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.
[1]
Raphael Yuster,et al.
Hardness and algorithms for rainbow connection
,
2008,
J. Comb. Optim..
[2]
Jiuying Dong,et al.
Sharp upper bound for the rainbow connection number of a graph with diameter 2
,
2011
.
[3]
David R. Karger,et al.
An Õ(n^{3/14})-Coloring Algorithm for 3-Colorable Graphs
,
1997,
Information Processing Letters.
[4]
J. A. Bondy,et al.
Graph Theory
,
2008,
Graduate Texts in Mathematics.
[5]
Xueliang Li,et al.
Rainbow connection of graphs with diameter 2
,
2012,
Discret. Math..
[6]
Garry L. Johns,et al.
Rainbow connection in graphs
,
2008
.