  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 by $rc(G)$, is the smallest number of
colors that are needed in order to make $G$ rainbow connected. In this paper,
we proved that $rc(G)\leq 3(n+1)/5$ for all $3$-connected graphs.
