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A total-colored graph is a graph $G$ such that both all edges and allvertices of $G$ are colored. A path in a total-colored graph $G$ is a totalrainbow path if its edges and internal vertices have distinct colors. Atotal-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 connectionnumber of $G$, denoted by $trc(G)$, is defined as the smallest number of colorsthat are needed to make $G$ total-rainbow connected. These concepts wereintroduced by Liu et al. Notice that for a connected graph $G$, $2diam(G)-1\leqtrc(G)\leq 2n-3$, where $diam(G)$ denotes the diameter of $G$ and $n$ is theorder of $G$. In this paper we show, for a connected graph $G$ of order $n$with minimum degree $\delta$, that $trc(G)\leq6n/{(\delta+1)}+28$ for$\delta\geq\sqrt{n-2}-1$ and $n\geq 291$, while $trc(G)\leq7n/{(\delta+1)}+32$for $16\leq\delta\leq\sqrt{n-2}-2$ and$trc(G)\leq7n/{(\delta+1)}+4C(\delta)+12$ for $6\leq\delta\leq15$, where$C(\delta)=e^{\frac{3\log({\delta}^3+2{\delta}^2+3)-3(\log3-1)}{\delta-3}}-2$.This implies that when $\delta$ is in linear with $n$, then the total rainbownumber $trc(G)$ is a constant. We also show that $trc(G)\leq 7n/4-3$ for$\delta=3$, $trc(G)\leq8n/5-13/5$ for $\delta=4$ and $trc(G)\leq3n/2-3$ for$\delta=5$. Furthermore, an example shows that our bound can be seen tight upto additive factors when $\delta\geq\sqrt{n-2}-1$.

Good upper bounds for the total rainbow connection of graphs