Rainbow vertex connection number of dense and sparse graphs

A vertex-colored graph G is rainbow connected, if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex connection number of a connected graph G, denoted rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex connected. In this paper, we show that rvc(G) <= k, if vertical bar E(G)vertical bar >= ((n-k)(2)) + k, for k = 2,3,n - 4,n - 5,n - 6. These bounds are sharp.