Matroid fragility and relaxations of circuit hyperplanes

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 $M_2$ is obtained from a matroid $M_1$ by relaxing a circuit-hyperplane and both $M_1$ and $M_2$ are $\mathbb{F}$-representable, then the branch-width of $M_1$ is bounded by a function depending only upon $\mathbb{F}$. Our main result is that the second conjecture implies the first.