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arXiv · 2507.09015

Unavoidable Minors of Matroids with Minimum Cocircuit Size Four

Matthew Mizell·James Oxley
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Abstract

In 1963, Halin and Jung proved that every simple graph with minimum degree at least four has $K_5$ or $K_{2,2,2}$ as a minor. Mills and Turner proved an analog of this theorem by showing that every $3$-connected binary matroid in which every cocircuit has size at least four has $F_7, M^*(K_{3,3}), M(K_5),$ or $ M(K_{2,2,2})$ as a minor. Generalizing these results, this paper proves that every simple matroid in which all cocircuits have at least four elements has as a minor one of nine matroids, seven of which are well known. All nine of these special matroids have rank at most five and have at most twelve elements.

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Matthew Mizell, James Oxley. 2025-07-11. Unavoidable Minors of Matroids with Minimum Cocircuit Size Four. https://arxiv.org/abs/2507.09015

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