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 2-connected 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 C 1 and C 2 are disjoint circuits satisfying r(C 1)+r(C 2)=r(C 1C 2), then |C 1|+|C 2|≤2(c-k + 1).

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Correspondence to Sean McGuinness.

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McGuinness, S. Circuits Through Cocircuits In A Graph With Extensions To Matroids. Combinatorica 25, 451–463 (2005).

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Mathematics Subject Classification (2000):

  • 05C38
  • 05B35