An edge coloring of a graph G is a mapping Á : E(G) ! N. The edge coloring Á is called strong if Á(e) 6= Á(e0) for any two edges e and e0 that are distance at most one apart. The minimum number of colors needed for a strong edge coloring of a graph G is called strong chromatic index of G and denoted by Â0 s(G). Strong edge colorings of graphs were introduced by Fouquet and Jolivet [1]. Clearly, Â0 s(G) ¸ ¢(G) for any graph G. In 1985, during a seminar in Prague, Erd˝os and Ne˘set˘ril put forward the following conjecture.Conjecture 1. For every graph G with maximum degree ¢,formulaErd˝os and Ne˘set˘ril provided a construction showing that Conjecture 1 is tight if it’s true. In 1997, using probabilistic method, Molloy and Reed [3] showed that Â0 s(G) · 1:9982¢ for a graph G with sufficiently large ¢. The currently best known upper bound for a graph G is 1:932¢, due to Bruhn and Joos [2]. The Hamming graph H(n;m) is the Cartesian product of n copies of the complete graph Km (n;m 2 N). Theorem 1. Let H(n;m) be a Hamming graph with m ¸ 2. Thenformulaand the upper bound is sharp for m = 2. Theorem 2. Let H(n; 3) be a Hamming graph. Thenformulaand the upper bound is sharp for n = 3.