Quaternary matroids are vf-safe

TL;DR

This paper extends the closure property of binary delta-matroids under vertex flips to quaternary matroids, demonstrating they are vf-safe and exploring implications for bicycle spaces and matroid representations.

Contribution

It generalizes the vf-safe property from GF(2) to GF(4) for delta-matroids, establishing new invariance results for quaternary matroids.

Findings

Quaternary matroids are vf-safe under vertex flips.

The matroid of a bicycle space of a quaternary matroid is representation-independent.

Extension of Vertigan's bicycle dimension result to quaternary matroids.

Abstract

Binary delta-matroids are closed under vertex flips, which consist of the natural operations of twist and loop complementation. In this note we provide an extension of this result from GF(2) to GF(4). As a consequence, quaternary matroids are "safe" under vertex flips (vf-safe for short). As an application, we find that the matroid of a bicycle space of a quaternary matroid is independent of the chosen representation. This extends a result of Vertigan [J. Comb. Theory B (1998)] concerning the bicycle dimension of quaternary matroids.