The strong chromatic index of C4-free graphs

The strong chromatic index of a graph G is the minimum number of induced matchings which partition E(G). In 1985, Erd} os and Ne set ril conjectured that the strong chromatic index of every graph of maximum degree is at most 5 4 2. In this paper, we use probabilistic method to prove an asymptotically better result for graphs which do not have a cycle of length 4. Also, we show that our bound is asymptotically best possible, up to a constant multiple. At the end, some open problems and a conjecture are stated.