Rainbow connection of graphs with diameter 2

A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the minimum integer $i$ for which there exists an $i$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. It is known that for a graph $G$ with diameter 2, to determine $rc(G)$ is NP-hard. So, it is interesting to know the best upper bound of $rc(G)$ for such a graph $G$. In this paper, we show that $rc(G)\leq 5$ if $G$ is a bridgeless graph with diameter 2, and that $rc(G)\leq k+2$ if $G$ is a connected graph of diameter 2 with $k$ bridges, where $k\geq 1$.
Comment: 10 pages