Frame Matroids, Toric Ideals, and a Conjecture of White

TL;DR

This paper extends the verification of White's conjecture on the generation of toric ideals by quadrics from graphic matroids to a broader class of frame matroids satisfying a linearity condition, including several important subclasses.

Contribution

It generalizes White's conjecture verification from graphic to certain frame matroids, broadening the understanding of toric ideal generation in matroid theory.

Findings

Toric ideal of these frame matroids is generated by quadrics.

Includes classes like bicircular and signed graphic matroids.

Extends White's conjecture verification to new matroid classes.

Abstract

Blasiak verified a conjecture of White for graphic matroids by showing that the toric ideal of a graphic matroid is generated by quadrics. In this paper, we extend this result to frame matroids satisfying a linearity condition. Such classes of matroids include graphic matroids, bicircular matroids, signed graphic matroids, and more generally frame matroids obtained from group-labelled graphs.