t-STRONG CLIQUES AND THE DEGREE-DIAMETER PROBLEM

For a graph G, L(G)t is the tth power of the line graph of G; that is, vertices of L(G)(t) are edges of G and two edges e, f epsilon E(G) are adjacent in L(G)(t) if G contains a path with at most t vertices that starts in a vertex of e and ends in a vertex of f. The distance-t chromatic index of G is the chromatic number of L(G)(t), and a t-strong clique in G is a clique in L(G)(t). Finding upper bounds for the distance-t chromatic index and t-strong clique are problems related to two famous problems: the conjecture of Erdos and Nesetril concerning the strong chromatic index, and the degree/diameter problem. We prove that the size of a t-strong clique in a graph with maximum degree Delta is at most 1.75(Delta)t + O (Delta(t-1)), and for bipartite graphs the upper bound is at most Delta(t) + O (Delta(t-1)). As a corollary, we obtain upper bounds of 1.881 Delta(t) + O (Delta(t-1)) and 1.9703 + O (Delta(t-1)) on the distance-t chromatic index of bipartite graphs and general graphs. We also show results for some special classes of graphs: K1,r-free graphs and graphs with a large girth.