Chromatic thresholds in sparse random graphs

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 δχ (H) := δχ (H, 1) was initiated in 1973 by Erdős and Simonovits. Recently δχ (H) was determined for all graphs H. It is known that δχ (H, p) = δχ (H) for all fixed p ∈ (0, 1), but that typically δχ (H, p) ≠ δχ (H) if p ≠ o(1). Here we study the problem for sparse random graphs. We determine δχ (H, p) for most functions p = p(n) when H ∈ {K3,C5}, and also for all graphs H with χ(H) ≠∈ {3, 4}. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 215–236, 2017.