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 $$\textit{pc}(G)$$ , 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 a 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 $$4\le \textit{pc}(G)+\textit{pc}(\overline{G})\le n$$ , and the upper bound holds if and only if G or $$\overline{G}$$ is the n-vertex tree with maximum degree $$n-2$$ .
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