### Abstract

The concept of color-bounded hypergraph is introduced here. It is a hypergraph (set system) with vertex set X and edge set E = {E_{1}, ..., E_{m}}, where each edge E_{i} is associated with two integers s_{i} and t_{i} such that 1 ≤ s_{i} ≤ t_{i} ≤ | E_{i} |. A vertex coloring φ : X → N is considered to be feasible if the number of colors occurring in E_{i} satisfies s_{i} ≤ | φ (E_{i}) | ≤ t_{i}, for all i ≤ m. Color-bounded hypergraphs generalize the concept of 'mixed hypergraphs' introduced by Voloshin [V. Voloshin, The mixed hypergraphs, Computer Science Journal of Moldova 1 (1993) 45-52], and a recent model studied by Drgas-Burchardt and Łazuka [E. Drgas-Burchardt, E. Łazuka, On chromatic polynomials of hypergraphs, Applied Mathematics Letters 20 (12) (2007) 1250-1254] where only lower bounds s_{i} were considered. We discuss the similarities and differences between our general model and the more particular earlier ones. An important issue is the chromatic spectrum-strongly related to the chromatic polynomial-which is the sequence whose kth element is the number of allowed colorings with precisely k colors (disregarding color permutations). Problems concerning algorithmic complexity are also considered.

Original language | English |
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Pages (from-to) | 4890-4902 |

Number of pages | 13 |

Journal | Discrete Mathematics |

Volume | 309 |

Issue number | 15 |

DOIs | |

Publication status | Published - Aug 6 2009 |

### Keywords

- Chromatic polynomial
- Feasible set
- Hypergraph
- Mixed hypergraph
- Uniquely colorable hypergraph
- Vertex coloring

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*309*(15), 4890-4902. https://doi.org/10.1016/j.disc.2008.04.019