Book contents
- Frontmatter
- Contents
- Preface
- 1 Counting planar maps, coloured or uncoloured
- 2 A survey of PPAD-completeness for computing Nash equilibria
- 3 Hypergraph Turán problems
- 4 Some new results in extremal graph theory
- 5 The cyclic sieving phenomenon: a survey
- 6 Order in building theory
- 7 Graphs, colours, weights and hereditary properties
- 8 Random geometric graphs
- 9 Transversals in latin squares: a survey
4 - Some new results in extremal graph theory
Published online by Cambridge University Press: 05 August 2011
- Frontmatter
- Contents
- Preface
- 1 Counting planar maps, coloured or uncoloured
- 2 A survey of PPAD-completeness for computing Nash equilibria
- 3 Hypergraph Turán problems
- 4 Some new results in extremal graph theory
- 5 The cyclic sieving phenomenon: a survey
- 6 Order in building theory
- 7 Graphs, colours, weights and hereditary properties
- 8 Random geometric graphs
- 9 Transversals in latin squares: a survey
Summary
Abstract
In recent years several classical results in extremal graph theory have been improved in a uniform way and their proofs have been simplified and streamlined. These results include a new Erdős-Stone-Bollobás theorem, several stability theorems, several saturation results and bounds for the number of graphs with large forbidden subgraphs.
Another recent trend is the expansion of spectral extremal graph theory, in which extremal properties of graphs are studied by means of eigenvalues of various matrices. One particular achievement in this area is the casting of the central results above in spectral terms, often with additional enhancement. In addition, new, specific spectral results were found that have no conventional analogs.
All of the above material is scattered throughout various journals, and since it may be of some interest, the purpose of this survey is to present the best of these results in a uniform, structured setting, together with some discussions of the underpinning ideas.
Introduction
The purpose of this survey is to give a systematic account of two recent lines of research in extremal graph theory. The first one, developed in [14],[15],[16],[63, 68], improves a number of classical results grouped around the theorem of Turán. The main progress is along the following three guidelines: replacing fixed parameters by variable ones; giving explicit conditions for the validity of the statements; developing and using tools of general scope.
- Type
- Chapter
- Information
- Surveys in Combinatorics 2011 , pp. 141 - 182Publisher: Cambridge University PressPrint publication year: 2011
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