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Central European Journal of Operations Research

, Volume 23, Issue 4, pp 867–876 | Cite as

Maximum number of colors in hypertrees of bounded degree

  • Csilla Bujtás
  • Zsolt Tuza
Original Paper

Abstract

The upper chromatic number \(\overline{\chi }(\mathcal{H})\) of a hypergraph \(\mathcal{H}=(X,\mathcal{E})\) is the maximum number of colors that can occur in a vertex coloring \(\varphi :X\rightarrow \mathbb {N}\) such that no edge \(E\in \mathcal{E}\) is completely multicolored. A hypertree (also called arboreal hypergraph) is a hypergraph whose edges induce subtrees on a fixed tree graph. It has been shown that on hypertrees it is algorithmically hard not only to determine exactly but also to approximate the value of \(\overline{\chi }\), unless \(\mathsf{P}=\mathsf{NP}\). In sharp contrast to this, here we prove that if the input is restricted to hypertrees \(\mathcal{H}\) of bounded maximum vertex degree, then \(\overline{\chi }(\mathcal{H})\) can be determined in linear time if an underlying tree is also given in the input. Consequently, \(\overline{\chi }\) on hypertrees is fixed parameter tractable in terms of maximum degree.

Keywords

Hypergraph Hypertree Arboreal hypergraph Vertex coloring C-coloring Upper chromatic number 

Mathematical Subject Classification

05C15 05C65 05C85 

Notes

Acknowledgments

Research supported in part by the Hungarian Scientific Research Fund, OTKA Grant T-81493, moreover by the European Union and Hungary, co-financed by the European Social Fund through the project TÁMOP-4.2.2.C-11/1/KONV-2012-0004—National Research Center for Development and Market Introduction of Advanced Information and Communication Technologies.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Computer Science and Systems TechnologyUniversity of PannoniaVeszprémHungary
  2. 2.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary

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