Edge ideals of Erdös-Rényi random graphs : Linear resolution, unmixedness and regularity

Banerjee, Arindam; Yogeshwaran, D.

doi:10.48550/arxiv.2007.08869

Cited by 2 publications

(3 citation statements)

References 19 publications

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“…Their construction is similar to that of Erdős-Rényi random graphs, and under certain parameters can recover our model. In a recent paper of Banerjee and Yogeshwaran [7] it is shown that the threshold for a coedge ideal to have a linear resolution coincides with the threshold for the ideal to have a linear presentation. They also establish precise threshold results regarding regularity for the case that reg(I G ) = 2.…”

Section: Introductionmentioning

confidence: 94%

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Random subcomplexes and Betti numbers of random edge ideals

Dochtermann¹,

Newman²

2021

Preprint

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We study asymptotic homological properties of random quadratic monomial ideals in a polynomial ring R = K[x 1 , . . . x n ], utilizing methods from the Erdős-Rényi model of random graphs. We consider coedge ideals I G for G ∼ G(n, p) and establish asymptotic results about the algebraic properties of R/I G . Our main results involve fixing the edge probability p = p(n) so that asymptotically almost surely the Krull dimension of R/I G is fixed. Under these conditions we establish various properties regarding the Betti table of R/I G , including sharp bounds on regularity and projective dimension, and distribution of nonzero normalized Betti numbers. These results extend work of Erman-Yang, who studied such ideals in the context of conjectured phenomena in the nonvanishing of asymptotic syzygies. Along the way we establish results regarding subcomplexes of random clique complexes as well as notions of higher-dimensional vertex k-connectivity that may be of independent interest.

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Section: Introductionmentioning

confidence: 94%

“…This implies that, unlike the sparse regime we consider here, R/I G will have more than one extremal Betti number when G ∼ G(n, p) and p is a constant. In addition, the regime p = 1 − λ/n for λ constant is considered in [7], where they establish results about the regularity of R/I G and prove it is order Θ(n).…”

Section: Now Let S Be a Fixed Set Ofmentioning

confidence: 99%

Random subcomplexes and Betti numbers of random edge ideals

Dochtermann¹,

Newman²

2021

Preprint

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show abstract

“…There are more recent interesting results on resolutions of ideals of random graphs, see for example [2,4,5,13,9,8,20].…”

Section: Relation To Papers On Edge Ideals Of Random Graphsmentioning

confidence: 99%

The regularity of almost all edge ideals

Engström¹,

Orlich²

2021

Preprint

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A fruitful contemporary paradigm in graph theory is that almost all graphs that do not contain a certain subgraph have common structural characteristics. The "almost" is crucial, without it there is no structure. In this paper we transfer this paradigm to commutative algebra and make use of deep graph theoretic results. A key tool are the critical graphs introduced by Balogh and Butterfield.We consider edge ideals I G of graphs and their Betti numbers. The numbers of the form β i,2i+2 constitute the "main diagonal" of the Betti table. It is well known that any Betti number β i,j (I G ) below (or equivalently, to the left of) this diagonal is always zero. We identify a certain "parabola" inside the Betti table and call parabolic Betti numbers the entries of the Betti table bounded on the left by the main diagonal and on the right by this parabola. Let β i,j be a parabolic Betti number on the r-th row of the Betti table, for r ≥ 3. Our main results state that almost all graphs G with β i,j (I G ) = 0 can be partitioned into r − 2 cliques and one independent set, and in particular for almost all graphs G with β i,j (I G ) = 0 the regularity of I G is r − 1.

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Edge ideals of Erdös-Rényi random graphs : Linear resolution, unmixedness and regularity

Cited by 2 publications

References 19 publications

Random subcomplexes and Betti numbers of random edge ideals

Random subcomplexes and Betti numbers of random edge ideals

The regularity of almost all edge ideals

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