Koszul Bipartite Graphs

Ohsugi, Hidefumi; Hibi, Takayuki

doi:10.1006/aama.1998.0615

Cited by 67 publications

(77 citation statements)

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“…The hierarchy (iii) ⇒ (ii) ⇒ (i) is known (e.g., [4]). The converse hierarchy is true for normal toric surfaces [5] and for affine semigroup rings arising from bipartite graphs [9]. On the other hand, it is shown in [10] that there exist a non-Koszul monomial curve whose toric ideal is generated by quadratic binomials and a Koszul monomial curve whose toric ideal has no quadratic Gröbner basis.…”

Section: Introductionmentioning

confidence: 99%

Toric Ideals Generated by Quadratic Binomials

1999

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A combinatorial criterion for the toric ideal arising from a finite graph to be generated by quadratic binomials is studied. Such a criterion guarantees that every Koszul algebra generated by squarefree quadratic monomials is normal. We present an example of a normal non-Koszul squarefree semigroup ring whose toric ideal is generated by quadratic binomials as well as an example of a non-normal Koszul squarefree semigroup ring whose toric ideal possesses no quadratic Gröbner basis. In addition, all the affine semigroup rings which are generated by squarefree quadratic monomials and which have 2-linear resolutions will be classified. Moreover, it is shown that the toric ideal of a normal affine semigroup ring generated by quadratic monomials is generated by quadratic binomials if its underlying polytope is simple.

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Section: Introductionmentioning

confidence: 99%

Toric Ideals Generated by Quadratic Binomials

1999

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“…Assume that J (P ) is planar. Then K[A P ] is isomorphic to K[A] where A is the vertex-edge incidence matrix of a bipartite graph (see [7]). Since A is unimodular [10], A P is also unimodular.…”

Section: Then G Is a Gröbner Bases Of I A ±mentioning

confidence: 99%

Centrally symmetric configurations of order polytopes

et al. 2015

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“…Essentially this problem has been studied in detail from an algebraic viewpoint in a series of papers by Ohsugi and Hibi [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Markov bases for two-way subtable sum problems

Hara

Takemura

Yoshida

2009

Journal of Pure and Applied Algebra

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It has been well-known that for two-way contingency tables with fixed row sums and column sums the set of square-free moves of degree two forms a Markov basis. However when we impose an additional constraint that the sum of a subtable is also fixed, then these moves do not necessarily form a Markov basis. Thus, in this paper, we show a necessary and sufficient condition on a subtable so that the set of square-free moves of degree two forms a Markov basis.Comment: 23 page

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Koszul Bipartite Graphs

Cited by 67 publications

References 1 publication

Toric Ideals Generated by Quadratic Binomials

Toric Ideals Generated by Quadratic Binomials

Centrally symmetric configurations of order polytopes

Markov bases for two-way subtable sum problems

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