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

Let G be a finite connected simple graph and IG the toric ideal of the edge ring K[G] of G. In the present paper, we study finite graphs G with the property that IG is generated by quadratic binomials and IG 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 IG 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.