Proper Connection Number of Graph Products

A path P in an edge-colored graph G is called a proper path if no two adjacent edges of P are colored the same, and G is proper connected if every two vertices of G are connected by a proper path in G. The proper connection number of a connected graph G, denoted by pc(G), is the minimum number of colors that are needed to make G proper connected. In this paper, we study the proper connection number on the lexicographic, strong, Cartesian, and direct products and present exact values or upper bounds for these products of graphs.