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 G are connected complementary graphs on n vertices, then trc(G) + trc(G) ≤ 2n when n ≥ 6 and trc(G) + trc (

An Updated Survey on Rainbow Connections of Graphs- A Dynamic Survey

“…This was solved recently by Li, Li, Magnant and Zhang in [88]. [143] Suppose G is a connected graph with n ′ inner vertices, and assume that there is a set of t vertex-disjoint cycles that cover all but s vertices of G. Then trc(G) ≤ n ′ + s + 2t − 1; moreover, the bound is sharp.…”