Chromatic thresholds in sparse random graphs

TL;DR

This paper investigates the chromatic thresholds in sparse random graphs, extending previous work to determine these thresholds for specific graphs and functions p(n), revealing new insights into graph coloring in probabilistic settings.

Contribution

The paper determines the chromatic thresholds $oldsymbol{ ext{delta}_ ext{chi}(H,p)}$ for most functions p(n) when H is a triangle, 5-cycle, or has chromatic number not in {3,4}, advancing understanding in sparse graph coloring.

Findings

Determined $ ext{delta}_ ext{chi}(H,p)$ for $H=K_3$ and $C_5$ across various $p(n)$.

Established $ ext{delta}_ ext{chi}(H,p)$ for all $H$ with $ ext{chi}(H) otin ext{{3,4}}$.

Showed that $ ext{delta}_ ext{chi}(H,p)$ equals $ ext{delta}_ ext{chi}(H)$ for fixed $p eq 0$, but differs when $p = o(1)$.

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 $\delta_\chi(H) :=\delta_\chi(H,1)$ was initiated in 1973 by Erd\H{o}s and Simonovits. Recently $\delta_\chi(H)$ was determined for all graphs $H$. It is known 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 $p = o(1)$. Here we study the problem for sparse random graphs. We determine $\delta_\chi(H,p)$ for most functions $p = p(n)$ when $H\in\{K_3,C_5\}$, and also for all graphs $H$ with $\chi(H) \not\in \{3,4\}$.