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On the complexity of rainbow coloring problems

Research output: Contribution to journalArticleScientificpeer-review


Original languageEnglish
Pages (from-to)38-48
JournalDiscrete Applied Mathematics
Early online date2016
Publication statusPublished - Sep 2018
Publication typeA1 Journal article-refereed


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 Strong Rainbow Vertex Coloring problem is View the MathML source-complete even on graphs of diameter 3, and also when the number of colors is restricted to 2. On the other hand, we show that if 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.

Publication forum classification

Field of science, Statistics Finland