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On the complexity of rainbow coloring problems

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EIBEN Eduard GANIAN Robert LAURI Juho

Druh Článek v odborném periodiku
Časopis / Zdroj Discrete Applied Mathematics
Fakulta / Pracoviště MU

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Citace
Doi http://dx.doi.org/10.1016/j.dam.2016.10.021
Klíčová slova parameterized complexity; rainbow coloring
Popis An edge-colored graph 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 , is the minimum number of colors needed to make 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 -complete even on graphs of diameter , and also when the number of colors is restricted to . 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.
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