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 $diam(G)$ 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 $δ$, that $trc(G)\leq6n/{(δ+1)}+28$ for $δ\geq\sqrt{n-2}-1$ and $n\geq 291$, while $trc(G)\leq7n/{(δ+1)}+32$ for $16\leqδ\leq\sqrt{n-2}-2$ and $trc(G)\leq7n/{(δ+1)}+4C(δ)+12$ for $6\leqδ\leq15$, where $C(δ)=e^{\frac{3\log(δ^3+2δ^2+3)-3(\log3-1)}{δ-3}}-2$. This implies that when $δ$ is in linear with $n$, then the total rainbow number $trc(G)$ is a constant. We also show that $trc(G)\leq 7n/4-3$ for $δ=3$, $trc(G)\leq8n/5-13/5$ for $δ=4$ and $trc(G)\leq3n/2-3$ for $δ=5$. Furthermore, an example shows that our bound can be seen tight up to additive factors when $δ\geq\sqrt{n-2}-1$.

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