The chromatic thresholds of graphs
Combinatorics
2011-08-09 v1
Abstract
The chromatic threshold delta_chi(H) of a graph H is the infimum of d>0 such that there exists C=C(H,d) for which every H-free graph G with minimum degree at least d|G| satisfies chi(G)<C. We prove that delta_chi(H) \in {(r-3)/(r-2), (2r-5)/(2r-3), (r-2)/(r-1)} for every graph H with chi(H)=r>2. We moreover characterise the graphs H with a given chromatic threshold, and thus determine delta_chi(H) for every graph H. This answers a question of Erd\H{o}s and Simonovits [Discrete Math. 5 (1973), 323-334], and confirms a conjecture of {\L}uczak and Thomass\'e [preprint (2010), 18pp].
Cite
@article{arxiv.1108.1746,
title = {The chromatic thresholds of graphs},
author = {Peter Allen and Julia Böttcher and Simon Griffiths and Yoshiharu Kohayakawa and Robert Morris},
journal= {arXiv preprint arXiv:1108.1746},
year = {2011}
}
Comments
37 pages, 2 figures