The chromatic number of very dense random graphs

TL;DR

This paper investigates the chromatic number of very dense random graphs, establishing concentration results, non-concentration phenomena, and a central limit theorem in specific density ranges.

Contribution

It extends understanding of chromatic number behavior in dense random graphs, especially when the largest independent set size is 3, providing new concentration and distribution results.

Findings

Chromatic number concentrates on an interval of length O(√μ₃).

There are infinitely many n where chromatic number is not concentrated on small intervals.

A central limit theorem holds for the chromatic number in certain density ranges.

Abstract

The chromatic number of a very dense random graph $G(n,p)$, with $p \ge 1 - n^{-c}$ for some constant $c > 0$, was first studied by Surya and Warnke, who conjectured that the typical deviation of $\chi(G(n,p))$ from its mean is of order $\sqrt{\mu_r}$, where $\mu_r$ is the expected number of independent sets of size $r$, and $r$ is maximal such that $\mu_r > 1$, except when $\mu_r = O(\log n)$. They moreover proved their conjecture in the case $n^{-2} \ll 1 - p = O(n^{-1})$. In this paper, we study $\chi(G(n,p))$ in the range $n^{-1}\log n \ll 1 - p \ll n^{-2/3}$, that is, when the largest independent set of $G(n,p)$ is typically of size 3. We prove in this case that $\chi(G(n,p))$ is concentrated on some interval of length $O(\sqrt{\mu_3})$, and for sufficiently `smooth' functions $p = p(n)$, that there are infinitely many values of $n$ such that $\chi(G(n,p))$ is not concentrated on any interval of size $o(\sqrt{\mu_3})$. We also show that $\chi(G(n,p))$ satisfies a central limit theorem in the range $n^{-1} \log n \ll 1 - p \ll n^{-7/9}$.