Rainbow connections and the parameter $\sigma_k$

For a graph $G$, we define $\sigma_k(G)=min \{d(u_1)+...+d(u_k)| u_1,...,u_k\in V(G), u_iu_j\not\in E(G),i\neq j, i,j\in\{1,...,n\}\}$, or simply denoted by $\sigma_k$. A edge-colored graph is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors, which was introduced by Chartrand et al. The rainbow connection of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow edge-connected. A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors, which was recently introduced by Krivelevich and Yuster. The 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. We prove that if $G$ is a connected graph of order $n$, then $rc(G)\leq 3k\frac{n-2}{\sigma_k+k}+6k-4$. Moreover, the bound is seen to be tight up to additive factors by a construction mentioned in introduction. And we also prove that if $G$ is a connected graph of order $n$, then $rvc(G)< \frac{7kn}{\sigma_k+k}+5k$ for $5k<\sigma_k< 8k$, and $rvc(G)< \frac{6kn}{\sigma_k+k}+5k $ for $\sigma_k\leq 5k$ and $\sigma_k\geq 8k$. Note that, in some graphs, $\sigma_k$ can be very large and it is monotone increasing on $k$. From the above results, we know that if $\sigma_k \geq n-k$ then the upper bounds of $rc(G)$ and $rvc(G)$ are constants.