Regularity of binomial edge ideals of Cohen-Macaulay bipartite graphs

Jayanthan, A. V.; Kumar, Arvind

doi:10.1080/00927872.2019.1596278

Cited by 31 publications

(20 citation statements)

References 16 publications

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“…Algebraic properties and invariants of binomial edge ideals have been studied by many authors, see [4, 16, 21]. In particular, establishing a relationship between Castelnuovo–Mumford regularity (simply regularity), projective dimension, Hilbert series of binomial edge ideals and combinatorial invariants associated with graphs is an active area of research, see [1, 11, 13, 18, 20]. In general, the algebraic invariants such as regularity and depth of

J_{G}

are hard to compute.…”

Section: Introductionmentioning

confidence: 99%

Depth and extremal Betti number of binomial edge ideals

Kumar

Sarkar

2020

Mathematische Nachrichten

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Let G be a simple graph on the vertex set [n] and let JG be the corresponding binomial edge ideal. Let G=v∗H be the cone of v on H. In this article, we compute all the Betti numbers of JG in terms of the Betti numbers of JH and as a consequence, we get the Betti diagram of wheel graph. Also, we study Cohen–Macaulay defect of S/JG in terms of Cohen–Macaulay defect of SH/JH and using this we construct a graph with Cohen–Macaulay defect q for any q≥1. We obtain the depth of binomial edge ideal of join of graphs. Also, we prove that for any pair (r,b) of positive integers with 1≤b

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J_{G}

are hard to compute.…”

Section: Introductionmentioning

confidence: 99%

Depth and extremal Betti number of binomial edge ideals

Kumar

Sarkar

2020

Mathematische Nachrichten

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“…Example 5. 15 In general, it is possible that the equivalent conditions of Proposition 5.14 hold, but G is not accessible. For instance, if G is the graph in Fig.…”

Section: Remark 510mentioning

confidence: 99%

“…In [5, Corollary 6.2], we proved that for bipartite graphs, binomial edge ideals are the same up to isomorphism as Lovász-Saks-Schrijver ideals in two sets of variables (see [14]), permanental edge ideals (see [14,Section 3]) and parity binomial edge ideals (see [17]), but this does not hold for non-bipartite graphs. Hence, even though Conjecture 1.1 would prove the field-independence of Cohen-Macaulayness In [15], Jayanthan and Kumar compute the regularity of Cohen-Macaulay binomial edge ideals of bipartite graphs using the explicit description of these graphs given in [5,Theorem 6.1 (c)]. By the proof of Corollary 6.9, these graphs are traceable.…”

Section: Further Remarks and Problemsmentioning

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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“…There are several attempts at this problem available for some families of graphs. Some papers in this direction are [8], [19], [20], [15], [2], [3], [21], [14], [1], [9], and [4]. In the latter, the authors introduce two combinatorial properties strictly related to the Cohen-Macaulayness of binomial edge ideals: accessibility and strongly unmixedness.…”

Section: Introductionmentioning

confidence: 99%

$(S_2)$-condition and Cohen-Macaulay binomial edge ideals

Lerda¹,

Mascia²,

Rinaldo³

et al. 2021

Preprint

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We describe the simplicial complex ∆ such that the initial ideal of J G is the Stanley-Reisner ideal of ∆. By ∆ we show that if J G is (S 2 ) then G is accessible. We also characterize all accessible blocks with whiskers of cycle rank 3 and we define a new infinite class of accessible blocks with whiskers for any cycle rank. Finally, by using a computational approach, we show that the graphs with at most 12 vertices whose binomial edge ideal is Cohen-Macaulay are all and only the accessible ones. https://mathworld.wolfram.com/HelmGraph.html.

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Regularity of binomial edge ideals of Cohen-Macaulay bipartite graphs

Cited by 31 publications

References 16 publications

Depth and extremal Betti number of binomial edge ideals

Depth and extremal Betti number of binomial edge ideals

Cohen–Macaulay binomial edge ideals and accessible graphs

$(S_2)$-condition and Cohen-Macaulay binomial edge ideals

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