Compressed polytopes, initial ideals and complete multipartite graphs

Ohsugi, Hidefumi; Hibi, Takayuki

doi:10.1215/ijm/1255984847

Cited by 46 publications

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“…Since the sum of the components of δ C i + δ C 16 is 9, it follows that one of C j and C k is C 16 and hence…”

Section: Lexicographic and Reverse Lexicographic Gröbner Basesmentioning

confidence: 99%

“…This contradicts the hypothesis that in < (I K 2,3 ) is generated by quadratic monomials. Hence each x 1 x i x 16 does not belong to in < (I K 2,3 ). Thus the initial monomial of each cubic binomial 16 ].…”

Section: Lexicographic and Reverse Lexicographic Gröbner Basesmentioning

confidence: 99%

“…Hence each x 1 x i x 16 does not belong to in < (I K 2,3 ). Thus the initial monomial of each cubic binomial 16 ]. Note that each f i belongs to R. Let x k (k ∈ {1, 5, 6, .…”

Section: Lexicographic and Reverse Lexicographic Gröbner Basesmentioning

confidence: 99%

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Lexicographic and reverse lexicographic quadratic Gröbner bases of cut ideals

Sakamoto

2021

Journal of Symbolic Computation

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Hibi conjectured that if a toric ideal has a quadratic Gröbner basis, then the toric ideal has either a lexicographic or a reverse lexicographic quadratic Gröbner basis. In this paper, we present a cut ideal of a graph that serves as a counterexample to this conjecture. We also discuss the existence of a quadratic Gröbner basis of a cut ideal of a cycle. Nagel and Petrović claimed that a cut ideal of a cycle has a lexicographic quadratic Gröbner basis using the results of Chifman and Petrović. However, we point out that the results of Chifman and Petrović used by Nagel and Petrović are incorrect for cycles of length greater than or equal to 6. Hence the existence of a quadratic Gröbner basis for the cut ideal of a cycle (a ring graph) is an open question. We also provide a lexicographic quadratic Gröbner basis of a cut ideal of a cycle of length less than or equal to 7.2010 Mathematics Subject Classification. 13P10.

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“…Since the sum of the components of δ C i + δ C 16 is 9, it follows that one of C j and C k is C 16 and hence…”

Section: Lexicographic and Reverse Lexicographic Gröbner Basesmentioning

confidence: 99%

Section: Lexicographic and Reverse Lexicographic Gröbner Basesmentioning

confidence: 99%

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Lexicographic and reverse lexicographic quadratic Gröbner bases of cut ideals

Sakamoto

2021

Journal of Symbolic Computation

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“…In particular the edge polytope of a finite connected bipartite graph is unimodular. The edge polytope of G 1 of Example 2.2 is compressed [10] but not unimodular, and that of G 2 is unimodular. A combinatorial characterization of finite graphs G for which the edge polytope P G is compressed is given in [ We conclude the present paper with Example 3.3.…”

Section: Proof (A) Letmentioning

confidence: 99%

Special simplices and Gorenstein toric rings

Ohsugi

Hibi

2006

Journal of Combinatorial Theory, Series A

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“…It is known [6] that I G has a linear resolution if and only if the complementary graph G of G is chordal (see [11,Theorem 3.2]). It is proved [14] that I G is polymatroidal if and only if G is a complete multipartite graph. A tree is a finite connected graph with no cycle.…”

Section: Every Weakly Polymatroidal Ideal Has a Linear Resolutionmentioning

confidence: 99%

Weakly Polymatroidal Ideals

Kokubo

Hibi

2006

Algebra Colloq.

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The concept of the weakly polymatroidal ideal, which generalizes both the polymatroidal ideal and the prestable ideal, is introduced. A fundamental fact is that every weakly polymatroidal ideal has a linear resolution. One of the typical examples of weakly polymatroidal ideals arises from finite partially ordered sets. We associate each weakly polymatroidal ideal with a finite sequence, called the polymatroidal sequence, which will be useful for the computation of graded Betti numbers of weakly polymatroidal ideals as well as for the construction of weakly polymatroidal ideals.

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Compressed polytopes, initial ideals and complete multipartite graphs

Cited by 46 publications

References 0 publications

Lexicographic and reverse lexicographic quadratic Gröbner bases of cut ideals

Lexicographic and reverse lexicographic quadratic Gröbner bases of cut ideals

Special simplices and Gorenstein toric rings

Weakly Polymatroidal Ideals

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