21 research outputs found

    Gr\"obner bases of balanced polyominoes

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    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

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    Let GG be a finite connected simple graph and IGI_{G} the toric ideal of the edge ring K[G]K[G] of GG. In the present paper, we study finite graphs GG with the property that IGI_{G} is generated by quadratic binomials and IGI_{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 IGI_{G} to be generated by quadratic binomials and, by means of the computer search, we classify the finite graphs GG with the above property, up to 8 vertices.Comment: 11 pages, 17 figures, Typos corrected, Reference adde

    Toric Rings of Nonsimple Polyominoes

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    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