  Let $G$ be an edge-colored connected graph. A path of $G$ is called rainbow
if its every edge is colored by a distinct color. $G$ is called rainbow
connected if there exists a rainbow path between every two vertices of $G$. 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, then
$rc(G)\leq 2\alpha(G)-1$. Two examples $G$ are given to show that the upper
bound $2\alpha(G)-1$ is equal to the diameter of $G$, and therefore the best
possible since the diameter is a lower bound of $rc(G)$.
