Small cocircuits in minimally vertically $4$-connected matroids

TL;DR

The paper proves that minimally vertically 4-connected matroids with at least six elements generally contain small cocircuits, extending known results for lower connectivity levels and identifying specific exceptions.

Contribution

It establishes the existence of small cocircuits in minimally vertically 4-connected matroids, including binary cases, with a notable exception of a particular non-binary matroid.

Findings

Every minimally vertically 4-connected matroid with ≥6 elements has a 4-element cocircuit.

Such matroids also have a 5-element cocircuit containing a triangle, except for one non-binary case.

Binary cases always have a 4-element cocircuit.

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.