A Beginner’s Guide to Edge and Cover Ideals

Tuyl, Adam Van

doi:10.1007/978-3-642-38742-5_3

Cited by 52 publications

(30 citation statements)

References 60 publications

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“…If a vertex x i of D is a source (i.e., has only arrows leaving x i ) we shall always assume d i = 1 because in this case the definition of I(D) does not depend on the weight of x i . In the special case when d i = 1 for all i, we recover the edge ideal of the graph G which has been extensively studied in the literature [8,11,14,18,20,30,38,39,40,42]. A vertex-weighted digraph D is called Cohen-Macaulay (over the field K) if R/I(D) is a Cohen-Macaulay ring.…”

Section: Introductionmentioning

confidence: 99%

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Gimenez

Martinez-Bernal

Simis

et al. 2018

Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

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In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertexweighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen-Macaulay vertexweighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen-Macaulay and satisfy Alexander duality.

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Section: Introductionmentioning

confidence: 99%

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Gimenez

Martinez-Bernal

Simis

et al. 2018

Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

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“…Any W is a maximal independent set if and only if V \ W is a minimal vertex cover. We now use the fact that the associated primes of the edge ideal I(H) correspond to the minimal vertex covers of H (e.g., see the proof [27,Corollary 3.35] for edge ideals of graphs, which can be easily adapted to hypergraphs).…”

Section: Background Definitions and Resultsmentioning

confidence: 99%

The Waldschmidt constant for squarefree monomial ideals

et al. 2016

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Abstract. Given a squarefree monomial ideal I ⊆ R = k[x 1 , . . . , x n ], we show that α(I), the Waldschmidt constant of I, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of I. By applying results from fractional graph theory, we can then express α(I) in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of I. Moreover, expressing α(I) as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on α(I), thus verifying a conjecture of Cooper-Embree-Hà-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of P n with few components compared to n, and we find the Waldschmidt constant for the Stanley-Reisner ideal of a uniform matroid.

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“…As an application we recover the following fact. For all unexplained terminology and additional information we refer to [11,42] (for commutative algebra), [6,50,51] (for combinatorial optimization), [28] (for graph theory), and [15,21,31,62,66] (for the theory of powers of edge ideals of clutters and monomial ideals).…”

Section: Introductionmentioning

confidence: 99%

Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

et al. 2019

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In this paper we use polarization to study the behavior of the depth and regularity of a monomial ideal I, locally at a variable xi, when we lower the degree of all the highest powers of the variable xi occurring in the minimal generating set of I, and examine the depth and regularity of powers of edge ideals of clutters using combinatorial optimization techniques. If I is the edge ideal of an unmixed clutter with the max-flow min-cut property, we show that the powers of I have non-increasing depth and non-decreasing regularity. In particular edge ideals of unmixed bipartite graphs have non-decreasing regularity. We are able to show that the symbolic powers of the ideal of covers of the clique clutter of a strongly perfect graph have non-increasing depth. A similar result holds for the ideal of covers of a uniform ideal clutter.

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A Beginner’s Guide to Edge and Cover Ideals

Cited by 52 publications

References 60 publications

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

The Waldschmidt constant for squarefree monomial ideals

Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

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