Graph Colouring and the Probabilistic Method
Springer-Verlag GmbH
978-3-642-04015-3 (ISBN)
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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.
From the reviews:
"a ] We think that this well-written monograph will serve as a main reference on the subject for years to come."
JAnos BarAt, Acta Scientiarum Mathematicarum 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. a ] 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 45 (2), 2003
"This monograph provides an accessible and unified treatment of major advances made in graph colouring via the probabilistic method. a ] 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 987 (12), 2002
Colouring Preliminaries.- Probabilistic Preliminaries.- The First Moment Method.- The Lovasz Local Lemma.- The Chernoff Bound.- Hadwiger's Conjecture.- A First Glimpse of Total Colouring.- The Strong Chromatic Number.- Total Colouring Revisited.- Talagrand's Inequality and Colouring Sparse Graphs.- Azuma's Inequality and a Strengthening of Brooks'Theorem.- Graphs with Girth at Least Five.- Triangle-Free Graphs.- The List Colouring Conjecture.- A Structural Decomposition.- Omega,Delta, and Chi.- Near Optimal Total Colouring I: Sparse Graphs.- Near Optimal Total Colouring II: General Graphs.- Generalizations of the Local Lemma.- A Closer Look at Talagrand's Inequality.- Finding Fractional Colourings and Large Stable Sets.- Hardcore Distribution on Matchings.- The Asymptotics of Edge Colouring Multigraphs.- The Method of Conditional Expectation.- Algorithmic Aspects of the Local Lemma.
| Erscheint lt. Verlag | 4.12.2002 |
|---|---|
| Reihe/Serie | Algorithms and Combinatorics |
| Zusatzinfo | 19 schwarz-weisse Abbildungen |
| Verlagsort | Berlin |
| Sprache | englisch |
| Themenwelt | Informatik ► Theorie / Studium ► Algorithmen |
| Mathematik / Informatik ► Mathematik ► Graphentheorie | |
| Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik | |
| Schlagworte | algorithms • Computer • Computer Science • Graph • graph theory • Matching • Matchings |
| ISBN-10 | 3-642-04015-2 / 3642040152 |
| ISBN-13 | 978-3-642-04015-3 / 9783642040153 |
| Zustand | Neuware |
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