Gr\"obner bases of balanced polyominoes

We introduce balanced polyominoes and show that their ideal of inner minors is a prime ideal and has a quadratic Gr\"obner basis with respect to any monomial order, and we show that any row or column convex and any tree-like polyomino is simple and balanced

Many toric ideals generated by quadratic binomials possess no quadratic Gr\"obner bases

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 Gr\"obner 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.Comment: 11 pages, 17 figures, Typos corrected, Reference adde

Toric Rings of Nonsimple Polyominoes

Abstract It is known that toric ring of a simple polyomino is ring homomorphic to a edge ring of a weakly chordal bipartite graph. In this paper we identify the attached toric rings of nonsimple polyominoes which are of the form "rectangle minus rectangle". Mathematics Subject Classification: 13C05, 05E4