### Abstract

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 H_{c} = (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 H_{c} 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 language | English |
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Pages (from-to) | 737-751 |

Number of pages | 15 |

Journal | Journal of Combinatorial Optimization |

Volume | 25 |

Issue number | 4 |

DOIs | |

Publication status | Published - May 1 2013 |

### Keywords

- 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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## Cite this

*Journal of Combinatorial Optimization*,

*25*(4), 737-751. https://doi.org/10.1007/s10878-012-9559-7