Chromatic thresholds in dense random graphs1. Issue 2 (16th May 2017)

• Record Type: Journal Article
• Authors: Allen, Peter; Böttcher, Julia; Griffiths, Simon; Kohayakawa, Yoshiharu; Morris, Robert
• Is Part Of: Random structures & algorithms. Volume 51:Issue 2(2017)
• Page Start: 185
• Page End: 214
• Publication Date: 2017-05-16
• Subjects: random graphs -- chromatic threshold -- minimum degree; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20708
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract: The chromatic threshold δ χ ( 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 ) ⩾ d p n has bounded chromatic number. The study of the parameter δ χ ( H ) : = δ χ ( H, 1 ) was initiated in 1973 by Erdős and Simonovits, and was recently determined for all graphs H . In this paper we show that δ χ ( H, p ) = δ χ ( H ) for all fixed p ∈ ( 0, 1 ), but that typically δ χ ( H, p ) ≠ δ χ ( H ) if p = o ( 1 ) . We also make significant progress towards determining δ χ ( 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. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 185–214, 2017