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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 U E2) is precisely cd. The proof is based on probabilistic arguments.