eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-08-27
83:1
83:13
10.4230/LIPIcs.MFCS.2018.83
article
Rainbow Vertex Coloring Bipartite Graphs and Chordal Graphs
Heggernes, Pinar
1
Issac, Davis
2
Lauri, Juho
3
Lima, Paloma T.
1
van Leeuwen, Erik Jan
4
Department of Informatics, University of Bergen, Norway
Max Planck Institute for Informatics, Saarland Informatics Campus, Germany
Nokia Bell Labs, Dublin, Ireland
Department of Information and Computing Sciences, Utrecht University, The Netherlands
Given a graph with colors on its vertices, a path is called a rainbow vertex path if all its internal vertices have distinct colors. We say that the graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. We study the problem of deciding whether the vertices of a given graph can be colored with at most k colors so that the graph becomes rainbow vertex-connected. Although edge-colorings have been studied extensively under similar constraints, there are significantly fewer results on the vertex variant that we consider. In particular, its complexity on structured graph classes was explicitly posed as an open question.
We show that the problem remains NP-complete even on bipartite apex graphs and on split graphs. The former can be seen as a first step in the direction of studying the complexity of rainbow coloring on sparse graphs, an open problem which has attracted attention but limited progress. We also give hardness of approximation results for both bipartite and split graphs. To complement the negative results, we show that bipartite permutation graphs, interval graphs, and block graphs can be rainbow vertex-connected optimally in polynomial time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol117-mfcs2018/LIPIcs.MFCS.2018.83/LIPIcs.MFCS.2018.83.pdf
Rainbow coloring
graph classes
polynomial-time algorithms
approximation algorithms