Many toric ideals generated by quadratic binomials possess no quadratic Gröbner bases (Q402702)

The authors consider finite connected simple graphs \(G\) on the vertex set \(\{ 1, \dots, n \}\) and their edge rings \(K[G]\) over a field \(K\). In particular, they explicitely construct an infinite number of such nontrivial graphs \(G\) with the property that the toric ideal \(I_G\) of \(K[G]\) is generated by quadratic binomials and that \(I_G\) possesses no Gröbner basis consisting of quadratic binomials, and they classify (by means of an exhaustive computer search) all such (minimal) graphs with up to \(8\) vertices.NEWLINENEWLINEMore precisely, for a given graph \(G\), the authors consider the suspension of \(G\). The suspension of \(G\) can be constructed by introducing an additional vertex \(n+1\) and joining each of the vertices \(1, \dots, n\) of \(G\) with the new vertex \(n+1\) by an edge. The authors give a characterization for such a suspension graph to have the above property and use it to construct the infinitely many nontrivial graphs with the above property.