Nordhaus–Gaddum-Type Theorem for Total-Proper Connection Number of Graphs

Abstract

A graph is said to be total-colored if all the edges and the vertices of the graph are colored. A path P in a total-colored graph G is called a total-proper path if (1) any two adjacent edges of P are assigned distinct colors; (2) any two adjacent internal vertices of P are assigned distinct colors; and (3) any internal vertex of P is assigned a distinct color from its incident edges of P. The total-colored graph G is total-proper connected if any two distinct vertices of G are connected by a total-proper path. The total-proper connection number of a connected graph G, denoted by tpc(G), is the minimum number of colors that are required to make G total-proper connected. In this paper, we first characterize the graphs G on n vertices with $tpc(G)=n-1$. Based on this, we obtain a Nordhaus–Gaddum-type result for total-proper connection number. We prove that if G and $\overline{G}$ are connected complementary graphs on n vertices, then $6\le tpc(G)+tpc(\overline{G})\le n+2$. Examples are given to show that the lower bound is sharp for $n\ge 4$. The upper bound is reached for $n\ge 4$ if and only if G or $\overline{G}$ is the tree with maximum degree $n-2$.