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A bi-CM bipartite graph.
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In this paper we consider bi-Cohen-Macaulay graphs, and give a complete
classification of such graphs in the case they are bipartite or chordal.
General bi-Cohen-Macaulay graphs are classified up to separation. The
inseparable bi-Cohen-Macaulay graphs are determined. We establish a bijection
between the set of all trees and the set of inseparable b...
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Citations
... In other words, the bi-CM property of edge ideals does not depend on the field K. This also follows from the work of Herzog and Rahimi [26] [Corollary 1. Notice that x 1 > x 2 > x 3 > x 4 > x 5 is a perfect elimination order of G c , so that G c is chordal (Theorem 4). ...
In this paper, we give a new criterion for the Cohen–Macaulayness of vertex splittable ideals, a family of monomial ideals recently introduced by Moradi and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and provides an inductive way of checking the Cohen–Macaulay property. As a result, we obtain characterizations for Gorenstein, level and pseudo-Gorenstein vertex splittable ideals. Furthermore, we provide new and simpler combinatorial proofs of known Cohen–Macaulay criteria for several families of monomial ideals, such as (vector-spread) strongly stable ideals and (componentwise) polymatroidals. Finally, we characterize the family of bi-Cohen–Macaulay graphs by the novel criterion for the Cohen–Macaulayness of vertex splittable ideals.
... In other words, the bi-Cohen-Macaulay property of edge ideals does not depend on the field K. This also follows from the work of Herzog and Rahimi [19,Corollary 1.2 (d)] where other classifications of the bi-Cohen-Macaulay graphs are given. ...
The Cohen-Macaulay vertex splittable ideals are characterized. As a consequence, we recover several Cohen-Macaulay classifications of families of monomial ideals known in the literature by new simpler combinatorial proofs.
... Hence I ∆ is bi-CM if I ∆ and its Alexander dual I ∨ ∆ are CM. Recently, Herzog and Rahimi in [15] studied the bi-CM graphs and they called G is bi-CM if I(G) is bi-CM. They characterized all bi-CM bipartite graphs. ...
... The Herzog-Hibi's theorem [11] implies that the square-free monomial ideal I is bi-SCM if and only if I is a SCM ideal with componentwise linear. In this paper, we study bi-SCM bipartite graphs as a generalization of Herzog and Rahimi's theorem [15] and as consequence we classify all bi-SCM tree graphs. Furthermore, we determine the projective dimension of bi-SCM bipartite graphs. ...
... Set a = b + y 1 p, where a is an edge ideal of G , p = (x 1 , . . . , x n−1 ) and b is bi-CM (see [15,Theorem 3]). Since a = (b, y 1 ) ∩ p, we have a ∨ = (y 1 b ∨ , p ∨ ). ...
Let [Formula: see text] be the polynomial ring over a field [Formula: see text], [Formula: see text] be a graph on vertex set [Formula: see text] and [Formula: see text] be its edge ideal. In this paper, we study bi-sequentially Cohen–Macaulay (bi-SCM) bipartite graphs and as consequence we classify all bi-SCM tree graphs. Furthermore, if [Formula: see text] is bi-SCM then we determine the projective dimension of [Formula: see text]. Moreover, we give some examples.
... Theorem 4.1 gives precise conditions for when ∆ is both linear resolution and Cohen-Macaulay. These have been studied in the literature, for example [9,11]. For example, the Bi-Cohen-Macaulay bipartite graphs describe in [11] satisfy the conditions on the number of facets or minimal non-faces given above. ...
... These have been studied in the literature, for example [9,11]. For example, the Bi-Cohen-Macaulay bipartite graphs describe in [11] satisfy the conditions on the number of facets or minimal non-faces given above. In particular td(∆) ≥ 0. Furthermore, the equality occurs for some j if and only if ∆ is the clique complex of a chordal graph with exactly c + 1 maximal cliques of size n − c. ...
Fix a field $k$. When $\Delta$ is a simplicial complex on $n$ vertices with Stanley-Reisner ideal $I_\Delta$, we define and study an invariant called the $\textit{type defect}$ of $\Delta$. Except when $\Delta$ is of a single simplex, the type defect of $\Delta$, $\textrm{td}(\Delta)$, is the difference $ \dim_k \textrm{Tor}_c^S(S/ I_\Delta,k) - c$, where $c$ is the codimension of $\Delta$ and $S = k[x_1, \ldots x_n]$. We show that this invariant admits surprisingly nice properties. For example, it is well-behaved when one glues two complexes together along a face. Furthermore, $\Delta$ is Cohen-Macaulay if $\textrm{td}(\Delta) \leq 0$. On the other hand, if $\Delta$ is a simple graph (viewed as a one-dimensional complex), then $\textrm{td}(\Delta') \geq 0$ for every induced subgraph $\Delta'$ of $\Delta$ if and only if $\Delta$ is chordal. Requiring connected induced subgraphs to have type defect zero allows us to define a class of graphs that we call $\textit{treeish}$, 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.
... 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. ...
... As a complete classification of all bi-CM graphs seems to be impossible, they gave a classification of all bi-CM graphs up to separation. As they showed in [HR16,Theorem 11], the generic graph G T of a tree T provides a bi-CM inseparable model. In particular, G T is bi-CM. ...
... 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 ...
When $\mathcal{C}$ is a chordal clutter in the sense of Woodroofe or Emtander, we show that the complement clutter is edgewise strongly shellable. When $\mathcal{C}$ 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 $G_T$ 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. \end{abstract}
... Their boundaries are therefore triangulations of spheres, and this class of sphere triangulations comprehensively generalizes the class of Bier spheres [3]. The notion of separation is further investigated in [24] and in [1], which shows that separation corresponds to a deformation of the monomial ideal, and identifies the deformation directions in the cotangent cohomology it corresponds to. In [16] deformations of letterplace ideals L(2, P ) are computed when the Hasse diagram has the structure of a rooted tree. ...
To a natural number n, a finite partially ordered set P and a poset ideal in the poset of isotonian maps from P to the chain on n elements, we associate two monomial ideals, the letterplace ideal and the co-letterplace ideal . These ideals give a unified understanding of a number of ideals studied in monomial ideal theory in recent years. By cutting down these ideals by regular sequences of variable differences we obtain: multichain ideals and generalized Hibi type ideals, initial ideals of determinantal ideals, strongly stable ideals, d-partite d-uniform ideals, Ferrers ideals, edge ideals of cointerval d-hypergraphs, and uniform face ideals.
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 CMt 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 CMt simplicial complexes.
We give a generalization of Eagon-Reiner’s theorem relating Betti numbers of the Stanley-Reisner ideal of a simplicial complex and the CMt property of its Alexander dual. Then we characterize bi-CMt bipartite graphs and bi-CMt chordal graphs. These are generalizations of recent results due to Herzog and Rahimi.
