Peer-reviewedCC BY 4.0The Electronic Journal of Combinatorics
Unavoidable Minors of Matroids with Minimum Cocircuit Size Four
Abstract
In 1963, Halin and Jung proved that every simple graph with minimum degree at least four has K5 or K2,2,2 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*(K3,3), 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.
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BibTeX
@article{mizell2026unavoidable,
title = {{Unavoidable Minors of Matroids with Minimum Cocircuit Size Four}},
author = {Matthew Mizell and James Oxley},
journal = {The Electronic Journal of Combinatorics},
year = {2026},
month = oct,
doi = {10.37236/14364},
url = {https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p3},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Unavoidable Minors of Matroids with Minimum Cocircuit Size Four
AU - Matthew Mizell
AU - James Oxley
PY - 2026
DA - 2026/10/09
JO - The Electronic Journal of Combinatorics
DO - 10.37236/14364
UR - https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p3
AB - 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.
SN - 1077-8926
ER -