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)  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.
Bibliographical noteFunding 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 .
- Edge cover
- Matching theory
- Matroid intersection
- Matroid theory
- SeymourGoldberg conjecture
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics