Many toric ideals generated by quadratic binomials possess no quadratic Gr\"obner bases

TL;DR

This paper investigates finite graphs whose toric ideals are generated by quadratic binomials but lack quadratic Gr"obner bases, providing infinite examples and a classification for graphs up to 8 vertices.

Contribution

It introduces an infinite series of such graphs and offers a combinatorial characterization and classification for small graphs.

Findings

Identified an infinite series of graphs with the property.

Classified graphs up to 8 vertices with the property.

Provided a combinatorial criterion for quadratic binomial generation.

Abstract

Let $G$ be a finite connected simple graph and $I_{G}$ the toric ideal of the edge ring $K[G]$ of $G$. In the present paper, we study finite graphs $G$ with the property that $I_{G}$ is generated by quadratic binomials and $I_{G}$ possesses no quadratic Gr\"obner basis. First, we give a nontrivial infinite series of finite graphs with the above property. Second, we implement a combinatorial characterization for $I_{G}$ to be generated by quadratic binomials and, by means of the computer search, we classify the finite graphs $G$ with the above property, up to 8 vertices.