The strong chromatic number of a graph It is shown that there is an absolute constant c with the following property: For any two Summary: The strong chromatic number of a graph Noga Alon Abstract It is shown that there is an absolute constant c with the following property: For any two graphs G1 = (V, E1) and G2 = (V, E2) on the same set of vertices, where G1 has maximum degree at most d and G2 is a vertex disjoint union of cliques of size cd each, the chromatic number of the graph G = (V, E1 E2) is precisely cd. The proof is based on probabilistic arguments. 1 Introduction Let G = (V, E) be a graph on n vertices. If k divides n we say that G is strongly k-colorable if for any partition of V into pairwise disjoint sets Vi, each of cardinality k precisely, there is a proper k-vertex coloring of G in which each color class intersects each Vi by exactly one vertex. Notice that G is strongly k-colorable if and only if the chromatic number of any graph obtained from G by adding to it a union of vertex disjoint k-cliques (on the set V ) is k. If k does not divide n , we say that G is strongly k-colorable if the graph obtained from G by adding to it k n/k - n isolated vertices is strongly k-colorable. The strong chromatic number of a graph G , denoted by s(G), is the minimum k such that G is strongly k-colorable. As observed in [6] if G is strongly k-colorable then it is strongly k + 1- colorable as well, and hence s(G) is in fact the smallest k such that G is strongly s-colorable for all s k. Collections: Mathematics