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arXiv · 2110.13120

Small cocircuits in minimally vertically $4$-connected matroids

James Oxley·Zach Walsh
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Abstract

Halin proved that every minimally $k$-connected graph has a vertex of degree $k$. More generally, does every minimally vertically $k$-connected matroid have a $k$-element cocircuit? Results of Murty and Wong give an affirmative answer when $k \le 3$. We show that every minimally vertically $4$-connected matroid with at least six elements has a $4$-element cocircuit, or a $5$-element cocircuit that contains a triangle, with the exception of a specific non-binary $9$-element matroid. Consequently, every minimally vertically $4$-connected binary matroid with at least six elements has a $4$-element cocircuit.

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James Oxley, Zach Walsh. 2021-10-25. Small cocircuits in minimally vertically $4$-connected matroids. https://arxiv.org/abs/2110.13120

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