Unavoidable Minors of Matroids with Minimum Cocircuit Size Four
Matthew Mizell, James Oxley

TL;DR
This paper generalizes classical graph minor results to matroids, showing that simple matroids with all cocircuits of size at least four necessarily contain one of nine specific minors, all of small rank and size.
Contribution
It extends known theorems from graphs and binary matroids to a broader class of simple matroids with cocircuit size constraints.
Findings
Identifies nine specific minors for such matroids.
All these minors have rank at most five and at most twelve elements.
Provides a unifying framework for minors in matroids with cocircuit size conditions.
Abstract
In 1963, Halin and Jung proved that every simple graph with minimum degree at least four has or as a minor. Mills and Turner proved an analog of this theorem by showing that every -connected binary matroid in which every cocircuit has size at least four has or 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Algebra and Logic · graph theory and CDMA systems · Fuzzy and Soft Set Theory
