The strong chromatic index of a class of graphs

The strong chromatic index of a graph G is the minimum integer k such that the edge set of G can be partitioned into k induced matchings. Faudree et al. [R.J. Faudree, R.H. Schelp, A. Gyarfas, Zs. Tuza, The strong chromatic index of graphs, Ars Combin. 29B (1990) 205-211] proposed an open problem: If G is bipartite and if for each edge xy@?E(G), d(x)+d(y)@?5, then s@g^'(G)@?6. Let H"0 be the graph obtained from a 5-cycle by adding a new vertex and joining it to two nonadjacent vertices of the 5-cycle. In this paper, we show that if G (not necessarily bipartite) is not isomorphic to H"0 and d(x)+d(y)@?5 for any edge xy of G then s@g^'(G)@?6. The proof of the result implies a linear time algorithm to produce a strong edge coloring using at most 6 colors for such graphs.