On a question on graphs with rainbow connection number 2

For a connected graph $G$, the \emph{rainbow connection number $rc(G)$} of a graph $G$ was introduced by Chartrand et al. In "Chakraborty et al., Hardness and algorithms for rainbow connection, J. Combin. Optim. 21(2011), 330--347", Chakraborty et al. proved that for a graph $G$ with diameter 2, to determine $rc(G)$ is NP-Complete, and they left 4 open questions at the end, the last one of which is the following: Suppose that we are given a graph $G$ for which we are told that $rc(G)=2$. Can we rainbow-color it in polynomial time with $o(n)$ colors ? In this paper, we settle down this question by showing a stronger result that for any graph $G$ with $rc(G)=2$, we can rainbow-color $G$ in polynomial time by at most 5 colors.

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