The jump of the clique chromatic number of random graphs

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 Prałat noted that around p≈n−1/2$$ p\approx {n}^{-1/2} $$ the clique chromatic number of the random graph Gn,p$$ {G}_{n,p} $$ changes by nΩ(1)$$ {n}^{\Omega (1)} $$ when we increase the edge‐probability p$$ p $$ by no(1)$$ {n}^{o(1)} $$ , but left the details of this surprising “jump” phenomenon as an open problem. We settle this problem, that is, resolve the nature of this polynomial “jump” of the clique chromatic number of the random graph Gn,p$$ {G}_{n,p} $$ around edge‐probability p≈n−1/2$$ p\approx {n}^{-1/2} $$ . 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 Gn,p$$ {G}_{n,p} $$ up to logarithmic factors for any edge‐probability p$$ p $$ .