Cohen-Macaulay clutters with combinatorial optimization properties and parallelizations of normal edge ideals

Dupont, Luis A.; Villarreal, Rafael H.; Reyes, Enrique

doi:10.11606/issn.2316-9028.v3i1p61-75

Cited by 9 publications

(4 citation statements)

References 29 publications

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“…We conclude this section with a collection of results giving conditions under which Conjecture 4.33, or its equivalent statements mentioned above, is known to hold. For uniform clutters it suffices to prove Conjecture 4.33 for Cohen-Macaulay clutters [23]. Let P = (X, ≺) be a partially ordered set (poset for short) on the finite vertex set X and let G be its comparability graph.…”

Section: Stability Of Associated Primesmentioning

confidence: 99%

Edge Ideals: Algebraic and Combinatorial Properties

Morey¹,

Villarreal²

2012

Progress in Commutative Algebra 1

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Let C be a clutter and let I(C) ⊂ R be its edge ideal. This is a survey paper on the algebraic and combinatorial properties of R/I(C) and C, respectively. We give a criterion to estimate the regularity of R/I(C) and apply this criterion to give new proofs of some formulas for the regularity. If R/I(C) is sequentially Cohen-Macaulay, we present a formula for the regularity of the ideal of vertex covers of C and give a formula for the projective dimension of R/I(C). We also examine the associated primes of powers of edge ideals, and show that for a graph with a leaf, these sets form an ascending chain.We are interested in determining which families of clutters have the property that ∆ C is pure, Cohen-Macaulay, or shellable. These properties have been extensively studied, see [10,89,92,93,94,100,102,110,111] and the references there.The above definition of shellable is due to Björner and Wachs [6] and is usually referred to as nonpure shellable, although here we will drop the adjective "nonpure". Originally, the definition of shellable also required that the simplicial complex be pure, that is, all facets have the same dimension. We will say ∆ is pure shellable if it also satisfies this hypothesis. These properties are related to other important properties [10,102,110]:If a shellable complex is not pure, an implication similar to that above holds when Cohen-Macaulay is replaced by sequentially Cohen-Macaulay. Definition 2.2. Let R = K[x 1 , . . . , x n ]. A graded R-module M is called sequentially Cohen-Macaulay (over K) if there exists a finite filtration of graded R-modules (0) = M 0 ⊂ M 1 ⊂ · · · ⊂ M r = M

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Section: Stability Of Associated Primesmentioning

confidence: 99%

Edge Ideals: Algebraic and Combinatorial Properties

Morey¹,

Villarreal²

2012

Progress in Commutative Algebra 1

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“…[9, p. 53]). The notion of a parallelization was used in [6,15] to describe the symbolic Rees algebra of an edge ideal. This notion has its origin in combinatorial optimization and has been used to describe the max-flow min-cut property of clutters [5,22].…”

Section: Perfect Matchings and Persistence Of Associated Primesmentioning

confidence: 99%

Associated primes of powers of edge ideals

2011

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Abstract. Let G be a graph and let I be its edge ideal. Our main result shows that the sets of associated primes of the powers of I form an ascending chain. It is known that the sets of associated primes of I i and I i stabilize for large i. We show that their corresponding stable sets are equal. To show our main result we use a classical result of Berge from matching theory and certain notions from combinatorial optimization.

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“…This result has been recently extended to clutters using the notion of parallelization [7]. Let C be a clutter on the vertex set X = {x 1 , .…”

Section: Symbolic Rees Algebras Of Edge Idealsmentioning

confidence: 99%

“…Notice that x k i is a vertex, i.e., k is an index not an exponent. Proposition 3.8 ( [7]). Let C be a clutter and let Υ(C) be the blocker of…”

Section: Symbolic Rees Algebras Of Edge Idealsmentioning

confidence: 99%

Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones

Dupont,

Villarreal

2007

Preprint

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Let G be a simple graph and let I c (G) be its ideal of vertex covers. We give a graph theoretical description of the irreducible b-vertex covers of G, i.e., we describe the minimal generators of the symbolic Rees algebra of I c (G). Then we study the irreducible b-vertex covers of the blocker of G, i.e., we study the minimal generators of the symbolic Rees algebra of the edge ideal of G. We give a graph theoretical description of the irreducible binary b-vertex covers of the blocker of G. It is shown that they correspond to irreducible induced subgraphs of G. As a byproduct we obtain a method, using Hilbert bases, to obtain all irreducible induced subgraphs of G. In particular we obtain all odd holes and antiholes. We study irreducible graphs and give a method to construct irreducible b-vertex covers of the blocker of G with high degree relative to the number of vertices of G.

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Cohen-Macaulay clutters with combinatorial optimization properties and parallelizations of normal edge ideals

Cited by 9 publications

References 29 publications

Edge Ideals: Algebraic and Combinatorial Properties

Edge Ideals: Algebraic and Combinatorial Properties

Associated primes of powers of edge ideals

Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones

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