Stability of depth functions of cover ideals of balanced hypergraphs

Hang, Nguyen Thu

doi:10.1142/s0219498820500553

Cited by 2 publications

(2 citation statements)

References 18 publications

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“…To the best of our knowledge this is open even for bipartite graphs. Our next application extends the fact that the powers of I(G) ∨ , the ideal of covers of G, have non-increasing depth if G is bipartite [5,26,27]. An interesting example due to Kaiser, Stehlík, andŠkrekovski [38] shows that the powers of the ideal of covers of a graph does not always have non-increasing depth (Example 5.4), that is, part (b) of Corollary 5.3 fails for non-bipartite graphs.…”

Section: Introductionmentioning

confidence: 81%

“…The constant value of depth(R/I k ) for k ≫ 0 is called the limit depth of I and is denoted by lim k→∞ depth(R/I k ). There are some classes of monomial ideals with non-increasing depth and non-decreasing regularity [3,5,26,27,52]. A natural way to show these properties for a monomial ideal I is to prove the existence of a monomial f such that (I k+1 : f ) = I k for k ≥ 1.…”

Section: Edge Ideals Of Clutters With Non-increasing Depthmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 81%

Section: Edge Ideals Of Clutters With Non-increasing Depthmentioning

confidence: 99%

Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

et al. 2019

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A general formula for the index of depth stability of edge ideals

Lam,

Trung,

Trung

2024

Trans. Amer. Math. Soc.

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By a classical result of Brodmann, the function d e p t h R / I t depthR/I^t is asymptotically a constant, i.e. there is a number s s such that d e p t h R / I t = d e p t h R / I s depthR/I^t = depthR/I^s for t > s t > s . One calls the smallest number s s with this property the index of depth stability of I I and denotes it by d s t a b ( I ) dstab(I) . This invariant remains mysterious till now. The main result of this paper gives an explicit formula for d s t a b ( I ) dstab(I) when I I is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize d s t a b ( I ) dstab(I) explicitly. The formula expresses d s t a b ( I ) dstab(I) in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals.

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Stability of depth functions of cover ideals of balanced hypergraphs

Abstract: We prove that the depth functions of cover ideals of balanced hypergraph have the nonincreasing property. Furthermore, we also give a bound for the index of depth stability of these ideals.

Cited by 2 publications

References 18 publications

Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

A general formula for the index of depth stability of edge ideals

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