Proper connection numbers of complementary graphs

Abstract: 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 investigate the proper connection number of the complement of graph $G$ according to some constraints of $G$ itself. Also, we characterize the graphs on $n$ vertices that have proper connection number $n-2$. Using this result, we give a Nordhaus-Gaddum-type theorem for the proper connection number. We prove that if $G$ and $\overline{G}$ are both connected, then $4\le pc(G)+pc(\overline{G})\le n$, and the only graph attaining the upper bound is the tree with maximum degree $\Delta=n-2$.