STRONG CHROMATIC INDEX OF CLAW-FREE GRAPHS WITH EDGE WEIGHT SEVEN1

Let G be a graph and k a positive integer. A strong k-edge-coloring of G is a mapping phi : E(G) -> {1, 2, ... , k} such that for any two edges e and e' that are either adjacent to each other or adjacent to a common edge, phi(e) =6 phi(e'). The strong chromatic index of G is the minimum integer k such that G has a strong k-edge-coloring. The edge weight of G is defined to be max{d(u) + d(v) : uv is an element of E(G)}, where d(v) denotes the degree of v in G. In this paper, we prove that every claw-free graph with edge weight at most 7 has strong chromatic index at most 9, which is sharp.