Vertex colorings with a distance restriction

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. (C) 1998 Elsevier Science B.V. All rights reserved.