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Chromatic thresholds in dense random graphs

Allen, Peter and Böttcher, Julia and Griffiths, Simon and Kohayakawa, Yoshiharu and Morris, Robert (2016) Chromatic thresholds in dense random graphs. Random Structures & Algorithms. ISSN 1098-2418 (In Press)

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Abstract

The chromatic threshold δχ(H,p) of a graph H with respect to the random graph G(n,p) is the infimum over d>0 such that the following holds with high probability: the family of H-free graphs G⊂G(n,p) with minimum degree δ(G)≥dpn has bounded chromatic number. The study of the parameter δχ(H):=δχ(H,1) was initiated in 1973 by Erd\H{o}s and Simonovits, and was recently determined for all graphs H. In this paper we show that δχ(H,p)=δχ(H) for all fixed p∈(0,1), but that typically δχ(H,p)≠δχ(H) if p=o(1). We also make significant progress towards determining δχ(H,p) for all graphs H in the range p=n−o(1). In sparser random graphs the problem is somewhat more complicated, and is studied in a separate paper.

Item Type: Article
Official URL: http://onlinelibrary.wiley.com/journal/10.1002/(IS...
Additional Information: © 2016 Wiley Periodicals, Inc.
Subjects: Q Science > QA Mathematics
Sets: Departments > Mathematics
Date Deposited: 22 Jun 2016 14:16
Last Modified: 30 Nov 2016 09:45
Projects: 2010/09555-7, 2009/17831-7, 500016/2010-2, 308509/2007-2, 479032/2012-2, 303275/2013-8, 484154/2010-9
Funders: São Paulo Research Foundation, São Paulo Research Foundation, National Council for Scientific and Technological Development, National Council for Scientific and Technological Development , National Council for Scientific and Technological Development , National Council for Scientific and Technological Development , National Council for Scientific and Technological Development
URI: http://eprints.lse.ac.uk/id/eprint/66979

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