Proper connection number and 2-proper connection number of a graph

Abstract: A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored with one same color. An edge-colored graph is called $k$-proper connected if any two vertices of the graph are connected by $k$ internally pairwise vertex-disjoint proper paths in the graph. The $k$-proper connection number of a $k$-connected graph $G$, denoted by $pc_k(G)$, is defined as the smallest number of colors that are needed in order to make $G$ $k$-proper connected. For $k=1$, we write $pc(G)$ other than $pc_1(G)$, and call it the proper connection number of $G$. In this paper, we present an upper bound for the proper connection number of a graph $G$ in terms of the minimum degree of $G$, and give some sufficient conditions for a graph to have $2$-proper connection number two. Also, we investigate the proper connection numbers of dense graphs.