Tight Nordhaus–Gaddum-Type Upper Bound for Total-Rainbow 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 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).