Note on rainbow connection number of dense graphs

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. Following an idea of Caro et al., in this paper we also investigate the rainbow connection number of dense graphs. We show that for $k\ge 2$, if $G$ is a non-complete graph of order $n$ with minimum degree $\delta (G)\geq \frac{n}{2}-1+log_{k}{n}$,orminimumdegree-sum$\sigma_{2}(G)\geq n-2+2log_{k}{n}$,then$rc(G)\leq k$;if$G$isagraphoforder$n$ with diameter 2 and $\delta (G)\ge 2(1+lo{g}_{\frac{k^2}{3k-2}}k)lo{g}_{k}n$, then $rc(G)\le k$. We also show that if $G$ is a non-complete bipartite graph of order $n$ and any two vertices in the same vertex class have at least $2lo{g}_{\frac{k^2}{3k-2}}klo{g}_{k}n$ common neighbors in the other class, then $rc(G)\le k$.