Proper connection of graphs

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