On the rainbow vertex-connection

A vertex-colored graph is {\it rainbow vertex-connected} if any two vertices are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection} of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. Krivelevich and Yuster proved that if $G$ is a graph of order $n$ with minimum degree $\delta$, then $rvc(G)<11n/\delta$. In this paper, we show that $rvc(G)\leq 3n/(\delta+1)+5$ for $\delta\geq \sqrt{n-1}-1$ and $n\geq 290$, while $rvc(G)\leq 4n/(\delta+1)+5$ for $16\leq \delta\leq \sqrt{n-1}-2$ and $rvc(G)\leq 4n/(\delta+1)+C(\delta)$ for $6\leq\delta\leq 15$, where $C(\delta)=e^{\frac{3\log(\delta^3+2\delta^2+3)-3(\log 3-1)} {\delta-3}}-2$. We also prove that $rvc(G)\leq 3n/4-2$ for $\delta=3$, $rvc(G)\leq 3n/5-8/5$ for $\delta=4$ and $rvc(G)\leq n/2-2$ for $\delta=5$. Moreover, an example shows that when $\delta\geq \sqrt{n-1}-1$ and $\delta=3,4,5$, our bounds are seen to be tight up to additive factors.