On a question on graphs with rainbow connection number 2

TL;DR

This paper proves that any connected graph with rainbow connection number 2 can be efficiently rainbow-colored using at most 5 colors, answering a previously open question about coloring complexity.

Contribution

It establishes that graphs with rainbow connection number 2 can be rainbow-colored in polynomial time with a constant number of colors, specifically at most 5.

Findings

Polynomial-time rainbow coloring for graphs with rc(G)=2

Use of at most 5 colors for rainbow coloring

Addresses an open problem in graph coloring complexity

Abstract

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.