On the (di)graphs with (directed) proper connection number two

A coloring of a graph G is properly connected if every two vertices of G are the ends of a properly colored path. We study the complexity of computing the proper connection number (minimum number of colors in a properly connected coloring) for edge and vertex colorings, in undirected and directed graphs, respectively. First we disprove some conjectures of Magnant et al. (2016) on characterizing the strong digraphs with proper arc connection number at most two. Then, we prove that deciding whether a given digraph has proper arc connection number at most two is NP-complete. We initiate the study of proper vertex connectivity in digraphs and we prove similar results as for the arc version. Finally, we present polynomial-time recognition algorithms for bounded-treewidth graphs and bipartite graphs with proper edge connection number at most two.