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A path $P$ in an edge-colored graph $G$ is called a proper path if no twoadjacent edges of $P$ are colored the same, and $G$ is proper connected ifevery two vertices of $G$ are connected by a proper path in $G$. The properconnection number of a connected graph $G$, denoted by $pc(G)$, is the minimumnumber 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 graphson $n$ vertices that have proper connection number $n-2$. Using this result, wegive a Nordhaus-Gaddum-type theorem for the proper connection number. We provethat if $G$ and $\overline{G}$ are both connected, then $4\lepc(G)+pc(\overline{G})\le n$, and the only graph attaining the upper bound isthe tree with maximum degree $\Delta=n-2$.

Proper connection numbers of complementary graphs