Cliques and Chromatic Number in Inhomogenous Random Graphs

Abstract

In this paper, we study cliques and chromatic number of inhomogenous random graphs where the individual edge probabilities could be arbitrarily low. We use a recursive method to obtain estimates on the maximum clique size under a mild positive average edge density assumption. As a Corollary, we also obtain uniform bounds on the maximum clique size and chromatic number for homogenous random graphs for all ranges of the edge probability $p_n$ satisfying $\frac{1}{n^{\alpha_1}} \leq p_n \leq 1-\frac{1}{n^{\alpha_2}}$ for some positive constants $\alpha_1$ and $\alpha_2$.