Colorability of mixed hypergraphs and their chromatic inversions

Máté Hegyháti, Zsolt Tuza

Research output: Contribution to journalArticle

3 Citations (Scopus)


We solve a long-standing open problem concerning a discrete mathematical model, which has various applications in computer science and several other fields, including frequency assignment and many other problems on resource allocation. A mixed hypergraph H is a triple (X,C,D) where X is the set of vertices, and C and D are two set systems over X, the families of so-called C-edges and D-edges, respectively. A vertex coloring of a mixed hypergraph H is proper if every C-edge has two vertices with a common color and every D-edge has two vertices with different colors. A mixed hypergraph is colorable if it has at least one proper coloring; otherwise it is uncolorable. The chromatic inversion of a mixed hypergraph H = (X,C,D) is defined as Hc = (X,C,D). Since 1995, it was an open problem wether there is a correlation between the colorability properties of a hypergraph and its chromatic inversion. In this paper we answer this question in the negative, proving that there exists no polynomial-time algorithm (provided that P ≠ N P to decide whether both H and Hc are colorable, or both are uncolorable. This theorem holds already for the restricted class of 3-uniform mixed hypergraphs (i.e., where every edge has exactly three vertices). The proof is based on a new polynomial-time algorithm for coloring a special subclass of 3-uniform mixed hypergraphs. Implementation in C++ programming language has been tested. Further related decision problems are investigated, too.

Original languageEnglish
Pages (from-to)737-751
Number of pages15
JournalJournal of Combinatorial Optimization
Issue number4
Publication statusPublished - May 1 2013


  • Algorithmic complexity
  • Chromatic inversion
  • Hypergraph coloring
  • Mixed hypergraph

ASJC Scopus subject areas

  • Computer Science Applications
  • Discrete Mathematics and Combinatorics
  • Control and Optimization
  • Computational Theory and Mathematics
  • Applied Mathematics

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