Related papers: The edge ideals of $\bf{t}$-spread $d$-partite hyp…

In this paper, we study the componentwise linearity of edge ideals of weighted oriented graphs. We show that if $D$ is a weighted oriented graph whose edge ideal $I(D)$ is componentwise linear, then the underlying simple graph $G$ of $D$ is…

This paper analyzes the cohomological dimension of the generalized binomial edge ideal $\calJ_{K_m,G}$ for a complete $r$-partite graph $G$. Additionally, the Krull dimension, the depth, the Castelnuovo--Mumford regularity, the Hilbert…

We consider homogeneous binomial ideals $I=(f_1,\ldots,f_n)$ in $K[x_1, \ldots, x_n]$, where $f_i = a_i x_i^{d_i} - b_i m_i$ and $a_i \neq 0$. When such an ideal is a complete intersection, we show that the monomials which are not divisible…

In this article, we investigate the combinatorial and algebraic properties of the lcm-lattice associated with the edge ideal of a hypergraph. Let $\H$ be a hypergraph, $I(\H)$ its corresponding edge ideal in a polynomial ring in $n$…

The path ideal (of length t >=2) of a graph G is the monomial ideal, denoted I_t(G), whose generators correspond to the directed paths of length t in G. We study some of the algebraic properties of I_t(G) when G is a tree. We first show…

We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications,…

In the present paper, we aim to classify monomial ideals whose all matching powers are Cohen-Macaulay. We especially focus our attention on edge ideals. The Cohen-Macaulayness of the last matching power of an edge ideal is characterized,…

We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an…

Let $K$ be a field, $V$ a finite dimensional $K$-vector space and $E$ the exterior algebra of $V$. We analyze iterated mapping cone over $E$. If $I$ is a monomial ideal of $E$ with linear quotients, we show that the mapping cone…

Let $I$ be the edge ideal of a Cohen-Macaulay tree of dimension $d$ over a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_{d},y_1,\ldots,y_d]$. We prove that for all $t \ge 1$, $$\operatorname{depth} (S/I^t) = \operatorname{max} \{d -t + 1, 1…

We associate to every graph a linear program for packings of vertex disjoint paths. We show that the optimal primal and dual values of the corresponding integer program are the binomial grade and height of the binomial edge ideal of the…

Let $G$ be a simple graph and $I(X_G)=\varphi^{-1}(x_i^2-x_j^2 : i,j\in V_G)$, where $\varphi \colon K[E_G]\to K[V_G]$ is the homomorphism that sends an edge to the product of its vertices. The ideal $I(X_G)$ is Cohen--Macaulay,…

Let $G$ be a finite simple graph, and let $I(G)$ denote its edge ideal. In this paper, we investigate the asymptotic behavior of the syzygies of powers of edge ideals through the lens of homological shift ideals $\text{HS}_i(I(G)^k)$. We…

In this paper, we characterize the Cohen-Macaulayness of the second power $I(G_\omega)^2$ of the weighted edge ideal $I(G_\omega)$ when the underlying graph $G$ is a very well-covered graph. We also characterize the Cohen-Macaulayness of…

Each partition $\lambda = (\lambda_1, \lambda_2, ..., \lambda_n)$ determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed Ferrers ideal, is a squarefree monomial ideal that is generated by…

We introduce the concept of $t$-spread monomials and $t$-spread strongly stable ideals. These concepts are a natural generalization of strongly stable and squarefree strongly stable ideals. For the study of this class of ideals we use the…

We provide a new combinatorial approach to study the minimal free resolutions of edge ideals, that is, quadratic square-free monomial ideals. With this method we can recover most of the known results on resolutions of edge ideals with…

Let $I\subset R=K[x_1, \ldots, x_n]$ be a square-free monomial ideal, $\mathfrak{q}$ be a prime monomial ideal in $R$, $h$ be a square-free monomial in $R$ with $\mathrm{supp}(h) \cap (\mathrm{supp}(\mathfrak{q}) \cup…

We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones, and we show the extremal Betti numbers of some classes of Cohen-Macaulay binomial edge ideals: Cohen-Macaulay bipartite and fan graphs.…

The $\mathrm{v}$-number of a graded ideal $I\subseteq R$, denoted by $\mathrm{v}(I)$, is the minimum degree of a polynomial $f$ for which $I:f$ is a prime ideal. Jaramillo and Villarreal (J Combin Theory Ser A 177:105310, 2021) studied the…