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A graph is said to be total-colored if all the edges and the vertices of the graph are colored. A total-colored graph is total-rainbow connected if any two vertices of the graph are connected by a path whose edges and internal vertices have distinct colors. For a connected graph G, the total-rainbow connection number of G, denoted by trc(G), is the minimum number of colors required in a total-coloring of G to make G total-rainbow connected. In this paper, we first characterize the graphs having large total-rainbow connection numbers. Based on this, we obtain a Nordhaus–Gaddum-type upper bound for the total-rainbow connection number. We prove that if G and \(\overline{G}\) are connected complementary graphs on n vertices, then \(trc(G)+trc(\overline{G})\le 2n\) when \(n\ge 6\) and \(trc(G)+trc(\overline{G})\le 2n+1\) when \(n=5\). Examples are given to show that the upper bounds are sharp for \(n\ge 5\). This completely solves a conjecture in Ma (Res Math 70(1–2):173–182, 2016).

Tight Nordhaus–Gaddum-Type Upper Bound for Total-Rainbow Connection Number of Graphs