White's Conjecture for Paving Matroids

TL;DR

White's conjecture asks whether any two $n$-tuples of bases with the same multiset union can be connected by symmetric exchanges, equivalently describing generators of the matroid toric ideal $I_M$. The paper proves the conjecture for paving matroids by an induction on stressed-hyperplane relaxations, extending the approach of Han et al. from circuit-hyperplane relaxations to the broader stressed-hyperplane framework, and showing preservation of WC under relaxation to connect paving matroids to the uniform matroid $U_{r,E}$. The authors establish the degree-2 and degree-3 cases and develop a higher-degree induction that leverages this relaxation structure, enabling a full proof for paving matroids. This advances White's conjecture toward split matroids and sheds light on the algebraic structure of matroid toric ideals and their relation to matroid polytopes.

Abstract

White's conjecture asserts that any two tuples of matroid bases that have the same multi-set union can be transformed from one to another by symmetric exchanges; it also implies that the toric ideals of matroids are generated by the binomials encoding these exchanges. We prove White's conjecture for the class of paving matroids. Our strategy is to generalize the inductive argument using circuit-hyperplane relaxations in the recent work of Han et al. to stressed hyperplane relaxations.

Theorems & Definitions (36)

• Conjecture 1.1
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Proposition 2.2
• Definition 2.3