Regularity of powers of quadratic sequences with applications to binomial ideals

Jayanthan, A. V.; Kumar, Arvind; Sarkar, Rajib

doi:10.1016/j.jalgebra.2020.08.004

Cited by 20 publications

(37 citation statements)

References 32 publications

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“…As one can observe, for both binomial edge ideals and parity binomial edge ideals, results on the regularity of their powers are not abundant. The work in [8] and [17] are perhaps the only ones along this research line that we can find so far. That is the reason why we want to push forward a little bit.…”

Section: Introductionmentioning

confidence: 96%

“…In contrast, little is known about the related binomial ideals. Only recently, Jayanthan et al in [17] provided the exact formulas for the regularity of powers of binomial edge ideals of several simple graphs such as cycle graphs, star graphs and balloon graphs. And Ene et al in [8] studied the same problems for connected closed graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Notice that Raghavan in [31] introduced the notion of quadratic sequence, which generalized the d-sequence of Huneke in [15]. And the main result of Jayanthan et al in [17] is an upper bound for the regularity of powers of an ideal generated by a quadratic sequence. Inspired by their work, in this paper, we will provide exact formulas for the regularity of powers of I, when I is an almost complete intersection ideal generated by a d-sequence (Corollary 3.8).…”

Section: Introductionmentioning

confidence: 99%

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Regularity of powers of (parity) binomial edge ideals

Shen¹,

Zhu²

2021

Preprint

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In this paper, we provide exact formulas for the Castelnuovo-Mumford regularity of powers of an almost complete intersection ideal I which is generated by a homogeneous d-sequence. As applications, when I is an almost complete intersection, taking the form of the (parity) binomial edge ideal of a connected graph, we can describe explicitly formulas for reg(I t ) for t ≥ 2. The only exception is when I is the parity binomial edge ideal of a graph which is obtained by adding an edge between two disjoint odd cycles.

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Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Regularity of powers of (parity) binomial edge ideals

Shen¹,

Zhu²

2021

Preprint

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“…Exploiting the combinatorics of the underlying graph, many authors have studied algebraic and homological properties and invariants of these ideals, such as their regularity [16,19,21,28,29,31], depth [3,30], local cohomology [1], universal Gröbner basis [2] and licci property [9]. In particular, their primary decomposition and unmixedness can be characterized combinatorially.…”

Section: Introductionmentioning

confidence: 99%

“…Corollary 5 16. Let G be an accessible graph and v be a cut vertex of G such that J G\{v} is unmixed.…”

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confidence: 99%

Cohen–Macaulay binomial edge ideals and accessible graphs

2021

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The cut sets of a graph are special sets of vertices whose removal disconnects the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible graphs as the graphs with unmixed binomial edge ideal and whose cut sets form an accessible set system. We prove that the graphs whose binomial edge ideal is Cohen–Macaulay are accessible and we conjecture that the converse holds. We settle the conjecture for large classes of graphs, including chordal and traceable graphs, providing a purely combinatorial description of Cohen–Macaulayness. The key idea in the proof is to show that both properties are equivalent to a further combinatorial condition, which we call strong unmixedness.

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