Depths of Powers of the Edge Ideal of a Tree

Morey, Susan

doi:10.1080/00927870903286900

Cited by 61 publications

(58 citation statements)

References 18 publications

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“…Note that r ≥ 1 and r is finite. Note also that by [4,Lemma 3.3], at most one of the x i 's is not a leaf. Without loss of generality, assume that x i is a leaf for 1 ≤ i < r. Let I j be the ideal of the minor of G formed by deleting x 1 , .…”

Section: The Main Resultsmentioning

confidence: 99%

“…A careful examination of [4,Lemma 3.3] guarantees that any tree with diameter d ≥ 3 will contain at least two such vertices that are not themselves leaves, namely the neighbors of the two leaves of a path realizing the diameter. Call a vertex v of G a near leaf of G if v is not a leaf and N (v) contains at most one vertex that is not a leaf.…”

Section: Appendix: a Generalization Of The Main Theoremmentioning

confidence: 99%

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Stanley depth of powers of the edge ideal of a forest

Pournaki

Fakhari

Yassemi

2013

Proc. Amer. Math. Soc.

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Abstract. Let K be a field and S = K[x 1 , . . . , x n ] be the polynomial ring in n variables over the field K. Let G be a forest with p connected components G 1 , . . . , G p and let I = I(G) be its edge ideal in S. Suppose that d i is the diameter of G i , 1 ≤ i ≤ p, and consider d = max {d i | 1 ≤ i ≤ p}. Morey has shown that for every t ≥ 1, the quantity max+ p − 1, p is a lower bound for depth(S/I t ). In this paper, we show that for every t ≥ 1, the mentioned quantity is also a lower bound for sdepth(S/I t ). By combining this inequality with Burch's inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem.

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Section: The Main Resultsmentioning

confidence: 99%

Section: Appendix: a Generalization Of The Main Theoremmentioning

confidence: 99%

Stanley depth of powers of the edge ideal of a forest

Pournaki

Fakhari

Yassemi

2013

Proc. Amer. Math. Soc.

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“…A 123, 239-251, 2014), Kaiser et al provide a family of critically 3-chromatic graphs whose expansions do not result in critically 4-chromatic graphs and, thus, give counterexamples to a conjecture of Francisco et al (Discrete Math. 310, 2176-2182, 2010. The cover ideal of the smallest member of this family also gives a counterexample to the persistence and non-increasing depth properties.…”

mentioning

confidence: 93%

Squarefree Monomial Ideals that Fail the Persistence Property and Non-increasing Depth

Harbourne

Sun

2015

Acta Math Vietnam

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In a recent work (Kaiser et al., J. Comb. Theory Ser. A 123, 239-251, 2014), Kaiser et al. provide a family of critically 3-chromatic graphs whose expansions do not result in critically 4-chromatic graphs and, thus, give counterexamples to a conjecture of Francisco et al. (Discrete Math. 310, 2176-2182, 2010. The cover ideal of the smallest member of this family also gives a counterexample to the persistence and non-increasing depth properties. In this paper, we show that the cover ideals of all members of their family of graphs indeed fail to have the persistence and non-increasing depth properties.

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“…For a graph G, if every its connected component is nonbipartite, then we can see that dstab(I(G)) < dim R from [4]. In general, there is not an absolute bound of dstab(I(G)) even in the case G is a tree (see [20]). In this paper we will establish a bound of dstab(I(G)) for any graph G. In particular, dstab(I(G)) < dim R.…”

Section: Introductionmentioning

confidence: 99%

“…z be a path realizing the diameter of G. Then v is a leaf, u and w both are not leaves. By [20,Lemma 3.3] we have N G (u) = {w} ∪ L G (u). And now we prove n υ(G) − ε 0 (G) by the same way as in Case 2 in the proof of Lemma 5.3.…”

mentioning

confidence: 99%

Stability of depths of powers of edge ideals

Trung

2016

Journal of Algebra

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Abstract. Let G be a graph and let I := I(G) be its edge ideal. In this paper, we provide an upper bound of n from which depth R/I (G) n is stationary, and compute this limit explicitly. This bound is always achieved if G has no cycles of length 4 and every its connected component is either a tree or a unicyclic graph. IntroductionLet R = K[x 1 , . . . , x r ] be a polynomial ring over a field K and I a homogeneous ideal in R. Brodmann [2] showed that depth R/I n is a constant for sufficiently large n. Moreover limwhere ℓ(I) is the analytic spread of I. It was shown in [6, Proposition 3.3] that this is an equality when the associated graded ring of I is Cohen-Macaulay. We call the smallest number n 0 such that depth R/I n = depth R/I n 0 for all n n 0 , the index of depth stability of I, and denote this number by dstab(I). It is of natural interest to find a bound for dstab(I). As until now we only know effective bounds of dstab(I) for few special classes of ideals I, such as complete intersection ideals (see [5]), square-free Veronese ideals (see [8]), polymatroidal ideals (see [10]). In this paper we will study this problem for edge ideals.From now on, every graph G is assumed to be simple (i.e., a finite, undirected, loopless and without multiple edges) without isolated vertices on the vertex set V (G) = [r] := {1, . . . , r} and the edge set E(G) unless otherwise indicated. We associate to G the quadratic squarefree monomial idealwhich is called the edge ideal of G.If I is a polymatroidal ideal in R, Herzog and Qureshi proved that dstab(I) < dim R and they asked whether dstab(I) < dim R for all Stanley-Reisner ideals I in R (see [10]). For a graph G, if every its connected component is nonbipartite, then we can see that dstab(I(G)) < dim R from [4]. In general, there is not an absolute bound of dstab(I(G)) even in the case G is a tree (see [20]). In this paper we will establish a bound of dstab(I(G)) for any graph G. In particular, dstab(I(G)) < dim R.1991 Mathematics Subject Classification. 13D45, 05C90, 05E40, 05E45.

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Depths of Powers of the Edge Ideal of a Tree

Cited by 61 publications

References 18 publications

Stanley depth of powers of the edge ideal of a forest

Stanley depth of powers of the edge ideal of a forest

Squarefree Monomial Ideals that Fail the Persistence Property and Non-increasing Depth

Stability of depths of powers of edge ideals

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