### 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 |
---|---|

Pages (from-to) | 737-751 |

Number of pages | 15 |

Journal | Journal of Combinatorial Optimization |

Volume | 25 |

Issue number | 4 |

DOIs | |

Publication status | Published - May 2013 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Journal of Combinatorial Optimization*,

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

**Colorability of mixed hypergraphs and their chromatic inversions.** / Hegyháti, Máté; Tuza, Z.

Research output: Contribution to journal › Article

*Journal of Combinatorial Optimization*, vol. 25, no. 4, pp. 737-751. https://doi.org/10.1007/s10878-012-9559-7

}

TY - JOUR

T1 - Colorability of mixed hypergraphs and their chromatic inversions

AU - Hegyháti, Máté

AU - Tuza, Z.

PY - 2013/5

Y1 - 2013/5

N2 - 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.

AB - 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.

KW - Algorithmic complexity

KW - Chromatic inversion

KW - Hypergraph coloring

KW - Mixed hypergraph

UR - http://www.scopus.com/inward/record.url?scp=84877814014&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84877814014&partnerID=8YFLogxK

U2 - 10.1007/s10878-012-9559-7

DO - 10.1007/s10878-012-9559-7

M3 - Article

AN - SCOPUS:84877814014

VL - 25

SP - 737

EP - 751

JO - Journal of Combinatorial Optimization

JF - Journal of Combinatorial Optimization

SN - 1382-6905

IS - 4

ER -