Many toric ideals generated by quadratic binomials possess no quadratic Gr?bner bases

摘要

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 1G is generated by quadratic binomials and I_G possesses no quadratic Gr?bner 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.