## Abstract

The edge covering number of a hypergraph A is β(A)=min|B|:B⊆A, ∪B=∪A. The paper studies a conjecture on the edge covering number of the intersection of two matroids. For two natural numbers k,ℓ, let f(k,ℓ) be the maximal value of β(M∩N) over all pairs of matroids M,N such that β(M)=k and β(N)=ℓ. In (Aharoni and Berger, 2006) [1] the first two authors proved that f(k,ℓ)≤2max(k,ℓ) and conjectured that f(k,k)=k+1 and f(k,ℓ)=ℓ when ℓ>k. In this paper we prove that f(k,k)<k+1, f(2,2)=3 and f(2,3)≤4. We also form a conjecture on the edge covering number of 2-polymatroids that is a common extension of the above conjecture and the GoldbergSeymour conjecture, and prove its first non-trivial case.

Original language | English |
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Pages (from-to) | 81-85 |

Number of pages | 5 |

Journal | Discrete Mathematics |

Volume | 312 |

Issue number | 1 |

DOIs | |

State | Published - 6 Jan 2012 |

### Bibliographical note

Funding Information:The work of the first author was supported by the Discount Bank chair, by GIF grant no. 2011507 and by BSF grant no. 2006099 . The work of the second author was supported by BSF grant no. 2006099 .

## Keywords

- Edge cover
- Hypergraphs
- Matching theory
- Matroid intersection
- Matroid theory
- Polymatroids
- SeymourGoldberg conjecture

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics