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

Let $I=I(D)$ be the edge ideal of a weighted oriented graph $D$, let $G$ be the underlying graph of $D$, and let $I^{(n)}$ be the $n$-th symbolic power of $I$ defined using the minimal primes of $I$. We prove that $I^2=I^{(2)}$ if and only…

Let $G$ be a simple graph on $n$ vertices and let $J_{G,m}$ be the generalized binomial edge ideal associated to $G$ in the polynomial ring $K[x_{ij}, 1\le i \le m, 1\le j \le n]$. We classify the Cohen-Macaulay generalized binomial edge…

We prove the multiplicity bounds conjectured by Herzog-Huneke-Srinivasan and Herzog-Srinivasan in the following cases: the strong conjecture for edge ideals of bipartite graphs, and the weaker Taylor bound conjecture for all quadratic…

Conca and Varbaro (Invent. Math. 221 (2020), no. 3) showed the equality of depth of a graded ideal and its initial ideal in a polynomial ring when the initial ideal is square-free. In this paper, we give some beautiful applications of this…

Binomial edge ideals IG of a graph G were introduced by [4]. They found some classes of graphs G with the property that IG is a Cohen-Macaulay ideal. This might happen only for few classes of graphs. A certain generalization of being…

Let $G$ be a finite simple graph on the vertex set $[n] = \{ 1, \ldots, n \}$ and $K[X, Y] = K[x_1, \ldots, x_n, y_1, \ldots, y_n]$ the polynomial ring in $2n$ variables over a field $K$ with each $\mathrm{deg} x_i = \mathrm{deg} y_j = 1$.…

A very well-covered graph is an unmixed graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. We study these graphs by means of Betti splittings and mapping cone constructions. We show…

We consider vector-spread Borel ideals. We show that these ideals have linear quotients and thereby we determine the graded Betti numbers and the bigraded Poincar\'e series. A characterization of the extremal Betti numbers of such a class…

Inspired by the notion of K\"onig graphs we introduce graded ideals of K\"onig type with respect to a monomial order $<$. It is shown that if $I$ is of K\"onig type, then the Cohen--Macaulay property of $\ini_<(I)$ does not depend on the…

Let $G$ be a simple graph on $n$ vertices. Let $L_G \text{ and } \mathcal{I}_G \: $ denote the Lov\'asz-Saks-Schrijver(LSS) ideal and parity binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1,\ldots, x_n, y_1, \ldots,…

In this paper, we first give some sufficient criteria for normality of monomial ideals. As applications, we show that closed neighborhood ideals of complete bipartite graphs are normal, and hence satisfy the (strong) persistence property.…

Let $G$ be a graph with $n$ vertices, $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$ and $I(G)$ denote the edge ideal of $G$. For every collection $\mathcal{H}$ of connected graphs with…

Let $G$ be a finite simple graph, and $J_G$ denote the binomial edge ideal of $G$. In this article, we first compute the $\mathrm{v}$-number of binomial edge ideals corresponding to Cohen-Macaulay closed graphs. As a consequence, we obtain…

We study the depth of classes of binomial edge ideals and classify all closed graphs whose binomial edge ideal is Cohen--Macaulay.

We study unmixed and Cohen-Macaulay properties of the binomial edge ideal of some classes of graphs. We compute the depth of the binomial edge ideal of a generalized block graph. We also characterize all generalized block graphs whose…

Let $S = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$ and $I \subset S$ a homogeneous ideal of $S$ with $\dim S/I = d$. The Hilbert series of $S/I$ is of the form…

A celebrated theorem of Fr\"oberg gives a complete combinatorial classification of quadratic square-free monomial ideals with a linear resolution. A generalization of this theorem to higher degree square-free monomial ideals is an active…

Let $I(G)$ be the edge ideal of a gapfree graph $G$. An open conjecture of Nevo and Peeva states that $I(G)^q$ has linear resolution for $q\gg 0$. We present a promising approach to this challenging conjecture by investigating the stronger…

In this paper, we explain the regularity, projective dimension and depth of edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a $C_5$-free vertex decomposable graph $G$, $\T{reg}(R/I(G))= c_G$, where…

We classify the bipartite graphs $G$ whose binomial edge ideal $J_G$ is Cohen-Macaulay. The connected components of such graphs can be obtained by gluing a finite number of basic blocks with two operations. In this context we prove the…