## Abstract

Let τ(script H sign) be the cover number and ν({script H sign) be the matching number of a hypergraph script H sign. Ryser conjectured that every r-partite hypergraph script H sign satisfies the inequality τ(script H sign) ≤ (r-1) ν ({script H sign). This conjecture is open for all r ≥ 4. For intersecting hypergraphs, namely those with ν(script H sign) = 1, Ryser's conjecture reduces to τ(script H sign) ≤ r-1. Even this conjecture is extremely difficult and is open for all r ≥ 6. For infinitely many r there are examples of intersecting r-partite hypergraphs with τ(script H sign) = r-1, demonstrating the tightness of the conjecture for such r. However, all previously known constructions are not optimal as they use far too many edges. How sparse can an intersecting r-partite hypergraph be, given that its cover number is as large as possible, namely τ(script H sign) ≥ r-1? In this paper we solve this question for r ≤ 5, give an almost optimal construction for r = 6, prove that any r-partite intersecting hypergraph with τ(H) ≥ r - 1 must have at least (3- 1/√18)r(1-o(1)) ≈ 2.764r(1-o(1)) edges, and conjecture that there exist constructions with Θ(r) edges.

Original language | English |
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Pages (from-to) | 101-109 |

Number of pages | 9 |

Journal | Graphs and Combinatorics |

Volume | 25 |

Issue number | 1 |

DOIs | |

State | Published - May 2009 |

## Keywords

- Covering
- Hypergraph
- Intersecting
- Matching
- R -partite
- Ryser's conjecture

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics