Upper bounds involving parameter $σ_2$ for the rainbow connection

For a graph $G$, we define $σ_2(G)=min \{d(u)+d(v)| u,v\in V(G), uv\not\in E(G)\}$, or simply denoted by $σ_2$. 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. We prove that if $G$ is a connected graph of order $n$, then $rc(G)\leq 6\frac{n-2}{σ_2+2}+7$. Moreover, the bound is seen to be tight up to additive factors by a construction mentioned by Caro et al. 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 $rvc(G)\leq 8\frac{n-2}{σ_2+2}+10$ for $2\leq σ_2\leq 6, σ_2\geq 28$, while for $7 \leq σ_2\leq 8, 16\leq σ_2\leq 27$, $rvc(G)\leq \frac{10n-16}{σ_2+2}+10$, and for $9 \leq σ_2\leq 15, rvc(G)\leq \frac{10n-16}{σ_2+2}+A(σ_2)$ where $A(σ_2)= 63,41,27,20,16,13,11,$ respectively.

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