Upper bound involving parameter σ2 for the rainbow connection number

Let G be a connected graph of order n. The rainbow connection number rc(G) of G was introduced by Chartrand et al. Chandran et al. used the minimum degree δ of G and obtained an upper bound that rc(G) ≤ 3n/(δ +1)+3, which is tight up to additive factors. In this paper, we use the minimum degree-sum σ2 of G to obtain a better bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$rc(G) \leqslant \tfrac{{6n}} {{\sigma _2 + 2}} + 8$$\end{document}, especially when δ is small (constant) but σ2 is large (linear in n).