Edge ideals of finite simple graphs $G$ on $n$ vertices are the ideals $I(G)$ of the polynomial ring $S$ in $n$ variables generated by the quadratic monomials associated with the edges of $G$. In this paper, we consider the possible pairs of dimensions and depths of $S/I(G)$ for connected graphs with a fixed number of vertices. We discuss such pairs in the case where dimension is relatively large. As a corollary, we completely determine the pairs for connected graphs with small number of vertices. We also study the possible pairs for connected chordal graphs.
In this paper we study squarefree monomial ideals which have constant depth functions. Edge ideals, matroidal ideals and facet ideals of pure simplicial forests connected in codimension one with this property are classified.
If $J\subset I$ are two monomials ideals, we give a practical upper bound for the Stanley depth of $J/I$, which we call it the \emph{quasi-depth} of $J/I$. Also, we compute the quasi-depth of several classes of square free monomial ideals. We also study the Stanley depth of the monomial ideal associated of the independent sets of a graph.
We prove that the depth functions of cover ideals of balanced hyper- gragh have the non-increasing property. Furthermore, we also give a bound for the index of depth stability of these ideals.
Let $(R,\mm)$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We say $M$ has maximal depth if there is an associated prime $\pp$ of $M$ such that $\depth M=\dim R/\pp$. In this paper we study squarefree monomial ideals which have maximal depth. Edge ideals of cycle graphs, transversal polymatroidal ideals and high powers of connected bipartite graph with this property are classified.
In this paper, we study associated primes and depth of integral closures of powers of edge ideals. We provide sharp bounds on how big of powers for which the set of associated primes and the depth of integral closures of powers of edge ideals are stable.
In this paper, we obtain explicit formulas for the Hilbert series and Hilbert depth of squarefree Veronese ideals in a standard graded polynomial ring.
We study minimal free resolutions of edge ideals of bipartite graphs. We associate a directed graph to a bipartite graph whose edge ideal is unmixed, and give expressions for the regularity and the depth of the edge ideal in terms of invariants of the directed graph. For some classes of unmixed edge ideals, we show that the arithmetic rank of the ideal equals projective dimension of its quotient.
Let $I\subset S=\KK[x_1,...,x_n]$ be a lexsegment ideal, generated by monomials of degree $d$. The main aim of this paper is to characterize when the Hilbert depth of $I$ will be 1, in the standard graded case. In addition to this, we will give an estimate of depth of squarefree monomial ideals, generalizing a result of Popescu. We will also show that Stanley conjecture holds for squarefree stable ideals, in the multigraded case.
We study the limit and initial behavior of the numerical function $f(k)=\depth S/I^k$. General properties of this function together with concrete examples arising from combinatorics are discussed.
Let $I$ be the edge ideal of a clutter $\mathcal{C}$ in a polynomial ring $S$. In this paper, we present estimations of the Stanley depth of $I$ as well as the Stanley regularity of $S/I$, in terms of combinatorial data from the clutter $\mathcal{C}$.
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 generating set of $I$, and examine the depth and regularity of powers of edge ideals of clutters using combinatorial optimization techniques. If $I$ is the edge ideal of an unmixed clutter with the max-flow min-cut property, we show that the powers of $I$ have non-increasing depth and non-decreasing regularity. In particular edge ideals of unmixed bipartite grap…
Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,...,x_n]$ be the polynomial ring in $n$ variables over the field $\mathbb{K}$. For every monomial ideal $I\subset S$, We provide a recursive formula to determine a lower bound for the Stanley depth of $S/I$. We use this formula to prove the inequality ${\rm sdepth}(S/I)\geq {\rm size}(I)$ for a particular class of monomial ideals.
The aim of this paper is to study the Stanley depth of symbolic powers of a squarefree monomial ideal. We prove that for every squarefree monomial ideal $I$ and every pair of integers $k, s\geq 1$, the inequalities ${\rm sdepth} (S/I^{(ks)}) \leq {\rm sdepth} (S/I^{(s)})$ and ${\rm sdepth} (I^{(ks)}) \leq {\rm sdepth} (I^{(s)})$ hold. If moreover $I$ is unmixed of height $d$, then we show that for every integer $k\geq1$, ${\rm sdepth}(I^{(k+d)})\leq {\rm sdepth}(I^{(k)})$ and ${\rm sdepth}(S/I^{(k+d)})\leq {\rm sdepth}(S/I^{(k)})$. Finally, we consider the limit behavior of the Stanley depth o…
Let $I$ be an ideal of a polynomial algebra $S$ over a field generated by square free monomials of degree $\geq d$. If $I$ contains more monomials of degree $d$ than $(n-d)/(n-d+1)$ of the total number of square free monomials of $S$ of degree $d+1$ then $\depth_SI\leq d$, in particular the Stanley's Conjecture holds in this case.
The Hilbert depth of a module M is the maximum depth that occurs among all modules with the same Hilbert function as M. In this note we compute the Hilbert depths of the powers of the irrelevant maximal ideal in a standard graded polynomial ring.
Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Let $G$ be a graph with $n$ vertices. Assume that $I=I(G)$ is the edge ideal of $G$ and $J=J(G)$ is its cover ideal. We prove that ${\rm sdepth}(J)\geq n-ν_{o}(G)$ and ${\rm sdepth}(S/J)\geq n-ν_{o}(G)-1$, where $ν_{o}(G)$ is the ordered matching number of $G$. We also prove the inequalities ${\rm sdepth}(J^k)\geq {\rm depth}(J^k)$ and ${\rm sdepth}(S/J^k)\geq {\rm depth}(S/J^k)$, for every integer $k\gg 0$, when $G$ is a bipartite graph. Moreover, we provide an elementary p…
Let $G$ be a graph and let $I := I (G)$ be its edge ideal. In this paper, we provide an upper bound of $n$ from which $\depth R/ I(G)^n$ is stationary, and compute this limit explicitly. This bound is always achieved if $G$ has no cycles of length $4$ and every its connected component is either a tree or a unicyclic graph.
Let R = K[x1,...,xr] be a polynomial ring over a field K. Let G be a graph with vertex set {1,...,r} and let J be the cover ideal of G. We give a sharp bound for the stability index of symbolic depth function sdstab(J). In the case G is bipartite, it yields a sharp bound for the stability index of depth function dstab(J) and this bound is exact if G is a forest.
Assume that $G$ is a graph with edge ideal $I(G)$ and star packing number $α_2(G)$. We denote the $s$-th symbolic power of $I(G)$ by $I(G)^{(s)}$. It is shown that the inequality ${\rm depth} S/(I(G)^{(s)})\geq α_2(G)-s+1$ is true for every chordal graph $G$ and every integer $s\geq 1$. Moreover, it is proved that for any graph $G$, we have ${\rm depth} S/(I(G)^{(2)})\geq α_2(G)-1$.
Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. In 1982, R. Stanley associated a combinatorial invariant to any finitely generated $\mathbb{Z}^n$-graded $S$-module which is now called Stanley depth. Stanley conjectured that this invariant is an upper bound for the depth of module. Stanley's conjecture has been disproved by Duval et al. \cite{abcj}, and the counterexample is a quotient of squarefree monomial ideals. On the other hand, there are evidences showing that Stanley's inequality can be true for high powers of mono…
In [7] we obtained a formula for the Hilbert depth of squarefree Veronese ideals in a standard graded polynomial ring by relating it to the Hilbert depth of powers of the irrelevant maximal ideal. In this paper, we prove that these two Hilbert depth formulas are equivalent to each other. Our result reveals that there is a strong connection between these two classes of seemingly unrelated monomial ideals. We conjecture that their Stanley depths are equivalent as well.
Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Assume that $G$ is a graph with edge ideal $I(G)$. We prove that the modules $S/\overline{I(G)^k}$ and $\overline{I(G)^k}/\overline{I(G)^{k+1}}$ satisfy Stanley's inequality for every integer $k\gg 0$. If $G$ is a non-bipartite graph, we show that the ideals $\overline{I(G)^k}$ satisfy Stanley's inequality for all $k\gg 0$. For every connected bipartite graph $G$ (with at least one edge), we prove that ${\rm sdepth}(I(G)^k)\geq 2$, for any positive integer $k\leq {\rm girth}…
Let $I$ be the edge ideal of a connected non-bipartite graph and $R$ the base polynomial ring. Then $\operatorname{depth} R/I \ge 1$ and $\operatorname{depth} R/I^t = 0$ for $t \gg 1$. We give combinatorial conditions for $\operatorname{depth} R/I^t = 1$ for some $t$ in between and show that the depth function is non-increasing thereafter. Especially, the depth function quickly decreases to 0 after reaching 1. We show that if $\operatorname{depth} R/I = 1$ then $\operatorname{depth} R/I^2 = 0$ and if $\operatorname{depth} R/I^2 = 1$ then $\operatorname{depth} R/I^5 = 0$. Other similar results …
We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs which have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes and Toledo.
We find a class of block graphs whose binomial edge ideals have minimal regularity. As a consequence, we characterize the trees whose binomial edge ideals have minimal regularity. Also, we show that the binomial edge ideal of a block graph has the same depth as its initial ideal.
We give a complete characterization of graphs whose binomial edge ideal is licci. An important tool is a new general upper bound for the regularity of binomial edge ideals.
We study the relationship between depth and regularity of a homogeneous ideal I and those of (I,f) and I:f, where f is a linear form or a monomial. Our results has several interesting consequences on depth and regularity of edge ideals of hypegraphs and of powers of ideals.
We survey recent studies and results on the following problem: which numerical functions can be the depth function of powers and symbolic powers of homogeneous ideals.
Edge ideals of finite simple graphs are well-studied over polynomial rings. In this paper, we initiate the study of edge ideals over exterior algebras, specifically focusing on the depth and singular varieties of such ideals. We prove an upper bound on the depth of the edge ideal associated to a general graph and a more refined bound for bipartite graphs, and we show that both are tight. We also compute the depth of several large families of graphs including cycles, complete multipartite graphs, spider graphs, and Ferrers graphs. Finally, we focus on the effect whiskering a graph has on the de…
The aim of this work is to study the v-number of edge ideals of clutters and graphs. We relate the v-number with the regularity of edge ideals and study the combinatorial structure of the graphs whose edge ideals have their second symbolic power Cohen-Macaulay.
We classify all binomial edge ideals that are complete intersection and Cohen-Macaulay almost complete intersection. We also describe an algorithm and provide an implementation to compute primary decomposition of binomial edge ideals.
We compute the Betti numbers for all the powers of initial and final lexsegment edge ideals. For the powers of the edge ideal of an anti-$d-$path, we prove that they have linear quotients and we characterize the normally torsion-free ideals. We determine a class of non-squarefree ideals, arising from some particular graphs, which are normally torsion-free.
Let P = k[x_1, ..., x_n] be the polynomial ring in n variables. A homogeneous ideal I of P generated in degree d is called Gotzmann if it has the smallest possible Hilbert function out of all homogeneous ideals with the same dimension in degree d. The edge ideal of a simple graph G on vertices x_1, ..., x_n is the quadratic square-free monomial ideal generated by all x_i x_j where {x_i,x_j} is an edge of G. The only edge ideals that are Gotzmann are those edge ideals corresponding to star graphs.
Let $G$ be a graph with $n$ vertices and let $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$. 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 $\{{\rm sdepth}(S/J(G)^{(k)})\}_{k=1}^\infty$ and $\{{\rm sdepth}(J(G)^{(k)})\}_{k=1}^\infty$ are non-increasing and hence convergent. Suppose that $ν_{o}(G)$ denotes the ordered matching number of $G$. We show that for every integer $k\geq 2ν_{o}(G)-1$, the modules $J(G)^{(k)}$ and $S/J(G)^{(k)}$ satisfy the Stanley's inequality. We al…
In this paper, we study the Hankel edge ideals of graphs. We determine the minimal prime ideals of the Hankel edge ideal of labeled Hamiltonian and semi-Hamiltonian graphs, and we investigate radicality, being a complete intersection, almost complete intersection and set theoretic complete intersection for such graphs. We also consider the Hankel edge ideal of trees with a natural labeling, called rooted labeling. We characterize such trees whose Hankel edge ideal is a complete intersection, and moreover, we determine those whose initial ideal with respect to the reverse lexicographic order sa…
Let (P,<) be a finite poset and let G be its comparability graph. If cl(G) is the clutter of maximal cliques of G, we prove that cl(G) satisfies the max-flow min-cut property and that its edge ideal is normally torsion free. We prove that edge ideals of complete admissible uniform clutters are normally torsion free. The normality of a monomial ideal is expressed in terms of blocking polyhedra and the integer decomposition property. For edge ideals of clutters this property completely determine their normality
Let $G$ be a cycle or a complete bipartite graph. We show that the binomial edge ideal $J_{G}$ and its initial ideal with respect to the lexicographic order have the same extremal Betti number.
We obtain an improved lower bound for the regularity of the binomial edge ideals of trees. We prove an upper bound for the regularity of the binomial edge ideals of certain subclass of block-graphs. As a consequence we obtain sharp upper and lower bounds for the regularity of binomial edge ideals of a class of trees called lobsters. We also obtain precise expressions for the regularities of binomial edge ideals of certain classes of trees and block graphs.
Closed graphs have been characterized by Herzog et al. as the graphs whose binomial edge ideals have a quadratic Gröbner basis with respect to a diagonal term order. In this paper, we focus on a generalization of closed graphs, namely weakly-closed graphs (or co-comparability graphs). Build on some results about Knutson ideals of generic matrices, we characterize weakly closed graphs as the only graphs whose binomial edge ideals are Knutson ideals for a certain polynomial $f$. In doing so, we re-prove Matsuda's theorem about the F-purity of binomial edge ideals of weakly closed graphs in posit…
