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Let $G$ be a connected graph. The \emph{rainbow connection number $rc(G)$} ofa graph $G$ was recently introduced by Chartrand et al. Li et al. proved thatfor every bridgeless graph $G$ with diameter 2, $rc(G)\leq 5$. They gaveexamples for which $rc(G)\leq 4$. However, they could not show that the upperbound 5 is sharp. It is known that for a graph $G$ with diameter 2, todetermine $rc(G)$ is NP-hard. So, it is interesting to know the best upperbound of $rc(G)$ for such a graph $G$. In this paper, we use different way toobtain the same upper bound, and moreover, examples are given to show that theupper is best possible.

Sharp upper bound for the rainbow connection number of a graph with diameter 2