A path in an edge-colored graph is called proper if no two consecutive 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. In this paper, we consider two conjectures on the proper connection number of graphs. The first conjecture states that if G is a noncomplete graph with connectivity κ(G) = 2 and minimum degree δ(G) ≥ 3, then pc(G) = 2, posed by Borozan et al. (2012). We give a family of counterexamples to disprove this conjecture. However, from a result of Thomassen it follows that 3-edge-connected noncomplete graphs have proper connection number 2. Using this result, we can prove that if G is a 2-connected noncomplete graph with diam(G) = 3, then pc(G) = 2, which solves the second conjecture we want to mention, posed by Li and Magnant (2015).
Available at: http://works.bepress.com/colton_magnant/57/