Bounding the strong chromatic index of dense random graphs

For a graph 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,χ_s(G), is the smallest number of colors in a strong edge coloring of G. Palka (Austral. J. Combin. 18 (1998) 219-226), proved that if p=p(n)=Θ(n^-1), then with high probability, χ _s(G(n,p))=O(Δ(G(n,p))). Recently Vu (Combin. Probab. Comput. 11 (2002) 103-111), proved that if n^-1(lnn) ^1+δ≤p=p(n)≤n^-ε for any 0<ε,δ<1, then with high probability, χ _s(G(n,p))=O((pn)²/ln(pn)). In this note, we prove that if p=p(n)>n^-ε for all ε>0, then with b=(1-p) ^-1, with high probability, (1-o(1))(pn2/log _bn)≤χ_s(G(n,p))≤(2+o(1))(pn2/log_bn).