Tight asymptotics of clique-chromatic numbers of dense random graphs

TL;DR

This paper determines the precise asymptotic behavior of the clique chromatic number in dense random graphs, confirming the conjecture that it is approximately half the logarithm of the number of vertices, with a tight concentration result.

Contribution

It proves the conjectured constant for the clique chromatic number of dense random graphs and establishes a tight concentration around this value.

Findings

Clique chromatic number is approximately (1/2) log_2 n.

Confirmed the conjecture on the constant factor in the asymptotics.

Established a tight concentration result around the asymptotic value.

Abstract

The clique chromatic number of a graph is the minimum number of colors required to assign to its vertex set so that no inclusion maximal clique is monochromatic. McDiarmid, Mitsche and Pra\l at proved that the clique chromatic number of the binomial random graph $G\left(n,\frac{1}{2}\right) $ is at most $\left(\frac{1}{2}+o(1)\right)\log_2n$ with high probability. Alon and Krivelevich showed that it is greater than $\frac{1}{2000}\log_2n$ with high probability and suggested that the right constant in front of the logarithm is $\frac{1}{2}.$ We prove their conjecture and, beyond that, obtain a tight concentration result: whp $\chi_c\left(G\left(n,1/2\right)\right) = \frac{1}{2}\log_2 n - \Theta\left(\ln\ln n\right).$