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∈ V(G), uv∉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)≤ 6n-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)≤ 8n-2/σ_2+2+10 for 2≤σ_2≤ 6, σ_2≥ 28, while for 7 ≤σ_2≤ 8, 16≤σ_2≤ 27, rvc(G)≤10n-16/σ_2+2+10, and for 9 ≤σ_2≤ 15, rvc(G)≤10n-16/σ_2+2+A(σ_2) where A(σ_2)= 63,41,27,20,16,13,11, respectively.