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 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) ≤ 3 if |E(G)| ≥ ( n−2 2 ) + 2, and rc(G) ≤ 4 if |E(G)| ≥ ( n−3 2 ) + 3. These bounds are sharp.