On the strong chromatic number of random graphs

TL;DR

This paper investigates the strong chromatic number of random graphs G(n, p), showing that in dense cases it concentrates on the maximum degree plus one, with some results for sparser graphs.

Contribution

It establishes the concentration of the strong chromatic number for dense random graphs and provides initial results for sparser cases.

Findings

Strong chromatic number concentrates on maximum degree plus one in dense graphs.

Results extend understanding of coloring properties in random graph models.

Weaker bounds are obtained for sparse random graphs.

Abstract

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 V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex k-coloring of G with each color appearing exactly once in each V_i. In the case when k does not divide n, G is defined to be strongly k-colorable if the graph obtained by adding k \lceil n/k \rceil - 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 G(n, p). In the dense case when p >> n^{-1/3}, we prove that the strong chromatic number is a.s. concentrated on one value \Delta+1, where \Delta is the maximum degree of the graph. We also obtain several weaker results for sparse random graphs.