Unavoidable Minors of Matroids with Minimum Cocircuit Size Four

2025/07/11 by Matthew Mizell, Mizell, Matthew, James Oxley +1
Computer Science · Decision Sciences · Engineering · #Advanced Algebra and Logic #Combinatorics (math.CO) #FOS: Mathematics #Fuzzy and Soft Set Theory #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2507.09015

openalex publication_date 2025/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 1963, Halin and Jung proved that every simple graph with minimum degree at least four has K₅ 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 F₇, M^*(K_3,3), M(K₅), 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.

Citations

• On the relative importance of excluded minors

Related

• Small cocircuits in minimally vertically 4-connected matroids
• Detecting minors in matroids through triangles
• Characterizing binary matroids with no P₉-minor
• The internally 4-connected binary matroids with no M(K5\e)-minor
• Excluding Kuratowski graphs and their duals from binary matroids