On the Strong Chromatic Number of Random Graphs

Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colorable if for every partition of V (G) into disjoint sets V1 ∪ . . . ∪ Vr, all of size exactly k, there exists a proper vertex k-coloring of G with each color appearing exactly once in each Vi. In the case when k does not divide n, G is defined to be strongly k-colorable if the graph obtained by adding k ⌈ n k ⌉ − n isolated vertices is strongly k-colorable. The strong chromatic number of G is the minimum k for which G is strongly k-colorable. In this paper, we study the behavior of this parameter for the random graph Gn,p. In the dense case when p n, we prove that the strong chromatic number is a.s. concentrated on one value ∆ + 1, where ∆ is the maximum degree of the graph. We also obtain several weaker results for sparse random graphs.