Clique coloring of dense random graphs

TL;DR

This paper proves that the clique chromatic number of dense random graphs G(n,1/2) is asymptotically logarithmic in n, resolving a previously open problem about its growth rate.

Contribution

It establishes a tight bound of (log n) for the clique chromatic number of G(n,1/2), confirming the conjecture and completing the understanding of its asymptotic behavior.

Findings

Clique chromatic number is (log n) with high probability.

Resolved a conjecture about the asymptotic growth of clique chromatic number.

Provides tight bounds confirming the order of the clique chromatic number.

Abstract

The clique chromatic number of a graph G=(V,E) is the minimum number of colors in a vertex coloring so that no maximal (with respect to containment) clique is monochromatic. We prove that the clique chromatic number of the binomial random graph G=G(n,1/2) is, with high probability, \Omega(log n). This settles a problem of McDiarmid, Mitsche and Pralat who proved that it is O(log n) with high probability.