Ring graphs and complete intersection toric ideals

Gitler, Isidoro; Reyes, Enrique; Villarreal, Rafael H.

doi:10.1016/j.disc.2009.03.020

Cited by 51 publications

(54 citation statements)

References 14 publications

(19 reference statements)

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“…, s. In particular, we recover Corollary 8.6 without using Proposition 5.3 and the volumes. For a study of simple graphs with complete intersection toric edge ideals, see [2,15,38].…”

Section: The J -Multiplicity Of Edge Ideals and Toric Edge Idealsmentioning

confidence: 99%

Generalized multiplicities of edge ideals

2017

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We explore connections between the generalized multiplicities of square-free monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and polyhedral geometry. For instance, we show the j -multiplicity is multiplicative over the connected components of a hypergraph, and we explicitly relate the j -multiplicity of the edge ideal of a properly connected uniform hypergraph to the Hilbert-Samuel multiplicity of its special fiber ring. In addition, we provide general bounds for the generalized multiplicities of the edge ideals and compute these invariants for classes of uniform hypergraphs.2010 Mathematics Subject Classification. 13H15, 13D40, 52B20.

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“…, s. In particular, we recover Corollary 8.6 without using Proposition 5.3 and the volumes. For a study of simple graphs with complete intersection toric edge ideals, see [2,15,38].…”

Section: The J -Multiplicity Of Edge Ideals and Toric Edge Idealsmentioning

confidence: 99%

Generalized multiplicities of edge ideals

2017

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“…In Theorem 2.2 we prove that if b is a binomial in P D then, there exists a binomial b ′ associated to a cycle whose monomials divide the monomials of b. Using this result we prove that the primitive binomials in P D are the ones associated with cycles and we recover the result in [6] that P D is generated by the binomials corresponding to cycles without chords.…”

Section: Toric Ideals Of Oriented Graphsmentioning

confidence: 68%

Complete intersection toric ideals of oriented graphs and chorded-theta subgraphs

2013

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Let G = (V, E) be a finite, simple graph. We consider for each oriented graph G O associated to an orientation O of the edges of G, the toric ideal P GO . In this paper we study those graphs with the property that P GO is a binomial complete intersection, for all O. These graphs are called CIO graphs. We prove that these graphs can be constructed recursively as clique-sums of cycles and/or complete graphs. We introduce the chorded-theta subgraphs and their transversal triangles. Also we establish that the CIO graphs are determined by the property that each chorded-theta has a transversal triangle. As a consequence, we obtain that the tournaments hold this property. Finally we explicitly give the minimal forbidden induced subgraphs that characterize these graphs, these families of graphs are: prisms, pyramids, thetas and a particular family of wheels that we call θ−partial wheels.

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“…A graph G is called a ring graph if it satisfies one of the following equivalent conditions (see [11]):…”

Section: Basic Properties Of the Cayley Sum Graph Of Idealsmentioning

confidence: 99%

Cayley Sum Graphs of Ideals of A commutative ring

Afkhami¹,

Barati²,

Khashyarmanesh³

et al. 2014

J. Aust. Math. Soc.

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Let R be a commutative ring, I(R) be the set of all ideals of R and S be a subset of I * (R) = I(R) \ {0}. We define a Cayley sum digraph of ideals of R, denoted by − − → Cay + (I(R), S ), as a directed graph whose vertex set is the set I(R) and, for every two distinct vertices I and J, there is an arc from I to J, denoted by I −→ J, whenever I + K = J, for some ideal K in S . Also, the Cayley sum graph Cay + (I(R), S ) is an undirected graph whose vertex set is the set I(R) and two distinct vertices I and J are adjacent whenever I + K = J or J + K = I, for some ideal K in S . In this paper, we study some basic properties of the graphs − − → Cay + (I(R), S ) and Cay + (I(R), S ) such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of Cay + (I(R), S ) and also we provide some characterization for rings R whose Cayley sum graphs have genus one.2010 Mathematics subject classification: primary 05C10; secondary 05C69, 13A15.

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Ring graphs and complete intersection toric ideals

Cited by 51 publications

References 14 publications

Generalized multiplicities of edge ideals

Generalized multiplicities of edge ideals

Complete intersection toric ideals of oriented graphs and chorded-theta subgraphs

Cayley Sum Graphs of Ideals of A commutative ring

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