Graph Colouring and the Probabilistic Method

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.

The book covers a topic of central interest to discrete mathematics.- The authors are two of the very best on this topic. Includes supplementary material: sn.pub/extras

Contenu

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.