Rainbow Connection Number and Independence Number of a Graph

path in an edge-colored graph is called rainbow if any two edges of the path have distinct colors. An edge-colored graph is called rainbow connected if there exists a rainbow path between every two vertices of the graph. For a connected graph G , the minimum number of colors that are needed to make G rainbow connected is called the rainbow connection number of G , denoted by rc( G ). In this paper, we investigate the relation between the rainbow connection number and the independence number of a graph. We show that if G is a connected graph without pendant vertices, then rc(G)≤ 2α (G)-1 . An example is given showing that the upper bound 2α (G)-1 is equal to the diameter of G , and so the upper bound is sharp since the diameter of G is a lower bound of rc(G) .