Circuits Through Cocircuits In A Graph With Extensions To Matroids

We show that for any k-connected graph having cocircumference c∗, there is a cycle which intersects every cocycle of size c∗ − k+2 or greater. We use this to show that in a 2connected graph, there is a family of at most c∗ cycles for which each edge of the graph belongs to at least two cycles in the family. This settles a question raised by Oxley. A certain result known for cycles and cocycles in graphs is extended to matroids. It is shown that for a k-connected regular matroid having circumference c≥2k if C1 and C2 are disjoint circuits satisfying r(C1)+r(C2)=r(C1∪C2), then |C1|+ |C2|≤2(c−k+1).