Cover Ideals Research Articles

Results on the normality of square-free monomial ideals and cover ideals under some graph operations

In this paper, we introduce techniques for producing normal square-free monomial ideals from old such ideals. These techniques are then used to investigate the normality of cover ideals under some graph operations. Square-free monomial ideals that come out as linear combinations of two normal ideals are shown to be not necessarily normal; under such a case we investigate the integral closedness of all powers of these ideals.

Freiman Cover Ideals of Unmixed Bipartite Graphs

An equigenerated monomial ideal $$I$$ in the polynomial ring $$R=k[z_1,\ldots, z_n]$$ is a Freiman ideal if $$\mu(I^2)=\ell(I)\mu(I)-{\ell(I)\choose 2}$$ , where $$\ell(I)$$ is the analytic spread of $$I$$ and $$\mu(I)$$ is the number of the minimal generators of $$I$$ . In this paper we classify all simple connected unmixed bipartite graphs whose cover ideals are Freiman ideals.

On the minimal free resolution of symbolic powers of cover ideals of graphs

For any graph G G , assume that J ( G ) J(G) is the cover ideal of G G . Let J ( G ) ( k ) J(G)^{(k)} denote the k k th symbolic power of J ( G ) J(G) . We characterize all graphs G G with the property that J ( G ) ( k ) J(G)^{(k)} has a linear resolution for some (equivalently, for all) integer k ≥ 2 k\geq 2 . Moreover, it is shown that for any graph G G , the sequence ( r e g ( J ( G ) ( k ) ) ) k = 1 ∞ \big ({\mathrm {reg}}(J(G)^{(k)})\big )_{k=1}^{\infty } is nondecreasing. Furthermore, we compute the largest degree of minimal generators of J ( G ) ( k ) J(G)^{(k)} when G G is either an unmixed of a claw-free graph.

Symbolic powers of cover ideals of graphs and Koszul property

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

Algebraic Properties of Edge Ideals and Cover Ideals of Unbalanced Crown Graphs

In this paper, we introduce the unbalanced crown graphs $${\mathcal {C}}_{m,n}$$ , and compute all graded Betti numbers of edge ideals of these graphs in terms of associated numerical data. As a consequence, we determine the regularity and the projective dimension of edge ideals of these graphs. Also, the Hilbert series of the cover ideals of these graphs is obtained.

Symbolic Powers of Certain Cover Ideals of Graphs

In this paper, we compute the regularity and Hilbert series of symbolic powers of the cover ideal of a graph $G$ when $G$ is either a crown graph or a complete multipartite graph. We also compute the multiplicity of symbolic powers of cover ideals in terms of the number of edges.

Regularity, Rees algebra, and Betti numbers of certain cover ideals

Let $$S={\textsf {k}}[X_1,\dots , X_n]$$ be a polynomial ring, where $${\textsf {k}}$$ is a field. This article deals with the defining ideal of the Rees algebra of a squarefree monomial ideal generated in degree $$n-2$$ . As a consequence, we prove that Betti numbers of powers of the cover ideal of the complement graph of a tree do not depend on the choice of the tree. Further, we study the regularity and Betti numbers of powers of cover ideals associated to certain graphs.

Symbolic powers of vertex cover ideals

Let [Formula: see text] be a finite simple graph and [Formula: see text] denote its vertex cover ideal in a polynomial ring over a field [Formula: see text]. In this paper, we show that all symbolic powers of vertex cover ideals of certain vertex-decomposable graphs have linear quotients. Using these results, we give various conditions on a subset [Formula: see text] of the vertices of [Formula: see text] so that all symbolic powers of vertex cover ideals of [Formula: see text], obtained from [Formula: see text] by adding a whisker to each vertex in [Formula: see text], have linear quotients. For instance, if [Formula: see text] is a vertex cover of [Formula: see text], then all symbolic powers of [Formula: see text] have linear quotients. Moreover, we compute the Castelnuovo–Mumford regularity of symbolic powers of certain vertex cover ideals.

On quasi-equigenerated and Freiman cover ideals of graphs

A quasi-equigenerated monomial ideal I in the polynomial ring is a Freiman ideal if where l(I) is the analytic spread of I and is the number of minimal generators of I. Freiman ideals are special since there exists an exact formula computing the minimal number of generators of any of their powers. In this work, we address the question of characterizing which cover ideals of simple graphs are Freiman.

Bass numbers of local cohomology of cover ideals of graphs

We develop splitting techniques to study the Lyubeznik numbers of cover ideals of graphs which allow us to describe them for large families of graphs including forests, cycles, wheels and cactus graphs. More generally, we are able to compute all the Bass numbers and the shape of the injective resolution of local cohomology modules by considering the connected components of all the induced subgraphs. Indeed, our method gives us a very simple criterion for the vanishing of these local cohomology modules in terms of the number of connected components of the induced subgraphs.

Associated Primes of Powers of Monomial Ideals: A Survey

Let <i>R</i> be a commutative Noetherian ring and <i>I</i> be an ideal of <i>R</i>. We say that <i>I</i> satisfies the persistence property if Ass<i>_R</i>(<i>R/I^k</i>) ⊆ Ass<i>_R</i>(<i>R/I^k+1</i>) for all positive integers k, where Ass_R(R/I ) denotes the set of associated prime ideals of <i>I</i>. In addition, an ideal I has the strong persistence property if (<i>I^k+1</i>: <i>_RI</i>) = <i>I^k</i> for all positive integers k. Also, an ideal <i>I</i> is called normally torsion-free if Ass<i>_R</i>(<i>R/I^k</i>) ⊆ Ass<i>_R</i>(<i>R/I</i>) for all positive integers k. In this paper, we collect the latest results in associated primes of powers of monomial ideals in three concepts, i.e., the persistence property, strong persistence property, and normally torsion-freeness. Also, we present some classes of monomial ideals such that are none of edge ideals, cover ideals, and polymatroidal ideals, but satisfy the persistence property and strong persistence property. In particular, we study the Alexander dual of path ideals of unrooted starlike trees. Furthermore, we probe the normally torsion-freeness of the Alexander dual of some path ideals which are related to banana trees. We close this paper with exploring the normally torsion-freeness under some monomial operations.

On the cover ideals of chordal graphs

The independence complex of a chordal graph is known to be shellable due to a result of Van Tuyl and Villarreal. This is equivalent to the fact that cover ideal of a chordal graph has linear quotients. We use this result to obtain recursive formulas for the Betti numbers of cover ideals of chordal graphs. Also, we give a new proof of their result which yields different shellings of the independence complex.

Syzygies, Betti Numbers, and Regularity of Cover Ideals of Certain Multipartite Graphs

Let G be a finite simple graph on n vertices. Let J G ⊂ K [ x 1 , … , x n ] be the cover ideal of G. In this article, we obtain syzygies, Betti numbers, and Castelnuovo–Mumford regularity of J G s for all s ≥ 1 for certain classes of graphs G.

Normality of Cover Ideals of Graphs and Normality Under Some Operations

We produce a procedure for constructing new normal monomial ideals from other ideals that are assumed to be normal. This enables us to prove that if the cover ideal of a graph G is normal, then the cover ideal of the graph H is normal as well, where the graph H is obtained by connecting all vertices in G with a new vertex. We use these ideas to explore the normality of the cover ideals of some imperfect graphs. Also, we investigate the normality under expansion, this leads us to generalize the work of Al-Ayyoub (Rocky Mountain J Math 39:1–9, 2009). Furthermore, we investigate the normality under more operations such as weighting, polarization, localization, contraction, and deletion.

Asymptotic invariants of ideals with Noetherian symbolic Rees algebra and applications to cover ideals

Let I be an ideal whose symbolic Rees algebra is Noetherian. For m≥1, the m-th symbolic defect, sdefect(I,m), of I is defined to be the minimal number of generators of the module I(m)Im. We prove that sdefect(I,m) is eventually quasi-polynomial as a function in m. We compute the symbolic defect explicitly for certain monomial ideals arising from graphs, termed cover ideals. We go on to give a formula for the Waldschmidt constant, an asymptotic invariant measuring the growth of the degrees of generators of symbolic powers, for ideals whose symbolic Rees algebra is Noetherian.

Stability of depth functions of cover ideals of balanced hypergraphs

We prove that the depth functions of cover ideals of balanced hypergraph have the nonincreasing property. Furthermore, we also give a bound for the index of depth stability of these ideals.

Associated primes of powers of cover ideals under graph operations

An ideal I in a commutative Noetherian ring R has the strong persistence property if for all positive integers t. In this article, we focus on how some graph operations affect the strong persistence property of the cover ideals. As applications, we show that the cover ideals of some imperfect graphs satisfy the strong persistence property.

Regularity of symbolic powers of cover ideals of graphs

Let G be a graph which belongs to either of the following classes: (i) bipartite graphs, (ii) unmixed graphs, or (iii) claw–free graphs. Assume that J(G) is the cover ideal G and $$J(G)^{(k)}$$ is its k-th symbolic power. We prove that $$\begin{aligned} k\mathrm{deg}(J(G))\le \mathrm{reg}(J(G)^{(k)})\le (k-1)\mathrm{deg}(J(G))+|V(G)|-1. \end{aligned}$$ We also determine families of graphs for which the above inequalities are equality.

Regularity of powers of cover ideals of unimodular hypergraphs

Let H be a unimodular hypergraph over the vertex set [n] and let J(H) be the cover ideal of H in the polynomial ring R=K[x1,…,xn]. We show that regJ(H)s is a linear function in s for all s⩾r⌈n2⌉+1 where r is the rank of H. Moreover for every i, ai(R/J(H)s) is also a linear function in s for s⩾n2.

Depth and Stanley depth of symbolic powers of cover ideals of graphs

Let G be a graph with n vertices and let S=K[x1,…,xn] be the polynomial ring in n variables over a field K. Assume that J(G) is the cover ideal of G and J(G)(k) is its k-th symbolic power. We prove that the sequences {sdepth(S/J(G)(k))}k=1∞ and {sdepth(J(G)(k))}k=1∞ are non-increasing and hence convergent. Suppose that νo(G) denotes the ordered matching number of G. We show that for every integer k≥2νo(G)−1, the modules J(G)(k) and S/J(G)(k) satisfy the Stanley's inequality. We also provide an alternative proof for [9, Theorem 3.4] which states that depth(S/J(G)(k))=n−νo(G)−1, for every integer k≥2νo(G)−1.