Circuits Through Cocircuits In A Graph With Extensions To Matroids |
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Authors: | Sean McGuinness |
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Institution: | (1) 17 London Road, Syosset, NY 11791, USA |
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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 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 C1 and C2 are disjoint circuits satisfying r(C1)+r(C2)=r(C1∪C2), then |C1|+|C2|≤2(c-k + 1). |
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Keywords: | 05C38 05B35 |
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