Related papers: Squarefree powers of closed neighborhood ideals

We introduce a family of squarefree monomial ideals associated to finite simple graphs, whose monomial generators correspond to closed neighborhood of vertices of the underlying graph. Any such ideal is called the closed neighborhood ideal…

Let $G$ be a finite simple graph and let $I(G)$ denote its edge ideal. For $q \ge 1$, the $q$-th squarefree power $I(G)^{[q]}$ is generated by squarefree monomials corresponding to matchings of size $q$ in $G$. We denote by…

Squarefree powers of edge ideals are intimately related to matchings of the underlying graph. In this paper we give bounds for the regularity of squarefree powers of edge ideals, and we consider the question of when such powers are linearly…

Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive…

We study the asymptotic behavior of the Castelnuovo-Mumford regularity along chains of graded ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups. A linear upper bound for the regularity of…

We consider the closed neighborhood ideal of square of the path graph and study some of its algebraic and homological invariants. We compute the height, the projective dimension and the Castelnuovo-Mumford regularity. We prove that these…

We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or…

Assume that $G$ is a graph with edge ideal $I(G)$ and matching number ${\rm match}(G)$. For every integer $s\geq 1$, we denote the $s$-th squarefree power of $I(G)$ by $I(G)^{[s]}$. It is shown that for every positive integer $s\leq {\rm…

In this article, we obtain an upper bound for the Castelnuovo-Mumford regularity of powers of an ideal generated by a homogeneous quadratic sequence in a polynomial ring in terms of the regularity of its related ideals and degrees of its…

Castelnuovo-Mumford regularity is a measure of algebraic complexity of an ideal. Regularity of monomial ideals can be investigated combinatorially. We use a simple graph decomposition and results from structural graph theory to prove,…

In this article, we study the squarefree powers of facet ideals associated with simplicial trees. Specifically, we examine the linearity of their minimal free resolution and their regularity. Additionally, we investigate when the first…

Let $G$ be a $(C_4, 2K_2)$-free graph with edge ideal $I(G)\subset \Bbbk[x_1,\dots , x_n]$. We show that $I(G)^s$ has linear resolution for every $s\geq 2$. Also, we show that every power of the vertex cover ideal of $G$ has linear…

In this paper we study the Castelnuovo-Mumford regularity of the path ideals of finite simple graphs. We find new upper bounds for various path ideals of gap free graphs. In particular we prove that the path ideals of gap free and claw…

Let $G$ be a finite simple graph and $NI(G)$ be the closed neighborhood ideal of $G$ in the polynomial ring $S=K[V(G)]$. In this paper, we study the Castelnuovo-Mumford regularity, projective dimension and Cohen-Macaulayness of this ideal.…

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal in a polynomial ring over a field $\mathbb{K}$. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regularity of symbolic powers of certain…

Let $G$ be a simple graph and $I(G)$ be its edge ideal. In this article, we study the Castelnuovo-Mumford regularity of symbolic powers of edge ideals of join of graphs. As a consequence, we prove Minh's conjecture for wheel graphs,…

Let $G$ be a finite simple graph and let $NI(G)$ denote the closed neighborhood ideal of $G$ in a polynomial ring $R$. We show that if $G$ is a forest, then the Castelnuovo-Mumford regularity of $R/NI(G)$ is the same as the matching number…

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{reg} I(G)$ and the asymptotic linear function $\text{reg} I(G)^q$, for $q \geq 1,$…

We study the topology of the lcm-lattice of edge ideals and derive upper bounds on the Castelnuovo-Mumford regularity of the ideals. In this context it is natural to restrict to the family of graphs with no induced 4-cycle in their…

We study the closed neighborhood ideals and the dominating ideals of graphs, in particular, of trees and cycles. We prove that the closed neighborhood ideals and the dominating ideals of trees are normally torsion-free. The closed…