### Abstract

A stably bounded hypergraphHis a hypergraph together with four color-bound functions s, t, a and b, each assigning positive integers to the edges. A vertex coloring ofHis considered proper if each edge E has at least s(E) and at most t(E) different colors assigned to its vertices, moreover each color occurs on at most b(E) vertices of E, and there exists a color which is repeated at least a(E) times inside E. The lower and the upper chromatic number of H is the minimum and the maximum possible number of colors, respectively, over all proper colorings. An interval hypergraph is a hypergraph whose vertex set allows a linear ordering such that each edge is a set of consecutive vertices in this order. We study the time complexity of testing colorability and determining the lower and upper chromatic numbers. A complete solution is presented for interval hypergraphs without overlapping edges. Complexity depends both on problem type and on the combination of color-bound functions applied, except that all the three coloring problems are NP-hard for the function pair a, b and its extensions. For the tractable classes, lineartime algorithms are designed. It also depends on problem type and function set whether complexity jumps from polynomial to NP-hard if the instance is allowed to contain overlapping intervals. Comparison is facilitated with three handy tables which also include further structure classes.

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

Pages (from-to) | 1965-1977 |

Number of pages | 13 |

Journal | Discrete Mathematics |

Volume | 313 |

Issue number | 19 |

DOIs | |

Publication status | Published - 2013 |

### Fingerprint

### Keywords

- Algorithmic complexity
- Chromatic number
- Color-bounded hypergraph
- Feasible set
- Hypergraph coloring
- Interval hypergraph
- Mixed hypergraph
- Stably bounded hypergraph
- Upper chromatic number

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*313*(19), 1965-1977. https://doi.org/10.1016/j.disc.2012.09.020

**Color-bounded hypergraphs, VI : Structural and functional jumps in complexity.** / Bujtás, Csilla; Tuza, Z.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 313, no. 19, pp. 1965-1977. https://doi.org/10.1016/j.disc.2012.09.020

}

TY - JOUR

T1 - Color-bounded hypergraphs, VI

T2 - Structural and functional jumps in complexity

AU - Bujtás, Csilla

AU - Tuza, Z.

PY - 2013

Y1 - 2013

N2 - A stably bounded hypergraphHis a hypergraph together with four color-bound functions s, t, a and b, each assigning positive integers to the edges. A vertex coloring ofHis considered proper if each edge E has at least s(E) and at most t(E) different colors assigned to its vertices, moreover each color occurs on at most b(E) vertices of E, and there exists a color which is repeated at least a(E) times inside E. The lower and the upper chromatic number of H is the minimum and the maximum possible number of colors, respectively, over all proper colorings. An interval hypergraph is a hypergraph whose vertex set allows a linear ordering such that each edge is a set of consecutive vertices in this order. We study the time complexity of testing colorability and determining the lower and upper chromatic numbers. A complete solution is presented for interval hypergraphs without overlapping edges. Complexity depends both on problem type and on the combination of color-bound functions applied, except that all the three coloring problems are NP-hard for the function pair a, b and its extensions. For the tractable classes, lineartime algorithms are designed. It also depends on problem type and function set whether complexity jumps from polynomial to NP-hard if the instance is allowed to contain overlapping intervals. Comparison is facilitated with three handy tables which also include further structure classes.

AB - A stably bounded hypergraphHis a hypergraph together with four color-bound functions s, t, a and b, each assigning positive integers to the edges. A vertex coloring ofHis considered proper if each edge E has at least s(E) and at most t(E) different colors assigned to its vertices, moreover each color occurs on at most b(E) vertices of E, and there exists a color which is repeated at least a(E) times inside E. The lower and the upper chromatic number of H is the minimum and the maximum possible number of colors, respectively, over all proper colorings. An interval hypergraph is a hypergraph whose vertex set allows a linear ordering such that each edge is a set of consecutive vertices in this order. We study the time complexity of testing colorability and determining the lower and upper chromatic numbers. A complete solution is presented for interval hypergraphs without overlapping edges. Complexity depends both on problem type and on the combination of color-bound functions applied, except that all the three coloring problems are NP-hard for the function pair a, b and its extensions. For the tractable classes, lineartime algorithms are designed. It also depends on problem type and function set whether complexity jumps from polynomial to NP-hard if the instance is allowed to contain overlapping intervals. Comparison is facilitated with three handy tables which also include further structure classes.

KW - Algorithmic complexity

KW - Chromatic number

KW - Color-bounded hypergraph

KW - Feasible set

KW - Hypergraph coloring

KW - Interval hypergraph

KW - Mixed hypergraph

KW - Stably bounded hypergraph

KW - Upper chromatic number

UR - http://www.scopus.com/inward/record.url?scp=84884818022&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884818022&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2012.09.020

DO - 10.1016/j.disc.2012.09.020

M3 - Article

VL - 313

SP - 1965

EP - 1977

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 19

ER -