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- W1993114645 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 $delta$, then $rvc(G)<11n/delta$. In this paper, we show that $rvc(G)leq 3n/(delta+1)+5$ for $deltageq sqrt{n-1}-1$ and $ngeq 290$, while $rvc(G)leq 4n/(delta+1)+5$ for $16leq deltaleq sqrt{n-1}-2$ and $rvc(G)leq 4n/(delta+1)+C(delta)$ for $6leqdeltaleq 15$, where $C(delta)=e^{frac{3log(delta^3+2delta^2+3)-3(log 3-1)} {delta-3}}-2$. 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 $deltageq sqrt{n-1}-1$ and $delta=3,4,5$, our bounds are seen to be tight up to additive factors." @default.
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- W1993114645 date "2013-01-01" @default.
- W1993114645 modified "2023-09-27" @default.
- W1993114645 title "On the rainbow vertex-connection" @default.
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- W1993114645 doi "https://doi.org/10.7151/dmgt.1664" @default.
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