Proper connection number and graph products

Abstract

A path $P$ in an edge-colored graph $G$ is called \emph{a proper path} if no two adjacent edges of $P$ are colored the same, and $G$ is \emph{proper connected} if every two vertices of $G$ are connected by a proper path in $G$. The \emph{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 lexicographical, strong, Cartesian, and direct product and present several upper bounds for these products of graphs.