Coloring mixed hypertrees

  • Authors:
  • Daniel Král';Jan Kratochvíl;Andrzej Proskurowski;Heinz-Jürgen Voss

  • Affiliations:
  • Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Malostranské nám. 25, 118 00 Prague, Czech Republic3;Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Malostranské nám. 25, 118 00 Prague, Czech Republic3;Department of Computer and Information Science, University of Oregon, Eugene, OR, USA;Technische Universität Dresden, Germany

  • Venue:
  • Discrete Applied Mathematics - Special issue: Efficient algorithms
  • Year:
  • 2006

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Abstract

A mixed hypergraph is a triple (V,C,D) where V is its vertex set and C and D are families of subsets of V, called C-edges and D-edges, respectively. For a proper coloring, we require that each C-edge contains two vertices with the same color and each D-edge contains two vertices with different colors. The feasible set of a mixed hypergraph is the set of all k's for which there exists a proper coloring using exactly k colors. A hypergraph is a hypertree if there exists a tree such that the edges of the hypergraph induce connected subgraphs of the tree. We prove that feasible sets of mixed hypertrees are gap-free, i.e., intervals of integers, and we show that this is not true for precolored mixed hypertrees. The problem to decide whether a mixed hypertree can be colored by k colors is NP-complete in general; we investigate complexity of various restrictions of this problem and we characterize their complexity in most of the cases.