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


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.


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

Mathematical Subject Classification

05C15 05C65 05C85 



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.


  1. Arocha JL, Bracho J, Neumann-Lara V (1992) On the minimum size of tight hypergraphs. J Graph Theory 16:319–326zbMATHMathSciNetCrossRefGoogle Scholar
  2. Bacsó G, Tuza Zs (2008) Upper chromatic number of finite projective planes. J Comb Des 16(3):221–230zbMATHCrossRefGoogle Scholar
  3. Berge C (1989) Hypergraphs. North-Holland, AmsterdamzbMATHGoogle Scholar
  4. Bujtás Cs, Tuza Zs (2011) Maximum number of colors: C-coloring and related problems. J Geom 101:83–97Google Scholar
  5. Bujtás Cs, Tuza Zs (2013) Approximability of the upper chromatic number of hypergraphs, manuscriptGoogle Scholar
  6. Král’ D (2004) On feasible sets of mixed hypergraphs. Electron J Comb 11 , #R19, 14 ppGoogle Scholar
  7. Král’ D, Kratochvíl J, Voss H-J (2003) Mixed hypergraphs with bounded degree: edge-coloring of mixed multigraphs. Theoret Comput Sci 295:263–278zbMATHMathSciNetCrossRefGoogle Scholar
  8. Sperner E (1928) Ein Satz über Untermengen einer endlichen Menge. Math Z 27(1):544–548zbMATHMathSciNetCrossRefGoogle Scholar
  9. Sterboul F (1973) A new combinatorial parameter. In: Hajnal A, et al (eds) Infinite and Finite Sets, Colloq. Math. Soc. J. Bolyai 10, Vol. III, Keszthely. (North-Holland/American Elsevier, 1975), pp 1387–1404Google Scholar
  10. Voloshin VI (1993) The mixed hypergraphs. Comput Sci J Moldova 1:45–52MathSciNetGoogle Scholar
  11. Voloshin VI (2002) Coloring mixed hypergraphs: theory, algorithms and applications, vol 17. Fields Institute Monographs, American Mathematics Society, USAGoogle Scholar

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

Personalised recommendations