Tight asymptotics of clique-chromatic numbers of dense random graphs

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).

The jump of the clique chromatic number of random graphs

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.Comment: 14 page

Clique coloring of binomial random graphs

A clique colouring of a graph is a colouring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colours in such a colouring is the clique chromatic number. In this paper, we study the asymptotic behaviour of the clique chromatic number of the random graph G(n,p) for a wide range of edge-probabilities p=p(n). We see that the typical clique chromatic number, as a function of the average degree, forms an intriguing step function