Note on the rainbow connection numbers of graphs with diameter 2
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Let G be a connected graph. The 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) ≤ 5. They gave examples for which rc(G) ≤ 4. However, they could not show that the upper bound 5 is sharp. 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.
[1] Xueliang Li,et al. Rainbow connection of graphs with diameter 2 , 2012, Discret. Math..
[2] Garry L. Johns,et al. Rainbow connection in graphs , 2008 .