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

Let $G$ be a connected graph. The \emph{rainbow connection number $rc(G)$} of a graph $G$ was recently introduced by Chartrand et al. Li et al. proved that for every bridgeless graph $G$ with diameter 2, $rc(G)\leq 5$. They gave examples for which $rc(G)\leq 4$. However, they could not show that the upper bound 5 is sharp. 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 use different way to obtain the same upper bound, and moreover, examples are given to show that the upper is best possible.

No reviews yet.

No comments yet.