### Abstract

The upper chromatic number (Formula presented.) of a hypergraph (Formula presented.) is the maximum number of colors that can occur in a vertex coloring φ:X→N such that no edge E∈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 χ¯, unless P=NP. In sharp contrast to this, here we prove that if the input is restricted to hypertrees H of bounded maximum vertex degree, then χ¯(H) can be determined in linear time if an underlying tree is also given in the input. Consequently, χ¯ on hypertrees is fixed parameter tractable in terms of maximum degree.

Original language | English |
---|---|

Pages (from-to) | 867-876 |

Number of pages | 10 |

Journal | Central European Journal of Operations Research |

Volume | 23 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 1 2015 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Management Science and Operations Research

### Cite this

*Central European Journal of Operations Research*,

*23*(4), 867-876. https://doi.org/10.1007/s10100-014-0357-4

**Maximum number of colors in hypertrees of bounded degree.** / Bujtás, Csilla; Tuza, Z.

Research output: Contribution to journal › Article

*Central European Journal of Operations Research*, vol. 23, no. 4, pp. 867-876. https://doi.org/10.1007/s10100-014-0357-4

}

TY - JOUR

T1 - Maximum number of colors in hypertrees of bounded degree

AU - Bujtás, Csilla

AU - Tuza, Z.

PY - 2015/12/1

Y1 - 2015/12/1

N2 - The upper chromatic number (Formula presented.) of a hypergraph (Formula presented.) is the maximum number of colors that can occur in a vertex coloring φ:X→N such that no edge E∈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 χ¯, unless P=NP. In sharp contrast to this, here we prove that if the input is restricted to hypertrees H of bounded maximum vertex degree, then χ¯(H) can be determined in linear time if an underlying tree is also given in the input. Consequently, χ¯ on hypertrees is fixed parameter tractable in terms of maximum degree.

AB - The upper chromatic number (Formula presented.) of a hypergraph (Formula presented.) is the maximum number of colors that can occur in a vertex coloring φ:X→N such that no edge E∈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 χ¯, unless P=NP. In sharp contrast to this, here we prove that if the input is restricted to hypertrees H of bounded maximum vertex degree, then χ¯(H) can be determined in linear time if an underlying tree is also given in the input. Consequently, χ¯ on hypertrees is fixed parameter tractable in terms of maximum degree.

KW - Arboreal hypergraph

KW - C-coloring

KW - Hypergraph

KW - Hypertree

KW - Upper chromatic number

KW - Vertex coloring

UR - http://www.scopus.com/inward/record.url?scp=84957957545&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957957545&partnerID=8YFLogxK

U2 - 10.1007/s10100-014-0357-4

DO - 10.1007/s10100-014-0357-4

M3 - Article

VL - 23

SP - 867

EP - 876

JO - Central European Journal of Operations Research

JF - Central European Journal of Operations Research

SN - 1435-246X

IS - 4

ER -