2021
DOI: 10.1007/s10801-021-01088-w
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Cohen–Macaulay binomial edge ideals and accessible graphs

Abstract: 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 … Show more

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Cited by 14 publications
(25 citation statements)
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“…This class of ideals was introduced independently in [15] and [23] and has been extensively studied in the last decade. In [4,Example 7.6], the first three authors exhibit a graph G such that the Betti numbers of J G depend on the field. However, it is still unknown whether the projective dimension or the regularity of J G may depend on the characteristic.…”
Section: Examples and Questionsmentioning
confidence: 99%
“…This class of ideals was introduced independently in [15] and [23] and has been extensively studied in the last decade. In [4,Example 7.6], the first three authors exhibit a graph G such that the Betti numbers of J G depend on the field. However, it is still unknown whether the projective dimension or the regularity of J G may depend on the characteristic.…”
Section: Examples and Questionsmentioning
confidence: 99%
“…Specifically, they have proved that J G is strongly unmixed ⇒ J G is Cohen-Macaulay ⇒ G is accessible. Moreover, they conjectured [5,Conjecture 1.1] on the equivalency of these three properties. In [17], the authors showed if R/J G satisfies Serre's condition S 2 , then G is accessible.…”
Section: Introductionmentioning
confidence: 99%
“…We are interested to classify those G for which J G is Cohen-Macaulay. Although, several works have been done in this direction (see [1], [3], [4], [5], [8], [9], [12], [17], [15], [21], [22]), but full characterization of Cohen-Macaulay binomial edge ideals is still widely open.…”
Section: Introductionmentioning
confidence: 99%
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“…, y n ], where K is a field. The connection between combinatorial invariants associated to G and algebraic invariants associated to J G has been of intense study in the past one decade (see [1,3,4,6,9,10,14,16,17,18] for a partial list). In [1], Banerjee and Núñez Betancourt proved that for a connected non-complete graph G, depth(S/J G ) ≤ n + 2 − κ(G), where κ(G) = min{|T | : T ⊂ [n] and G [n]\T is disconnected} is the graph connectivity.…”
Section: Introductionmentioning
confidence: 99%