The chromatic number of squares of random graphs

Abstract

The Erdös-Rényi model is a simple and widely studied model for generating random graphs. Given a positive integer n and a real p between 0 and 1, G(n, p) is the distribution over n-vertex graphs obtained by including, for every unordered pair {u, v} of vertices, the edge uv in the edge set of G independently with probability p. The square of a graph G, denoted by G2, is the graph obtained from G by also adding an edge between every pair of vertices that share at least one common neighbor. A proper k-coloring of a graph G is a function f that assigns to every vertex of G a color f(v) from the set {1, . . . , k} such that no two neighbouring vertices get the same color, and the chromatic number of a graph G is the minimum k so that G has a k-coloring. In a recent article, Cheng, Maji and Pothen [3] consider squares of sparse Erdős-Rényi graphs G(n, p) with p = Θ(1/n) as interesting benchmark instances to evaluate parallel algorithms that color the input graph. These authors prove that if G is sampled from G(n, p) with p = Θ(1/n) then, with high probability, the chromatic number of G2 lies between Ω ( log n log log n ) and O(log n). In this