Related papers: Comparing Powers of Edge Ideals

Let $D$ be a weighted oriented graph with the underlying graph $G$ and $I(D), I(G) $ be the edge ideals corresponding to $D$ and $G$ respectively. We show that the regularity of edge ideal of a certain class of weighted oriented graph…

Let $G$ be a finite simple graph and $J(G)$ denote its cover ideal in a polynomial ring over a field $\mathbb{K}$. In this paper, we show that all symbolic powers of cover ideals of certain vertex decomposable graphs have linear quotients.…

Let $G$ be a finite simple graph and $J(G)$ denote its vertex cover ideal in a polynomial ring over a field. % $\mathbb{K}$. The $k$-th symbolic power of $J(G)$ is denoted by $J(G)^{(k)}$. In this paper, we give a criteria for cover ideals…

Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms…

Let $\varphi : S = k[y_0,..., y_n] \to R = k[y_0,...,y_n]$ be given by $y_i \to f_i$ where $f_0,...,f_n$ is an $R$-regular sequence of homogeneous elements of the same degree. A recent paper shows for ideals, $I_\Delta \subseteq S$, of…

A graded ideal $I$ in $\mathbb{K}[x_1,\ldots,x_n]$, where $\mathbb{K}$ is a field, is said to have almost maximal finite index if its minimal free resolution is linear up to the homological degree $\mathrm{pd}(I)-2$, while it is not linear…

Assume that $G$ is a graph with cover ideal $J(G)$. For every integer $k\geq 1$, we denote the $k$-th symbolic power of $J(G)$ by $J(G)^{(k)}$. We provide a sharp upper bound for the regularity of $J(G)^{(k)}$ in terms of the star packing…

We study the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. It follows from the theorems of Dirac and of Koml\'os, Sark\"ozy, and Szemer\'edi…

Let $R = \mathbb{K}[x_1, \ldots, x_n]$ be a polynomial ring over a field $\mathbb{K}$, and let $I \subseteq R$ be a monomial ideal of height $h$. We provide a formula for the multiplicity of the powers of $I$ when all the primary ideals of…

In this paper we use polarization to study the behavior of the depth and regularity of a monomial ideal $I$, locally at a variable $x_i$, when we lower the degree of all the highest powers of the variable $x_i$ occurring in the minimal…

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…

The resurgence and asymptotic resurgence of an ideal in a polynomial ring are two statistics which measure the relationship between its regular and symbolic powers. We address two aspects of resurgence which can be studied via asymptotic…

In this paper we provide some exact formulas for the regularity of powers of edge ideals of vertex-weighted oriented cycles and vertex-weighted unicyclic graphs. These formulas are functions of the weight of vertices and the number of…

Let $I$ be a graded ideal of a standard graded polynomial ring $S$ with coefficients in a field $K$. The asymptotic behaviour of the $\text{v}$-number of the powers of $I$ is investigated. Natural lower and upper bounds which are linear…

Let $n\geq 2$ be an integer, and let $i\in\{0,...,n-1\}$. An $i$-th dimension edge in the $n$-dimensional hypercube $Q_n$ is an edge ${v_1}{v_2}$ such that $v_1,v_2$ differ just at their $i$-th entries. The parity of an $i$-th dimension…

Fermat ideals define planar point configurations that are closely related to the intersection locus of the members of a specific pencil of curves. These ideals have gained recent popularity as counterexamples to some proposed containments…

Let $R$ be a formal power series ring over a field, with maximal ideal $\mathfrak m$, and let $I$ be an ideal of $R$ such that $R/I$ is Artinian. We study the iterated socles of $I$, that is the ideals which are defined as the largest ideal…

We present a close relationship between matching number, covering numbers and their fractional versions in combinatorial optimization and ordinary powers, integral closures of powers, and symbolic powers of monomial ideals. This…

Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G) \subset S$ the edge ideal of a finite graph $G$ on $n$ vertices. Given a vector $\mathfrak{c}\in\mathbb{N}^n$ and an integer $q\geq 1$, we…

We show that the Conjecture of Harbourne and Huneke, $I^{(Nr-(N-1))} \subset M^{(r-1)(N-1)}I^{r}$ holds for ideals of generic (simple) points in $\PP^3$. As a result, for such ideals we prove the following bounds, which can be recognized as…