On Rainbow Connection of Strongly Regular Graphs

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 $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. We prove if $G$ is a connected strongly $r$-regular graph and $r\ge 600$, then $rc(G)\le 3$. Specially, there is a constant $c$ such that $rc(G)\le c$ for any connected strongly regular graph $G$.