Vertex colorings with a distance restriction

Abstract

Let d, k be any two positive integers with k > d > 0. We consider a k-coloring of a graph G such that the distance between each pair of vertices in the same color-class is at least d. Such graphs are said to be ( k, d)-colorable. The object of this paper is to determine the maximum size of ( k, 3)-colorable, ( k, 4)-colorable, and ( k, k − 1)-colorable graphs. Sharp results are obtained for both ( k, 3)-colorable and ( k, k − 1)-colorable graphs, while the results obtained for ( k, 4)-colorable graphs are close to the truth.