On the rainbow vertex-connection

Abstract

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 $δ$, then $rvc(G)<11n/δ$. In this paper, we show that $rvc(G)\leq 3n/(δ+1)+5$ for $δ\geq \sqrt{n-1}-1$ and $n\geq 290$, while $rvc(G)\leq 4n/(δ+1)+5$ for $16\leq δ\leq \sqrt{n-1}-2$ and $rvc(G)\leq 4n/(δ+1)+C(δ)$ for $6\leqδ\leq 15$, where $C(δ)=e^{\frac{3\log(δ^3+2δ^2+3)-3(\log 3-1)} {δ-3}}-2$. We also prove that $rvc(G)\leq 3n/4-2$ for $δ=3$, $rvc(G)\leq 3n/5-8/5$ for $δ=4$ and $rvc(G)\leq n/2-2$ for $δ=5$. Moreover, an example shows that when $δ\geq \sqrt{n-1}-1$ and $δ=3,4,5$, our bounds are seen to be tight up to additive factors.