### Abstract

Let ℋ = (X, ℰ) be a hypergraph with vertex set X and edge set ℰ. A C-coloring of ℋ is a mapping φ:X→ ℕ such that |φ(E)|<|E| holds for all edges E∈Escr; (i.e. no edge is multicolored). We denote by x(ℋ)the maximum number |φ(X)| of colors in a C-coloring. Let further α(ℋ) denote the largest cardinality of a vertex set S⊆ Xthat contains no E∈ ℰ,and τ(ℋ) = |X|-α(ℋ) the minimum cardinality of a vertex set meeting all E∈ℰ. The hypergraph ℋ is called C-perfect if x(ℋ') = α(ℋ') holds for every induced subhypergraph ℋ ⊆ℋ.lf ℋ is not C-perfect but all of its proper induced subhypergraphs are, then we say that it is minimally C-imperfect. We prove that for all r, k∈ ℕ there exists a finite upper bound h(r, k)onthe number of minimally C-imperfect hypergraphs ℋ with τ(ℋ) ≤ k and without edges of more than r vertices. We give a characterization of minimally C-imperfect hypergraphs that have τ = 2, which also characterizes implicitly the C-perfect ones with τ = 2. From this result we derive an infinite family of new constructions that are minimally C-imperfect. A characterization of minimally C-imperfect circular hypergraphs is presented, too.

Original language | English |
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Pages (from-to) | 132-149 |

Number of pages | 18 |

Journal | Journal of Graph Theory |

Volume | 64 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 2010 |

### Keywords

- C-perfect hypergraph
- Circular hypergraph
- Mixed hypergraph
- Transversal number
- Upper chromatic number
- Vertex coloring

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

*Journal of Graph Theory*,

*64*(2), 132-149. https://doi.org/10.1002/jgt.20444