Ore-type and Dirac-type theorems for matroids

Let M be a connected binary matroid having no F 7 ∗ -minor. Let A ∗ be a collection of cocircuits of M. We prove there is a circuit intersecting all cocircuits of A ∗ if either one of two things hold:(i) y two disjoint cocircuits A 1 ∗ and A 2 ∗ in A ∗ it holds that r ∗ ( A 1 ∗ ) + r ∗ ( A 2 ∗ ) > r ∗ ( A 1 ∗ ∪ A 2 ∗ ) . y two disjoint cocircuits A 1 ∗ and A 2 ∗ in A ∗ it holds that r ( A 1 ∗ ) + r ( A 2 ∗ ) > r ( M ) . (ii) implies Oreʹs Theorem, a well-known theorem giving sufficient conditions for the existence of a hamilton cycle in a graph. As an application of part (i), it is shown that if M is a k-connected regular matroid and has cocircumference c ∗ ⩾ 2 k , then there is a circuit which intersects each cocircuit of size c ∗ − k + 2 or greater. o extend a theorem of Dirac for graphs by showing that for any k-connected binary matroid M having no F 7 ∗ -minor, it holds that for any k cocircuits of M there is a circuit which intersects them.