MANY TORIC IDEALS GENERATED BY QUADRATIC BINOMIALS POSSESS NO QUADRATIC GROBNER BASES (SUMMARY) (Towards new development of mathematics via computational algebra system)

Abstract

This is a brief summary of Hibi Nishiyama Ohsugi-Shikama [6]. Let G be a finite connected simple graph and I_{G} the toric ideal of the edge ring 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.