Proper connection of graphs

An edge-colored graph GG is kk-proper connected if every pair of vertices is connected by kk internally pairwise vertex-disjoint proper colored paths. The kk-proper connection number of a connected graph GG, denoted by pck(G)pck(G), is the smallest number of colors that are needed to color the edges of GG in order to make it kk-proper connected. In this paper we prove several upper bounds for pck(G)pck(G). We state some conjectures for general and bipartite graphs, and we prove them for the case when k=1k=1. In particular, we prove a variety of conditions on GG which imply pc1(G)=2pc1(G)=2.