near Optimal Total Colouring I: Sparse Graphs 　18 Near Optimal Total Colouring II: General Graphs Part VII Sharpening our Tools 　19 Generalizations of the Local Lemma 　20 A Closer Look at Talagrand's Inequality Part VIII Colour Assignment via Fractional Colouring 　21 Finding Fractional Colourings and Large Stable Sets 　22 Hard-Core Distributions on Matchings 　23 The Asymptotics of Edge Colouring Multigraphs Part IX Algorithmic Aspects 　24 The Method of Conditional Expectations 　25 Algorithmic Aspects of the Local Lemma References Index

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