On the Rainbow Vertex - Connection 1

A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertexconnected. It was proved that ifG is a graph of order n with minimum degree δ, then rvc(G) < 11n/δ. In this paper, we show that rvc(G) ≤ 3n/(δ+1)+5 for δ ≥ √ n− 1 − 1 and n ≥ 290, while rvc(G) ≤ 4n/(δ + 1) + 5 for 16 ≤ δ ≤ √ n− 1 − 2 and rvc(G) ≤ 4n/(δ + 1) + C(δ) for 6 ≤ δ ≤ 15, where C(δ) = e 3 log(δ3+2δ2+3)−3(log 3−1) δ−3 − 2. We also prove that rvc(G) ≤ 3n/4 − 2 for δ = 3, rvc(G) ≤ 3n/5 − 8/5 for δ = 4 and rvc(G) ≤ n/2 − 2 for δ = 5. Moreover, an example constructed by Caro et al. shows that when δ ≥ √ n− 1−1 and δ = 3, 4, 5, our bounds are seen to be tight up to additive constants.