Separating cocircuits in binary matroids

Small cocircuits in matroids

We prove that, for any positive integers k, n, and q, if M is a simple matroid that has neither a U2,q+2nor an M(Kn)minor and M has sufficiently large rank, then M has a cocircuit of size at most r(M)/k.

Intersections of circuits and cocircuits in binary matroids

Oxley has shown that if, for some k >_-4, a matroid M has a k-element set that is the intersection of a circuit and a cocircuit, then M has a 4-element set that is the intersection of a circuit and a cocircuit. We prove that, under the above hypothesis, for k I> 6, a binary matroid will also have a 6-element set that is the intersection of a circuit and a cocircuit. In addition, we determine ex...

Regular Matroids with Graphic Cocircuits

In this paper we examine the effect of removing cocircuits from regular matroids and we focus on the case in which such a removal always results in a graphic matroid. The first main result, given in section 3, is that a regular matroid with graphic cocircuits is signed-graphic if and only if it does not contain two specific minors. This provides a useful connection between graphic, regular and ...

Disjoint cocircuits in matroids with large rank

We prove that, for any positive integers n; k and q; there exists an integer R such that, if M is a matroid with no MðKnÞor U2;qþ2-minor, then either M has a collection of k disjoint cocircuits or M has rank at most R: Applied to the class of cographic matroids, this result implies the edge-disjoint version of the Erdös–Pósa Theorem. r 2002 Elsevier Science (USA). All rights reserved. AMS 1991 ...

Partitioning Matroids With Only Small Cocircuits