Two Sufficient Conditions for 2-Connected Graphs to Have Proper Connection Number 2

The proper connection number of a graph G, denoted by pc(G), is the minimum number of colors needed to color the edges of G so that every pair of distinct vertices of G is connected by a path in which no two adjacent edges of the path receive the same color. A connected graph G is k-PC if pc(G) ≤ k. In the literature, it is shown that every 3-edge-connected graph is 2-PC and every 2-edge-connected graph is 3-PC. We show in this paper that every 2-connected graph such that each edge is contained in a cycle of length at most 4 is 2-PC, and every 2-connected graph with minimum degree at least 3 such that each edge is contained in a cycle of length at most 6 is 2-PC.