Bi-Cohen-Macaulay graphs

Herzog, Jürgen; Rahimi, Ahad

doi:10.37236/5557

Cited by 7 publications

(6 citation statements)

References 14 publications

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“…bi-CM t , etc.). We are now going to give two characterizations of bi-CM t bipartite graphs and bi-CM t chordal graphs generalizing results of Herzog and Rahimi on bi-Cohen-Macaulay bipartite graphs and bi-Cohen-Macaulay chordal graphs (see [16]). The first one is as follows.…”

Section: Any Bipartite CM T Graph Is Obtained By Such a Replacement Of Complete Bipartite Graphs In A Unique Bipartite Cohen-macaulay Gramentioning

confidence: 99%

A Brief Survey on Pure Cohen–Macaulayness in a Fixed Codimension

et al. 2021

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A concept of Cohen-Macaulay in codimension t is defined and characterized for arbitrary finitely generated modules and coherent sheaves by Miller, Novik, and Swartz in 2011. Soon after, Haghighi, Yassemi, and Zaare-Nahandi defined and studied CM t simplicial complexes, which is the pure version of the abovementioned concept and naturally generalizes both Cohen-Macaulay and Buchsbaum properties. The purpose of this paper is to survey briefly recent results of CM t simplicial complexes.

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Section: Any Bipartite CM T Graph Is Obtained By Such a Replacement Of Complete Bipartite Graphs In A Unique Bipartite Cohen-macaulay Gramentioning

confidence: 99%

A Brief Survey on Pure Cohen–Macaulayness in a Fixed Codimension

et al. 2021

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“…Theorem 1.3 gives precise conditions for when ∆ both has a linear resolution and is Cohen-Macaulay. These have been studied in the literature, for example [8,11]. For example, the Bi-Cohen-Macaulay bipartite graphs described in [11] satisfy the conditions on the number of facets or minimal non-faces given in Theorem 1.3.…”

Section: Ideals With Linear Resolutionmentioning

confidence: 99%

“…These have been studied in the literature, for example [8,11]. For example, the Bi-Cohen-Macaulay bipartite graphs described in [11] satisfy the conditions on the number of facets or minimal non-faces given in Theorem 1.3. Corollary 4.2.…”

Section: Ideals With Linear Resolutionmentioning

confidence: 99%

The type defect of a simplicial complex

Dao

Schweig

2019

Journal of Combinatorial Theory, Series A

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Fix a field k. When ∆ is a simplicial complex on n vertices with Stanley-Reisner ideal I ∆ , we define and study an invariant called the type defect of ∆. Except when ∆ is a single simplex, the type defect of ∆, td(∆), is the difference dim Tor S c (S/I ∆ , k) − c, where c is the codimension of ∆ and S = k[x 1 , . . . x n ]. We show that this invariant admits surprisingly nice properties. For example, it is wellbehaved when one glues two complexes together along a face. Furthermore, ∆ is Cohen-Macaulay if td(∆) ≤ 0. On the other hand, if ∆ is a simple graph (viewed as a one-dimensional complex), then td(∆ ) ≥ 0 for every induced subgraph ∆ of ∆ if and only if ∆ is chordal. Requiring connected induced subgraphs to have type defect zero allows us to define a class of graphs that we call treeish, and which we generalize to simplicial complexes. We then extend some of our chordality results to higher dimensions, proving sharp lower bounds for most Betti numbers of ideals with linear resolution, and classifying when equalities occur. As an application, we prove sharp lower bounds for Betti numbers of graded ideals (not necessarily monomial) with linear resolution. Definition 1.1. We define the modified type of ∆ to be: mtype(∆) = dim Tor S c (S/I ∆ , k) − dim Tor S c (S/I ∆ , k) c

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“…On the other hand, after [HR16], we can associate a special graph G T to the tree T . The vertices of the graph G T is given by VpG T q " x i,j , x j,i | ti, ju P EpT q ( .…”

Section: Perfect Elimination Order Chordality and Ess Graphsmentioning

confidence: 99%

“…Recently, Herzog and Rahimi [HR16] studied the bi-Cohen-Macaulay (abbreviated as bi-CM) property of finite simple graphs. Recall that a simplicial complex ∆ is called bi-CM, if both ∆ and its Alexander dual complex ∆ _ are CM.…”

Section: Introductionmentioning

confidence: 99%

Edgewise strongly shellable clutters

Guo

Shen

2018

J. Algebra Appl.

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When C is a chordal clutter in the sense of Woodroofe or Emtander, we show that the complement clutter is edgewise strongly shellable. When C is indeed a finite simple graph, we study various characterizations of chordal graphs from the point of view of strong shellability. In particular, the generic graph GT of a tree is shown to be bi-strongly shellable. We also characterize edgewise strongly shellable bipartite graphs in terms of constructions from upward sequences.2010 Mathematics Subject Classification. 05E40, 05E45, 13C14, 05C65.

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Bi-Cohen-Macaulay graphs

Cited by 7 publications

References 14 publications

A Brief Survey on Pure Cohen–Macaulayness in a Fixed Codimension

A Brief Survey on Pure Cohen–Macaulayness in a Fixed Codimension

The type defect of a simplicial complex

Edgewise strongly shellable clutters

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