Expected Chromatic Number of Random Subgraphs

Abstract

Given a graph G and p ∈ [0, 1], let Gp denote the random subgraph of G obtained by keeping each edge independently with probability p. Alon, Krivelevich, and Sudokov. We prove a new spectral lower bound on E[χ(Gp)], as progress towards Bukh's conjecture. We also propose the stronger conjecture that for any fixed p ≤ 1/2, among all graphs of fixed chromatic number, E[χ(Gp)] is minimized by the complete graph. We prove this stronger conjecture when G is planar or χ(G) < 4. We also consider weaker lower bounds on E[χ(Gp)] proposed in a recent paper by Shinkar [17]; we answer two open questions posed in [17] negatively and propose a possible refinement of one of them.

On the Chromatic Number in the Stochastic Block Model

“…In particular, the appearance of "balanced" independent sets in G implies the existence of a "balanced" independent set (with the same size proportion s/n) in its subgraph G ∼ G(n , P ) where n = ny for any 0 y x. Repeating the arguments of (5) for such G , we conclude that whp s can not exceed…”

“…Determining the chromatic number of a random subgraph G p for a general graph G is a much harder problem; see, for example, [4][5][6]29,34]. In particular, Bukh asks [6] whether for any graph G, there exists a positive constant c such that Eχ(G 1/2 ) c log(χ(G)) χ(G).…”