Circuits Through Cocircuits In A Graph With Extensions To Matroids

Abstract

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).

Ore-type and Dirac-type theorems for matroids

“…In [6], the following theorem was shown for regular matroids: Theorem 6.1. Let M be a k-connected regular matroid where c(M) 2k.…”