The jump of the clique chromatic number of random graphs

TL;DR

This paper investigates the dramatic change in the clique chromatic number of random graphs around a critical edge probability, providing a detailed resolution of this phenomenon and precise estimates.

Contribution

It resolves the open problem by characterizing the polynomial 'jump' of the clique chromatic number near p ≈ n^{-1/2} and extends previous methods beyond Janson's inequality.

Findings

The clique chromatic number exhibits a polynomial jump around p ≈ n^{-1/2}.

The authors determine the clique chromatic number up to logarithmic factors for all p.

The proof introduces new approximation and concentration techniques beyond existing inequalities.

Abstract

The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid, Mitsche and Pralat noted that around p \approx n^{-1/2} the clique chromatic number of the random graph G_{n,p} changes by n^{\Omega(1)} when we increase the edge-probability p by n^{o(1)}, but left the details of this surprising phenomenon as an open problem. We settle this problem, i.e., resolve the nature of this polynomial `jump' of the clique chromatic number of the random graph G_{n,p} around edge-probability p \approx n^{-1/2}. Our proof uses a mix of approximation and concentration arguments, which enables us to (i) go beyond Janson's inequality used in previous work and (ii) determine the clique chromatic number of G_{n,p} up to logarithmic factors for any edge-probability p.