The jump of the clique chromatic number of random graphs

05/25/2021

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by   Lyuben Lichev, et al.

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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 ≈n^-1/2 the clique chromatic number of the random graph G_n,p changes by n^Ω(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 ≈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.