On the Complexity of Rainbow Coloring Problems

TL;DR

This paper investigates the computational complexity of various rainbow coloring problems in graphs, establishing NP-completeness results and providing efficient algorithms for special graph classes and fixed parameters.

Contribution

It proves NP-completeness of the Strong Rainbow Vertex Coloring problem on diameter 3 graphs and offers linear-time algorithms for fixed color counts on graphs with bounded treewidth and vertex cover number.

Findings

NP-completeness of Strong Rainbow Vertex Coloring on diameter 3 graphs

Linear-time algorithms for fixed color counts on bounded treewidth graphs

Linear-time algorithms for exact rainbow connection numbers on graphs with bounded vertex cover

Abstract

An edge-colored graph $G$ is said to be rainbow connected if between each pair of vertices there exists a path which uses each color at most once. The rainbow connection number, denoted by $rc(G)$, is the minimum number of colors needed to make $G$ rainbow connected. Along with its variants, which consider vertex colorings and/or so-called strong colorings, the rainbow connection number has been studied from both the algorithmic and graph-theoretic points of view. In this paper we present a range of new results on the computational complexity of computing the four major variants of the rainbow connection number. In particular, we prove that the \textsc{Strong Rainbow Vertex Coloring} problem is $NP$-complete even on graphs of diameter $3$. We show that when the number of colors is fixed, then all of the considered problems can be solved in linear time on graphs of bounded treewidth. Moreover, we provide a linear-time algorithm which decides whether it is possible to obtain a rainbow coloring by saving a fixed number of colors from a trivial upper bound. Finally, we give a linear-time algorithm for computing the exact rainbow connection numbers for three variants of the problem on graphs of bounded vertex cover number.