Good upper bounds for the total rainbow connection of graphs

Abstract

A total-colored graph is a graph $G$ such that both all edges and all vertices of $G$ are colored. A path in a total-colored graph $G$ is a total rainbow path if its edges and internal vertices have distinct colors. A total-colored graph $G$ is total-rainbow connected if any two vertices of $G$ are connected by a total rainbow path of $G$. The total rainbow connection number of $G$, denoted by $trc(G)$, is defined as the smallest number of colors that are needed to make $G$ total-rainbow connected. These concepts were introduced by Liu et al. Notice that for a connected graph $G$, $2diam(G)-1\le trc(G)\le 2n-3$, where $diam(G)$ denotes the diameter of $G$ and $n$ is the order of $G$. In this paper we show, for a connected graph $G$ of order $n$ with minimum degree $\hat{I}´$, that $trc(G)\le 6n/(\hat{I}´+1)+28$ for $\hat{I}´\ge \sqrt{n-2}-1$ and $n\ge 291$, while $trc(G)\le 7n/(\hat{I}´+1)+32$ for $16\le \hat{I}´\le \sqrt{n-2}-2$ and $trc(G)\le 7n/(\hat{I}´+1)+4C(\hat{I}´)+12$ for $6\le \hat{I}´\le 15$, where $C(\hat{I}´)=e^{\frac{3\log (\hat{I}{´}^3+2\hat{I}{´}^2+3)-3(\log 3-1)}{\hat{I}´-3}}-2$. This implies that when $\hat{I}´$ is in linear with $n$, then the total rainbow number $trc(G)$ is a constant. We also show that $trc(G)\le 7n/4-3$ for $\hat{I}´=3$, $trc(G)\le 8n/5-13/5$ for $\hat{I}´=4$ and $trc(G)\le 3n/2-3$ for $\hat{I}´=5$. Furthermore, an example shows that our bound can be seen tight up to additive factors when $\hat{I}´\ge \sqrt{n-2}-1$.

8 pages