Note on directed proper connection number of a random graph

For an arc-colored digraph D, we say D is properly strongly connected, if for any ordered pair of vertices (x, y), D contains a directed path from x to y such that any adjacent arcs in that path have distinct colors. The directed proper connection number (pc) over right arrow (D) of a digraph D, is the minimum number of colors to make D properly strongly connected. Let D (n, p) denote the random digraph model, in which every arc of a digraph is chosen with probability p independently from other arcs. We prove that if p = {log n + log log n + lambda (n)} / n, then with high probability, (pc) over right arrow (D (n, p)) = 2, where lambda(n) tends to infinite. (c) 2019 Elsevier Inc. All rights reserved.