The edge covering number of the intersection of two matroids

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 Goldberg–Seymour conjecture, and prove its first non-trivial case.