Graphs with small total rainbow connection number

Abstract

A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we investigate graphs with small total rainbow connection number. First, for a connected graph G, we prove that $${\text{trc(G) = 3 if}}\left( {\begin{array}{*{20}{c}} {n - 1} \\ 2 \end{array}} \right) + 1 \leqslant \left| {{\text{E(G)}}} \right| \leqslant \left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right) - 1$$ , and $${\text{trc(G)}} \leqslant {\text{6 if }}\left| {{\text{E(G)}}} \right| \geqslant \left( {\begin{array}{*{20}{c}} {n - 2} \\ 2 \end{array}} \right) + 2$$ . Next, we investigate the total rainbow connection numbers of graphs G with |V(G)| = n, diam(G) ≥ 2, and clique number ω(G) = n − s for 1 ≤ s ≤ 3. In this paper, we find Theorem 3 of [Discuss. Math. Graph Theory, 2011, 31(2): 313–320] is not completely correct, and we provide a complete result for this theorem.