Chromatic thresholds in dense random graphs

Allen, Peter ORCID: 0000-0001-6555-3501, Böttcher, Julia ORCID: 0000-0002-4104-3635, Griffiths, Simon, Kohayakawa, Yoshiharu and Morris, Robert (2017) Chromatic thresholds in dense random graphs. Random Structures and Algorithms, 51 (2). 185 - 214. ISSN 1042-9832

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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:	https://onlinelibrary.wiley.com/journal/10982418
Additional Information:	© 2017 Wiley Periodicals, Inc.
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	22 Jun 2016 14:16
Last Modified:	01 Oct 2024 03:44
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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