Many toric ideals generated by quadratic binomials possess no quadratic Gröbner bases

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 Gröbner 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.

11 pages, 17 figures, Typos corrected, Reference added

Cited by in corpus (2)

• Toric ideals associated with gap-free graphs
• Strong Koszulness of toric rings associated with stable set polytopes of trivially perfect graphs