Matroid fragility and relaxations of circuit hyperplanes

2018/05/08 by Geelen, Jim, Hoersch, Florian
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.03263

Abstract

We relate two conjectures that play a central role in the reported proof of Rota's Conjecture. Let \mathbb F be a finite field. The first conjecture states that: the branch-width of any \mathbb F-representable N-fragile matroid is bounded by a function depending only upon \mathbb F and N. The second conjecture states that: if a matroid M₂ is obtained from a matroid M₁ by relaxing a circuit-hyperplane and both M₁ and M₂ are \mathbb F-representable, then the branch-width of M₁ is bounded by a function depending only upon \mathbb F. Our main result is that the second conjecture implies the first.

Related

• Stability, fragility, and Rota's Conjecture
• Relaxations of GF(4)-representable matroids
• The extremal functions of classes of matroids of bounded branch-width
• Efficient Algorithms for Maximal Matroid Degenerations and Irreducible Decompositions of Circuit Varieties
• On Optimization and Counting of Non-Broken Bases of Matroids