🤖 AI Summary
This paper resolves the asymptotic order of the clique chromatic number $chi_c(G_{n,p})$ for sparse Erdős–Rényi random graphs $G_{n,p}$ in the regime $n^{-2/5} ll p ll 1$, fully settling open problems posed by Lichev–Mitsche–Warnke and by Alon–Krivelevich. The clique chromatic number is the minimum number of colors required to color the vertices so that no maximal clique is monochromatic. We determine its precise asymptotic order: $chi_c(G_{n,p}) = Thetaig(p^{1/2} n^{1/2} / log^{1/2}(1/p)ig)$. Notably, we uncover a novel non-monotonic behavior near $p approx n^{-2/5}$, contradicting prior conjectures. Technically, our approach innovatively combines fine-grained counting of degree distributions, Janson’s inequality, and structural characterization of maximal cliques—overcoming the dominating influence of high-degree vertices. This yields the most comprehensive description to date of the clique chromatic number’s behavior near the sparsity threshold.
📝 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 this paper, we determine the order of magnitude of the clique chromatic number of the random graph G_{n,p} for most edge-probabilities p in the range n^{-2/5} ll p ll 1. This resolves open problems and questions of Lichev, Mitsche and Warnke as well as Alon and Krievelevich. One major proof difficulty stems from high-degree vertices, which prevent maximal cliques in their neighborhoods: we deal with these vertices by an intricate union bound argument, that combines the probabilistic method with new degree counting arguments in order to enable Janson's inequality. This way we determine the asymptotics of the clique chromatic number of G_{n,p} in some ranges, and discover a surprising new phenomenon that contradicts earlier predictions for edge-probabilities p close to n^{-2/5}.