## Abstract

A transversal in a hypergraph H is a subset of vertices that has a nonempty intersection with every edge of H. A transversal family F of H is a family of (not necessarily distinct) transversals of H. The effective transversal-ratio of the family F is the ratio of the number of sets in F over the maximum times r_{ F } any element appears in F. The fractional disjoint transversal number FDT(H) is the supremum of the effective transversal-ratio taken over all transversal families. That is, FDT(H)=sup_{F}|F|∕r_{ F }. Using a connection with not-all-equal 3-SAT, we prove that if H is a 3-regular 3-uniform hypergraph, then FDT(H)≥2, which proves a known conjecture. Using probabilistic arguments, we prove that for all k≥3, if H is a k-regular k-uniform hypergraph, then FDT(H)≥1∕(1−([Formula presented])[Formula presented]^{[Formula presented]}), and that this bound is essentially best possible.

Original language | English |
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Journal | Discrete Mathematics |

Volume | 340 |

Issue number | 10 |

Pages (from-to) | 2349-2354 |

ISSN | 0012-365X |

DOIs | |

Publication status | Published - 2017 |

## Keywords

- 3SAT
- Hypergraph
- Transversal