On two conjectures of Maurer concerning basis graphs of matroids

TL;DR

This paper characterizes complexes related to basis graphs of matroids, proving a version of Maurer's conjecture and clarifying the conditions needed for such characterizations, with implications for even Δ-matroids.

Contribution

It proves a version of Maurer's conjecture on basis graphs of matroids and clarifies the redundancy of conditions in their characterization.

Findings

Characterization of 2-dimensional complexes associated with basis graphs of matroids.

Proof of a version of Maurer's conjecture on these complexes.

Identification of local properties with positive-curvature-like aspects.

Abstract

We characterize 2-dimensional complexes associated canonically with basis graphs of matroids as simply connected triangle-square complexes satisfying some local conditions. This proves a version of a (disproved) conjecture by Stephen Maurer (Conjecture 3 of S. Maurer, Matroid basis graphs I, JCTB 14 (1973), 216-240). We also establish Conjecture 1 from the same paper about the redundancy of the conditions in the characterization of basis graphs. We indicate positive-curvature-like aspects of the local properties of the studied complexes. We characterize similarly the corresponding 2-dimensional complexes of even $\Delta$-matroids.