The universal Gr\"obner basis of a binomial edge ideal

TL;DR

This paper proves that for binomial edge ideals, the universal Gr"obner basis and Graver basis are identical, describes this basis via graph paths, and conjectures a similar result for parity binomial edge ideals.

Contribution

It establishes the equality of the universal Gr"obner basis and Graver basis for binomial edge ideals and provides a graph-based description, advancing understanding of their algebraic structure.

Findings

Universal Gr"obner basis and Graver basis coincide for binomial edge ideals

Basis described in terms of paths in the underlying graph

Conjecture and partial proof for parity binomial edge ideals in complete graphs

Abstract

We show that the universal Gr\"obner basis and the Graver basis of a binomial edge ideal coincide. We provide a description for this basis set in terms of certain paths in the underlying graph. We conjecture a similar result for a parity binomial edge ideal and prove this conjecture for the case when the underlying graph is the complete graph.