Minimum degree condition for proper connection number 2

Abstract

A path in an edge-colored graph is called a proper path if no two adjacent edges of the path receive 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. Recently, Li and Magnant in [8] posed the following conjecture: If G is a connected noncomplete graph of order n≥5 and minimum degree δ(G)≥n/4, then pc(G)=2. In this paper, we show that this conjecture is true except for two small graphs on 7 and 8 vertices, respectively. As a byproduct we obtain that if G is a connected bipartite graph of order n≥4 with δ(G)≥n+68, then pc(G)=2.