Strong edge colorings of uniform graphs

For a graph G = (V (G), E(G)), a strong edge coloring of G is an edge coloring in which every color class is an induced matching. The strong chromatic index of G, χs(G), is the smallest number of colors in a strong edge coloring of G. The strong chromatic index of the random graph G(n, p) was considered in [3], [4], [12], and [16]. In this paper, we consider χs(G) for a related class of graphs G known as uniform or -regular graphs. In particular, we prove that for 0 < d < 1, all (d, )-regular bipartite graphs G = (U ∪ V,E) with |U | = |V | ≥ n0(d, ) satisfy χs(G) ≤ ζ( )∆(G)2, where ζ( )→ 0 as → 0 (this order of magnitude is easily seen to be best possible). Our main tool in proving this statement is a powerful packing result of Pippenger and Spencer [11].