Strong Cliques in Claw-Free Graphs

For a graph G, L(G)(2) is the square of the line graph of G - that is, vertices of L(G)(2) are edges of G and two edges e, f is an element of EoGTHORN are adjacent in L(G)(2) if at least one vertex of e is adjacent to a vertex of f and e not equal f. The strong chromatic index of G, denoted by s'(G), is the chromatic number of L(G)(2). A strong clique in G is a clique in L(G())2. Finding a bound for the maximum size of a strong clique in a graph with given maximum degree is a problem connected to a famous conjecture of Erdos and Nes. etr.il concerning strong chromatic index of graphs. In this note we prove that a size of a strong clique in a claw-free graph with maximum degree triangle is at most triangle(2) + 1/2 triangle. This result improves the only known result 1:125 triangle(2) + triangle, which is a bound for the strong chromatic index of claw-free graphs.