Two rainbow connection numbers and the parameter $σ_k(G)$

Abstract: The rainbow connection number $rc(G)$ and the rainbow vertex-connection number $rvc(G)$ of a graph $G$ were introduced by Chartrand et al. and Krivelevich and Yuster, respectively. Good upper bounds in terms of minimum degree $\delta$ were reported by Chandran et al., Krivelevich and Yuster, and Li and Shi. However, if a graph has a small minimum degree $\delta$ and a large number of vertices $n$, these upper bounds are very large, linear in $n$. Hence, one may think to look for a good parameter to replace $\delta$ and decrease the upper bounds significantly. Such a natural parameter is $\sigma_k$. In this paper, for the rainbow connection number we prove that if $G$ is a connected graph of order $n$ with $k$ independent vertices, then $rc(G)\leq 3k\frac{n-2}{\sigma_k+k}+6k-4$. For the rainbow vertex-connection number, we prove that $rvc(G)\leq \frac{(4k+2k^{2})n}{\sigma_k+k}+5k $ if $\sigma_k\leq 7k$ and $\sigma_k\geq 8k$, and $rvc(G)\leq \frac{(\frac{38k}{9}+2k^{2})n}{\sigma_k+k}+5k$ if $7k<\sigma_k< 8k$. Examples are given showing that our bounds are much better than the existing ones, i.e., for the examples $\delta$ is very small but $\sigma_k$ is very large, and the bounds are $rc(G)< 9k-3$ and $rvc(G)\leq 9k+2k^{2}$ or $rvc(G)\leq\frac{83k}{9}+2k^{2}$, which imply that both $rc(G)$ and $rvc(G)$ can be upper bounded by constants from our upper bounds, but linear in $n$ from the existing ones.