Proper connection number of random graphs

Abstract

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 p c ( 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 ( n 2 ) pairs of vertices appears as an edge with probability p independent from other pairs. We prove that for sufficiently large n , p c ( G ( n , p ) ) ≤ 2 if p ≥ log ⁡ n + α ( n ) n , where α ( n ) → ∞ .