Rainbow connection number of dense graphs

Abstract: 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 $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this paper we show that $rc(G)\leq 3$, if $|E(G)|\geq {{n-2}\choose 2}+2$, and $rc(G)\leq 4$, if $|E(G)|\geq {{n-3}\choose 2}+3$. These bounds are sharp.