Proper connection number of random graphs.

A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored with the same color. For a connected graph G, the proper connection number pc(G) of G is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of G is connected by at least one proper path in G. In this paper, we show that almost all graphs have the proper connection number 2. More precisely, let G(n,p) denote the Erdös–Rényi random graph model, in which each of the (n2) pairs of vertices appears as an edge with probability p independent from other pairs. We prove that for sufficiently large n, pc(G(n,p))≤2 if p≥log⁡n+α(n)n, where α(n)→∞.