Abstract
A classical result of Dirac's shows that, for any two edges and any n−2 vertices in a simple n-connected graph, there is a cycle that contains both edges and all n−2 of the vertices. Oxley has asked whether, for any two elements and any n−2 cocircuits in an n-connected matroid, there is a circuit that contains both elements and that has a non-empty intersection with all n−2 of the cocircuits. By using Seymour's decomposition theorem and results of Oxley and Denley and Wu, we prove that a slightly stronger property holds for regular matroids.
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Mayhew, D. Circuits and Cocircuits in Regular Matroids. Graphs and Combinatorics 22, 383–389 (2006). https://doi.org/10.1007/s00373-006-0677-9
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DOI: https://doi.org/10.1007/s00373-006-0677-9