Good upper bounds for the total rainbow connection of graphs

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 $G$, $2diam(G)-1\leq trc(G)\leq 2n-3$,where$diam(G)$denotesthediameterof$G$and$n$ is the order of $G$. In this paper we show, for a connected graph $G$ of order $n$ with minimum degree $\delta$, that $trc(G)\le 6n\mathrm{/}(\delta +1)+28$ for $\delta \ge \sqrt{n-2}-1$ and $n\ge 291$, while $trc(G)\le 7n\mathrm{/}(\delta +1)+32$ for $16\le \delta \le \sqrt{n-2}-2$ and $trc(G)\le 7n\mathrm{/}(\delta +1)+4C(\delta )+12$ for $6\le \delta \le 15$, where $C(\delta )=e^{\frac{3\log ({\delta }^3+2{\delta }^2+3)-3(\log 3-1)}{\delta -3}}-2$. This implies that when $\delta$ is in linear with $n$, then the total rainbow number $trc(G)$ is a constant. We also show that $trc(G)\le 7n\mathrm{/}4-3$ for $\delta =3$, $trc(G)\le 8n\mathrm{/}5-13\mathrm{/}5$ for $\delta =4$ and $trc(G)\le 3n\mathrm{/}2-3$ for $\delta =5$. Furthermore, an example shows that our bound can be seen tight up to additive factors when $\delta \ge \sqrt{n-2}-1$.