Chromatic thresholds in dense random graphs

TL;DR

This paper investigates the chromatic thresholds of graphs within dense and sparse random graphs, establishing that these thresholds are consistent in dense cases but vary in sparser regimes, advancing understanding of graph coloring in random structures.

Contribution

It proves that the chromatic threshold in dense random graphs equals the deterministic threshold, and explores how this threshold behaves in sparser graphs, extending previous results.

Findings

Chromatic threshold equals deterministic threshold for all fixed p in (0,1)

Threshold behavior differs in sparser graphs where p = o(1)

Progress towards understanding thresholds for graphs with p = n^{-o(1)}

Abstract

The chromatic threshold $\delta_\chi(H,p)$ 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 $G \subset G(n,p)$ with minimum degree $\delta(G) \ge dpn$ has bounded chromatic number. The study of the parameter $\delta_\chi(H) := \delta_\chi(H,1)$ 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 $\delta_\chi(H,p) = \delta_\chi(H)$ for all fixed $p \in (0,1)$, but that typically $\delta_\chi(H,p) \ne \delta_\chi(H)$ if $p = o(1)$. We also make significant progress towards determining $\delta_\chi(H,p)$ 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.