Generalized cover ideals and the persistence property

Abstract

Let I be a square-free monomial ideal in R = k[x 1 , . . . , x n ], and consider the sets of associated primes Ass(I s ) for all integers s ≥ 1. Although it is known that the sets of associated primes of powers of I eventually stabilize, there are few results about the power at which this stabilization occurs (known as the index of stability). We introduce a family of square-free monomial ideals that can be associated to a finite simple graph G that generalizes the cover ideal construction. When G is a tree, we explicitly determine Ass(I s ) for all s ≥ 1. As consequences, not only can we compute the index of stability, we can also show that this family of ideals has the persistence property.

Normality and associated primes of Closed neighborhood ideals and dominating ideals

“…, x r f }, say u 3 = x 1 f . Then, we get x 3 We are in a position to state another main result of this section in the subsequent theorem. To see this, we have to use the corollary below as our main tool.…”

“…It has already been introduced, in [3], the notion of partial t-cover ideals of finite simple graphs. We first recall the definition of partial t-cover ideals in the following definition.…”

Symbolic powers of cover ideal of very well-covered and bipartite graphs

“…In this paper we study the minimal free resolution of the powers of cover ideal of graphs. Cover ideal of a graph is the Alexander dual of its edge ideal and has been studied by several authors (see e.g., [4], [6], [10], [11], [22]).…”