The chromatic number of dense random graphs

TL;DR

This paper advances understanding of the chromatic number in dense random graphs by providing tight bounds that narrow down its value, addressing a longstanding open problem in graph theory.

Contribution

It establishes the first bounds for the chromatic number of dense random graphs that match up to a small error term, refining previous asymptotic results.

Findings

New upper and lower bounds for $oldsymbol{ ext{chi}(G)}$

Bounds match up to an $o(1)$ term, narrowing the interval for the coloring rate

Answers a longstanding open question by Kang and McDiarmid

Abstract

The chromatic number $\chi(G)$ of a graph $G$ is defined as the minimum number of colours required for a vertex colouring where no two adjacent vertices are coloured the same. The chromatic number of the dense random graph $G \sim G(n,p)$ where $p \in (0,1)$ is constant has been intensively studied since the 1970s, and a landmark result by Bollob\'as in 1987 first established the asymptotic value of $\chi(G)$. Despite several improvements of this result, the exact value of $\chi(G)$ remains open. In this paper, new upper and lower bounds for $\chi(G)$ are established. These bounds are the first ones that match each other up to a term of size $o(1)$ in the denominator: they narrow down the colouring rate $n/\chi(G)$ of $G \sim G(n,p)$ to an explicit interval of length $o(1)$, answering a question of Kang and McDiarmid.