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, χs(G), is the smallest number of colors in a strong edge coloring of G. In [9], Z. Palka proved that if p = p(n) = Θ(n−1), then with high probability, χs(G(n, p)) = O(∆(G(n, p))). Recently in [12], V. Vu proved that if n−1(lnn)1+δ ≤ p = p(n) ≤ n−ε for any 0 < ε, δ < 1, then with high probability, χs(G(n, p)) = O((pn)/ 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)) p( n 2) logb n ≤ χs(G(n, p)) ≤ (2 + o(1)) p(n2) logb n .