### Abstract

For a mixed hypergraph H = (X, C, D) , where C and D are set systems over the vertex set X, a coloring is a partition of X into 'color classes' such that every C ∈ C meets some class in more than one vertex, and every D ∈ D has a nonempty intersection with at least two classes. The feasible set of H, denoted Φ(H), is the set of integers k such that H admits a coloring with precisely k nonempty color classes. It was proved by Jiang et al. [Graphs and Combinatorics 18 (2002), 309-318] that a set S of natural numbers is the feasible set of some mixed hypergraph if and only if either 1 ∉ S or S is an 'interval' {1, ..., k } for some integer k ≥ 1. In this note we consider r-uniform mixed hypergraphs, i.e. those with |C| = |D| = r for all C ∈ C and all D ∈ D, r ≥ 3. We prove that S is the feasible set of some r-uniform mixed hypergraph with at least one edge if and only if either S=1,..., k } for some natural number k ≥ r - 1, or S is of the form S = S′ ∪ S where S'' is any (possibly empty) subset of {r, r+1,...} and S′ is either the empty set or {r - 1} or an 'interval' {k, k + 1, ..., r - 1} for some k, 2 ≤ k ≤ r - 2. We also prove that all these feasible sets S ∌ 1 can be obtained under the restriction C = D , i.e. within the class of 'bi-hypergraphs'.

Original language | English |
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Pages (from-to) | 1-12 |

Number of pages | 12 |

Journal | Graphs and Combinatorics |

Volume | 24 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 1 2008 |

### Keywords

- Feasible set
- Mixed hypergraph
- Vertex coloring

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Graphs and Combinatorics*,

*24*(1), 1-12. https://doi.org/10.1007/s00373-007-0765-5