Research

4 pages, 213 KiB

Open AccessArticle

The Regularity of Edge Rings and Matching Numbers

by Jürgen Herzog and Takayuki Hibi

Mathematics 2020, 8(1), 39; https://doi.org/10.3390/math8010039 - 01 Jan 2020

Cited by 4 | Viewed by 1954

Let

K [G]

denote the edge ring of a finite connected simple graph G on

[d]

and

mat (G)

the matching number of G. It is shown that [...] Read more.

Let

K [G]

denote the edge ring of a finite connected simple graph G on

[d]

and

mat (G)

the matching number of G. It is shown that

reg (K [G]) \leq mat (G)

if G is non-bipartite and

K [G]

is normal, and that

reg (K [G]) \leq mat (G) - 1

if G is bipartite. Full article

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

4 pages, 237 KiB

Open AccessArticle

Odd Cycles and Hilbert Functions of Their Toric Rings

by Takayuki Hibi and Akiyoshi Tsuchiya

Mathematics 2020, 8(1), 22; https://doi.org/10.3390/math8010022 - 20 Dec 2019

Cited by 3 | Viewed by 1691

Abstract

Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that for each

1 \leq s \leq 5

, an O-sequence [...] Read more.

Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that for each

1 \leq s \leq 5

, an O-sequence

(h_{0}, h_{1}, \dots, h_{2 s - 1}) \in Z_{\geq 0}^{2 s}

satisfying the properties that (i)

h_{0} \leq h_{1} \leq \dots \leq h_{s - 1}

, (ii)

h_{2 s - 1} = h_{0}

,

h_{2 s - 2} = h_{1}

and (iii)

h_{2 s - 1 - i} = h_{i} + {(- 1)}^{i}

,

2 \leq i \leq s - 1

, can be the h-vector of a Cohen-Macaulay standard G-domain. Full article

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

18 pages, 291 KiB

Open AccessFeature PaperArticle

Syzygies, Betti Numbers, and Regularity of Cover Ideals of Certain Multipartite Graphs

by A. V. Jayanthan and Neeraj Kumar

Mathematics 2019, 7(9), 869; https://doi.org/10.3390/math7090869 - 19 Sep 2019

Cited by 2 | Viewed by 2073

Abstract

Let G be a finite simple graph on n vertices. Let

J_{G} \subset K [x_{1}, \dots, x_{n}]

be the cover ideal of G. In this article, we obtain syzygies, Betti numbers, and Castelnuovo–Mumford regularity of [...] Read more.

Let G be a finite simple graph on n vertices. Let

J_{G} \subset K [x_{1}, \dots, x_{n}]

be the cover ideal of G. In this article, we obtain syzygies, Betti numbers, and Castelnuovo–Mumford regularity of

J_{G}^{s}

for all

s \geq 1

for certain classes of graphs G. Full article

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

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18 pages, 324 KiB

Open AccessArticle

Algebraic Algorithms for Even Circuits in Graphs

by Huy Tài Hà and Susan Morey

Mathematics 2019, 7(9), 859; https://doi.org/10.3390/math7090859 - 17 Sep 2019

Cited by 1 | Viewed by 2074

Abstract

We present an algebraic algorithm to detect the existence of and to list all indecomposable even circuits in a given graph. We also discuss an application of our work to the study of directed cycles in digraphs. Full article

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

6 pages, 215 KiB

Open AccessFeature PaperArticle

Faces of 2-Dimensional Simplex of Order and Chain Polytopes

by Aki Mori

Mathematics 2019, 7(9), 851; https://doi.org/10.3390/math7090851 - 14 Sep 2019

Viewed by 1845

Abstract

Each of the descriptions of vertices, edges, and facets of the order and chain polytope of a finite partially ordered set are well known. In this paper, we give an explicit description of faces of 2-dimensional simplex in terms of vertices. Namely, it [...] Read more.

Each of the descriptions of vertices, edges, and facets of the order and chain polytope of a finite partially ordered set are well known. In this paper, we give an explicit description of faces of 2-dimensional simplex in terms of vertices. Namely, it will be proved that an arbitrary triangle in 1-skeleton of the order or chain polytope forms the face of 2-dimensional simplex of each polytope. These results mean a generalization in the case of 2-faces of the characterization known in the case of edges. Full article

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

13 pages, 261 KiB

Open AccessArticle

Cohen Macaulay Bipartite Graphs and Regular Element on the Powers of Bipartite Edge Ideals

by Arindam Banerjee and Vivek Mukundan

Mathematics 2019, 7(8), 762; https://doi.org/10.3390/math7080762 - 20 Aug 2019

Viewed by 2342

Abstract

In this article, we discuss new characterizations of Cohen-Macaulay bipartite edge ideals. For arbitrary bipartite edge ideals

I (G)

, we also discuss methods to recognize regular elements on

I {(G)}^{s}

for all

s \geq 1

in terms [...] Read more.

In this article, we discuss new characterizations of Cohen-Macaulay bipartite edge ideals. For arbitrary bipartite edge ideals

I (G)

, we also discuss methods to recognize regular elements on

I {(G)}^{s}

for all

s \geq 1

in terms of the combinatorics of the graph G. Full article

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

8 pages, 239 KiB

Open AccessArticle

Cohen-Macaulay and (S₂) Properties of the Second Power of Squarefree Monomial Ideals

by Do Trong Hoang, Giancarlo Rinaldo and Naoki Terai

Mathematics 2019, 7(8), 684; https://doi.org/10.3390/math7080684 - 31 Jul 2019

Cited by 1 | Viewed by 3445

Abstract

We show that Cohen-Macaulay and (S

_{2}

) properties are equivalent for the second power of an edge ideal. We give an example of a Gorenstein squarefree monomial ideal I such that

S / I^{2}

satisfies the Serre condition (S

_{2}

), [...] Read more.

We show that Cohen-Macaulay and (S

_{2}

) properties are equivalent for the second power of an edge ideal. We give an example of a Gorenstein squarefree monomial ideal I such that

S / I^{2}

satisfies the Serre condition (S

_{2}

), but is not Cohen-Macaulay. Full article

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

5 pages, 243 KiB

Open AccessArticle

Compatible Algebras with Straightening Laws on Distributive Lattices

by Daniel Bănaru and Viviana Ene

Mathematics 2019, 7(8), 671; https://doi.org/10.3390/math7080671 - 27 Jul 2019

Viewed by 1895

Abstract

We characterize the finite distributive lattices on which there exists a unique compatible algebra with straightening laws. Full article

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

13 pages, 327 KiB

Open AccessFeature PaperArticle

The Regularity of Some Families of Circulant Graphs

by Miguel Eduardo Uribe-Paczka and Adam Van Tuyl

Mathematics 2019, 7(7), 657; https://doi.org/10.3390/math7070657 - 22 Jul 2019

Cited by 5 | Viewed by 2990

Abstract

We compute the Castelnuovo–Mumford regularity of the edge ideals of two families of circulant graphs, which includes all cubic circulant graphs. A feature of our approach is to combine bounds on the regularity, the projective dimension, and the reduced Euler characteristic to derive [...] Read more.

We compute the Castelnuovo–Mumford regularity of the edge ideals of two families of circulant graphs, which includes all cubic circulant graphs. A feature of our approach is to combine bounds on the regularity, the projective dimension, and the reduced Euler characteristic to derive an exact value for the regularity. Full article

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

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12 pages, 310 KiB

Open AccessFeature PaperArticle

Toric Rings and Ideals of Stable Set Polytopes

by Kazunori Matsuda, Hidefumi Ohsugi and Kazuki Shibata

Mathematics 2019, 7(7), 613; https://doi.org/10.3390/math7070613 - 10 Jul 2019

Cited by 6 | Viewed by 2460

Abstract

In the present paper, we study the normality of the toric rings of stable set polytopes, generators of toric ideals of stable set polytopes, and their Gröbner bases via the notion of edge polytopes of finite nonsimple graphs and the results on their [...] Read more.

In the present paper, we study the normality of the toric rings of stable set polytopes, generators of toric ideals of stable set polytopes, and their Gröbner bases via the notion of edge polytopes of finite nonsimple graphs and the results on their toric ideals. In particular, we give a criterion for the normality of the toric ring of the stable set polytope and a graph-theoretical characterization of the set of generators of the toric ideal of the stable set polytope for a graph of stability number two. As an application, we provide an infinite family of stable set polytopes whose toric ideal is generated by quadratic binomials and has no quadratic Gröbner bases. Full article

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

17 pages, 315 KiB

Open AccessFeature PaperArticle

On the Stanley Depth of Powers of Monomial Ideals

by S. A. Seyed Fakhari

Mathematics 2019, 7(7), 607; https://doi.org/10.3390/math7070607 - 08 Jul 2019

Cited by 4 | Viewed by 2437

Abstract

In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s conjecture, and their counterexample is [...] Read more.

In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s conjecture, and their counterexample is a quotient of squarefree monomial ideals. On the other hand, there is evidence showing that Stanley’s inequality can be true for high powers of monomial ideals. In this survey article, we collect the recent results in this direction. More precisely, we investigate the Stanley depth of powers, integral closure of powers, and symbolic powers of monomial ideals. Full article

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

6 pages, 243 KiB

Open AccessArticle

Linear Maps in Minimal Free Resolutions of Stanley-Reisner Rings

by Lukas Katthän

Mathematics 2019, 7(7), 605; https://doi.org/10.3390/math7070605 - 06 Jul 2019

Viewed by 2040

Abstract

In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex

Δ

. Indeed, the differentials in the linear part are simply a compilation of restriction maps in [...] Read more.

In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex

Δ

. Indeed, the differentials in the linear part are simply a compilation of restriction maps in the simplicial cohomology of induced subcomplexes of

Δ

. Along the way, we also show that if a monomial ideal has at least one generator of degree 2, then the linear strand of its minimal free resolution can be written using only

\pm 1

coefficients. Full article

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

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