Ore-type and Dirac-type theorems for matroids

Let M be a connected binary matroid having no -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)For any two disjoint cocircuits and in A∗ it holds that .(ii)For any two disjoint cocircuits and in A∗ it holds that . Part (ii) implies Ore's 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∗⩾2k, then there is a circuit which intersects each cocircuit of size c∗−k+2 or greater.We also extend a theorem of Dirac for graphs by showing that for any k-connected binary matroid M having no -minor, it holds that for any k cocircuits of M there is a circuit which intersects them.