On the Strong Chromatic Number of Graphs

The strong chromatic number, χS(G), of an n-vertex graph G is the smallest number k such that after adding kdn/ke−n isolated vertices to G and considering any partition of the vertices of the resulting graph into disjoint subsets V1, . . . , Vdn/ke of size k each, one can find a proper k-vertex-coloring of the graph such that each part Vi, i = 1, . . . , dn/ke, contains exactly one vertex of each color. For any graph G with maximum degree ∆, it is easy to see that χS(G) ≥ ∆ + 1. Recently, Haxell proved that χS(G) ≤ 3∆−1. In this paper, we improve this bound for graphs with large maximum degree. We show that χS(G) ≤ 2∆ if ∆ ≥ n/6 and prove that this bound is sharp.