Associated primes of powers of cover ideals under graph operations

Nasernejad, Mehrdad; Khashyarmanesh, Kazem; Al-Ayyoub, Ibrahim

doi:10.1080/00927872.2018.1527920

Cited by 15 publications

(8 citation statements)

References 14 publications

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“…It has already been established in [15,Theorem 3.3] and [2, Theorem 1.10] that the cover ideals of odd cycle graphs are normal and have the (strong) persistence property. On the other hand, since any even cycle is a bipartite graph, its cover ideal is normally torsion-free, and so is normal and also has the (strong) persistence property.…”

Section: This Yields Thatmentioning

confidence: 99%

Normality and associated primes of Closed neighborhood ideals and dominating ideals

Nasernejad,

Bandari,

Roberts

2022

Preprint

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In this paper, we first give some criteria for normality of monomial ideals. As applications, we show that closed neighborhood ideals of complete bipartite graphs are normal, and hence satisfy the (strong) persistence property. We also prove that dominating ideals of complete bipartite graphs are nearly normally torsion-free. In addition, we show that dominating ideals of h-wheel graphs, under certain condition, are normal.

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Section: This Yields Thatmentioning

confidence: 99%

Normality and associated primes of Closed neighborhood ideals and dominating ideals

Nasernejad,

Bandari,

Roberts

2022

Preprint

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“…However, this behaviour changes when we consider the odd cycles. The cover ideals of odd cycles happen to be nearly normally torsion-free, see [15], but edge ideals of odd cycles do not admit such tamed behaviour for the set of their associated primes. Given these facts, it is natural to expect some irregularities for the closed neighborhood ideals and dominating ideals of even and odd cycles.…”

Section: On the Closed Neighborhood Ideals And Dominating Ideals Of C...mentioning

confidence: 99%

Dominating ideals and closed neighborhood ideals of graphs

Nasernejad¹,

Qureshi²,

Bandari³

et al. 2021

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We study the closed neighborhood ideals and the dominating ideals of graphs, in particular, of trees and cycles. We prove that the closed neighborhood ideals and the dominating ideals of trees are normally torsionfree. The closed neighborhood ideals and the dominating ideals of cycles fail to be normally torsion-free. However, we prove that the closed neighborhood ideals of cycles admit the (strong) persistence property and the dominating ideals of cycles are nearly normally torsion-free.

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“…Ratliff [24] proved that (I k+1 : I) = I k for all large k. There are some interesting classes of monomial ideals satisfies in the strong persistence property. These include normal ideals, edge ideals of graphs, vertex cover ideals of perfect graphs (or chordal graphs), polymatroidal ideals, vertex cover ideals of cycle graphs of odd orders, vertex cover ideals of wheel graphs of even orders and all square-free monomial ideals in R with n ≤ 4, see for details [24,20,7,8,15,23,25,28].…”

Section: Introductionmentioning

confidence: 99%

Strong persistence and associated prime of powers of monomial ideals

Mafi¹,

Saremi²

2022

Preprint

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Let R = K[x 1 , . . . , x n ] be the polynomial ring in n variables over a field K and I be a monomial ideal of degree d ≤ 2. We show that (I k+1 : I) = I k for all k ≥ 1 and we disprove a motivation question that was appeared in [5, Question 2.51] by providing of a counter-example. Also, by this counterexample, we give a negative answer to the question that depth function of squarefree monomial ideals are non-increasing.

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Associated primes of powers of cover ideals under graph operations

Cited by 15 publications

References 14 publications

Normality and associated primes of Closed neighborhood ideals and dominating ideals

Normality and associated primes of Closed neighborhood ideals and dominating ideals

Dominating ideals and closed neighborhood ideals of graphs

Strong persistence and associated prime of powers of monomial ideals

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