Given an arbitrary graph G, we study its basic covers algebra, which is the symbolic fiber cone of the Alexander dual of the edge ideal of G. Extending results of Villarreal and Benedetti-Constantinescu-Varbaro, valid only in the case when G is bipartite, we characterize in a combinatorial fashion the situations when: 1) the basic covers algebra is a domain, and 2) it is a domain and in addition (the edge ideal of) G is unmixed. It turns out that the last result gives a complete characterization of those graphs for which any symbolic power of the edge ideal is generated by monomials of the same degree.
Unmixed bipartite graphs have been characterized by Ravadra and Villarreal independently. Our aim in this paper is to characterize unmixed r-partite graphs under a certain condition, witch is a generalization of villarreal's theorem on bipartite graphs. Also we give some examples and counterexamples in relevance this subject.
The correspondence between unmixed bipartite graphs and sublattices of the oolean lattice is discussed. By using this correspondence, we show the existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs.
For an unmixed bipartite graph $G$ we consider the lattice of vertex covers $\mathcal{L}_G$ and compute depth, projective dimension and extremal Betti-numbers of $R/I(G)$ in terms of this lattice.
We compute the reduced Gröbner basis of the toric ideal with respect to a suitable monomial order and we study the Hilbert series of the vertex cover algebra $A(G)$, where $G$ is an unmixed bipartite graph without isolated vertices.
We study unmixed and Cohen-Macaulay properties of the binomial edge ideal of some classes of graphs. We compute the depth of the binomial edge ideal of a generalized block graph. We also characterize all generalized block graphs whose binomial edge ideals are Cohen-Macaulay and unmixed. So that we generalize the results of Ene, Herzog and Hibi on block graphs. Moreover, we study unmixedness and Cohen-Macaulayness of the binomial edge ideal of some graph products such as the join and corona of two graphs with respect to the original graphs'.
Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph whose edge ideal is $I(D)$. In this paper, we characterize the unmixed property of $I(D)$ for each one of the following cases: $G$ is an $SCQ$ graph; $G$ is a chordal graph; $G$ is a simplicial graph; $G$ is a perfect graph; $G$ has no $4$- or $5$-cycles; $G$ is a graph without $3$- and $5$-cycles; and ${\rm girth}(G)\geqslant 5$.
Let $D$ be a weighted oriented graph, whose underlying graph is $G$, and let $I(D)$ be its edge ideal. If $G$ has no $3$-, $5$-, or $7$-cycles, or $G$ is König, we characterize when $I(D)$ is unmixed. If $G$ has no $3$- or $5$-cycles, or $G$ is König, we characterize when $I(D)$ is Cohen--Macaulay. We prove that $I(D)$ is unmixed if and only if $I(D)$ is Cohen--Macaulay when $G$ has girth greater than $7$ or $G$ is König and has no $4$-cycles.
In a paper in 2008, Herzog, Hibi and Ohsugi introduced and studied the semigroup ring associated to the set of minimal vertex covers of an unmixed bipartite graph. In this paper we relate the dimension of this semigroup ring to the rank of the Boolean lattice associated to the graph.
For each Boolean graph $B_n$, it is proved that both $B_n$ and its complement graph $\overline{B_n}$ are vertex decomposable. It is also proved that $B_n$ is an unmixed graph, thus it is also Cohen-Macaulay.
An equigenerated monomial ideal $I$ in the polynomial ring $R=k[z_1,\ldots, z_n]$ is a Freiman ideal if $μ(I^2)=\ell(I)μ(I)-{\ell(I)\choose 2}$ where $\ell(I)$ is the analytic spread of $I$ and $μ(I)$ is the number of minimal generators of $I$. In this paper we classify all simple connected unmixed bipartite graphs whose cover ideals are Freiman ideals.
Let $G$ be an unmixed bipartite graph of dimension $d-1$. Assume that $K_{n,n}$, with $n\ge 2$, is a maximal complete bipartite subgraph of $G$ of minimum dimension. Then $G$ is Cohen-Macaulay in codimension $d-n+1$. This generalizes a characterization of Cohen-Macaulay bipartite graphs by Herzog and Hibi and a result of Cook and Nagel on unmixed Buchsbaum graphs. Furthermore, we show that any unmixed bipartite graph $G$ which is Cohen-Macaulay in codimension $t$, is obtained from a Cohen-Macaulay graph by replacing certain edges of $G$ with complete bipartite graphs. We provide some examples.
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 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.
We classify the bipartite graphs $G$ whose binomial edge ideal $J_G$ is Cohen-Macaulay. The connected components of such graphs can be obtained by gluing a finite number of basic blocks with two operations. In this context we prove the converse of a well-known result due to Hartshorne, showing that the Cohen-Macaulayness of these ideals is equivalent to the connectedness of their dual graphs. We study interesting properties also for non-bipartite graphs and in the unmixed case, constructing classes of bipartite graphs with $J_G$ unmixed and not Cohen-Macaulay.
We study the homological algebra of edge ideals of Erdös-Rényi random graphs. These random graphs are generated by deleting edges of a complete graph on $n$ vertices independently of each other with probability $1-p$. We focus on some aspects of these random edge ideals - linear resolution, unmixedness and algebraic invariants like the Castelnuovo-Mumford regularity, projective dimension and depth. We first show a double phase transition for existence of linear presentation and resolution and determine the critical windows as well. As a consequence, we obtain that except for a very specific ch…
Let $G=(V,E)$ be a connected simple graph, with $n$ vertices such that $S$ is its homogeneous monomial subring. We prove that if $S$ is normal and Gorenstein, then $G$ is unmixed with cover number $\lceil\frac{n}{2}\rceil$ and $G$ has a strong $\lceil\frac{n}{2}\rceil$-$τ$-reduction. Furthermore, if $n$ is even, then we show that $G$ is bipartite. Finally, if $S$ is normal and $G$ is unmixed whose cover number is $\lceil\frac{n}{2}\rceil$, we give sufficient conditions for $S$ to be Gorenstein.
We study weighted graphs and their "edge ideals" which are ideals in polynomial rings that are defined in terms of the graphs. We provide combinatorial descriptions of m-irreducible decompositions for the edge ideal of a weighted graph in terms of the combinatorics of "weighted vertex covers". We use these, for instance, to say when these ideals are m-unmixed. We explicitly describe which weighted cycles, suspensions, and trees are unmixed and which ones are Cohen-Macaulay, and we prove that all weighted complete graphs are Cohen-Macaulay.
The cut sets of a graph are special sets of vertices whose removal disconnects the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible graphs as the graphs with unmixed binomial edge ideal and whose cut sets form an accessible set system. We prove that the graphs whose binomial edge ideal is Cohen-Macaulay are accessible and we conjecture that the converse holds. We settle the conjecture for large classes of graphs, including chordal and traceable graphs, providing a purely combinatorial d…
Let $\mathcal{C}$ be a clutter with a perfect matching $e_1,...,e_g$ of König type and let $Δ_\mathcal{C}$ be the Stanley-Reisner complex of the edge ideal of $\mathcal{C}$. If all c-minors of $\mathcal{C}$ have a free vertex and $\mathcal{C}$ is unmixed, we show that $Δ_\mathcal{C}$ is pure shellable. We are able to describe, in combinatorial and algebraic terms, when $Δ_\mathcal{C}$ is pure. If $\mathcal{C}$ has no cycles of length 3 or 4, then it is shown that $Δ_\mathcal{C}$ is pure if and only if $Δ_\mathcal{C}$ is pure shellable (in this case $e_i$ has a free vertex for all $i$), and tha…
Let I=I(D) be the edge ideal of a weighted oriented graph D. We determine the irredundant irreducible decomposition of I. Also, we characterize the associated primes and the unmixed property of I. Furthermore, we give a combinatorial characterization for the unmixed property of I, when D is bipartite, D is a whisker or D is a cycle. Finally, we study the Cohen-Macaulay property of I.
We prove the following result, which is motivated by the recent work of Kurano and Roberts on Serre's positivity conjecture. Assume that (R,m) is a local ring with finitely-generated module M such that R/ann(M) is quasi-unmixed and contains a field, and that p and q are prime ideals in the support of M such that p is analytically unramified, p+q is m-primary and e(M_p)=e(M). Then dim(R/p)+dim(R/q)\leq dim(M). We also prove a similar theorem in a special case of mixed characteristic. Finally, we provide several examples to explain our assumptions as well as an example of a noncatenary, local do…
For a graph $G$, Bolognini et al. have shown $J_{G}$ is strongly unmixed $\Rightarrow$ $J_{G}$ is Cohen-Macaulay $\Rightarrow$ $G$ is accessible, where $J_{G}$ denotes the binomial edge ideals of $G$. Accessible and strongly unmixed properties are purely combinatorial. We give some motivations to focus only on blocks with whiskers for the characterization of all $G$ with Cohen-Macaulay $J_{G}$. We show that accessible and strongly unmixed properties of $G$ depend only on the corresponding properties of its blocks with whiskers and vice versa. Also, we give an infinite class of graphs whose bin…
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$$k{\rm deg}(J(G))\leq {\rm reg}(J(G)^{(k)})\leq (k-1){\rm deg}(J(G))+|V(G)|-1.$$We also determine families of graphs for which the above inequalities are equality.
Let G be a perfect graph and let J be its ideal of vertex covers. We show that the Rees algebra of J is normal and that this algebra is Gorenstein if G is unmixed. Then we give a description--in terms of cliques--of the symbolic Rees algebra and the Simis cone of the edge ideal of G.
In this paper we give new upper bounds on the regularity of edge ideals whose resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free monomial ideals of codimension two. We also discuss and connect these results to more classical topics in commutative algebra.
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…
We introduce a class of ideals generated by a set of 2-minors of $m\times n$-matrix of indeterminates indexed by a pair of graphs. This class of ideals is a natural common generalization of binomial edge ideals and ideals generated by adjacent minors. We determine the minimal prime ideals of such ideals and give a lower bound for their degree of nilpotency. In some special cases we compute their Gröbner basis and characterize unmixedness and Cohen--Macaulayness.
Given a finite simple graph one can associate the edge ideal. In this paper we prove that a graded Betti number of the edge ideal does not vanish if the original graph contains a set of complete bipartite subgraphs with some conditions. Also we give a combinatorial description for the projective dimension of the edge ideals of unmixed bipartite graphs.
For any graph $G$, assume that $J(G)$ is the cover ideal of $G$. Let $J(G)^{(k)}$ denote the $k$th symbolic power of $J(G)$. We characterize all graphs $G$ with the property that $J(G)^{(k)}$ has a linear resolution for some (equivalently, for all) integer $k\geq 2$. Moreover, it is shown that for any graph $G$, the sequence $\big({\rm reg}(J(G)^{(k)})\big)_{k=1}^{\infty}$ is nondecreasing. Furthermore, we compute the largest degree of minimal generators of $J(G)^{(k)}$ when $G$ is either an unmixed of a claw-free graph.
We introduce a conjecture about constructing critically (s+1)-chromatic graphs from critically s-chromatic graphs. We then show how this conjecture implies that any unmixed height two square-free monomial ideal I, i.e., the cover ideal of a finite simple graph, has the persistence property, that is, Ass(R/I^s) \subseteq Ass(R/I^{s+1}) for all s >= 1. To support our conjecture, we prove that the statement is true if we also assume that χ_f(G), the fractional chromatic number of the graph G, satisfies χ(G) -1 < χ_f(G) <= χ(G). We give an algebraic proof of this result.
A combinatorial property that characterizes Cohen-Macaulay binomial edge ideals has long been elusive. A recent conjecture ties the Cohen-Macaulayness of a binomial edge ideal $J_G$ to special disconnecting sets of vertices of its underlying graph $G$, called \textit{cut sets}. More precisely, the conjecture states that $J_G$ is Cohen-Macaulay if and only if $J_G$ is unmixed and the collection of the cut sets of $G$ is an accessible set system. In this paper we prove the conjecture theoretically for all graphs with up to $12$ vertices and develop an algorithm that allows to computationally che…
Let G be a simple undirected graph on n vertices, and let I(G) \subseteq R = k[x_1,...,x_n] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and implies Herzog, Hibi, and Zheng's theorem that a chordal graph is Cohen-Macaulay if and only if its edge ideal is unmixed. We also characterize the sequentially Cohen-Macaulay cycles and produce some ex…
A very well-covered graph is an unmixed graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. We study these graphs by means of Betti splittings and mapping cone constructions. We show that the cover ideals of Cohen-Macaulay very well-covered graphs are splittable. As a consequence, we compute explicitly the minimal graded free resolution of the cover ideals of such a class of graphs and prove that these graphs have homological linear quotients. Finally, we conjecture the same is true for each power of the cover ideal of a Cohen-Macaulay very…
Let $S$ be a polynomial ring over a field $K$, with a monomial order $\prec$, and let $I$ be an unmixed graded ideal of $S$. In this paper we study two functions associated to $I$: the minimum distance function $δ_I$ and the footprint function ${\rm fp}_I$. It is shown that $δ_I$ is positive and that ${\rm fp}_I$ is positive if the initial ideal of $I$ is unmixed. Then we show that if $I$ is radical and its associated primes are generated by linear forms, then $δ_I$ is strictly decreasing until it reaches the asymptotic value $1$. If $I$ is the edge ideal of a Cohen--Macaulay bipartite graph, …
Let $\mathbf{CCM}$ denote the class of closed graphs with Cohen-Macaulay binomial edge ideals and $\mathbf{PIG}$ denote the class of proper interval graphs. Then $\mathbf{CCM}\subseteq \mathbf{PIG}$. The $\mathbf{PIG}$-completion problem is a classical problem in molecular biology as well as in graph theory and this problem is known to be NP-hard. In this paper, we study the $\mathbf{CCM}$-completion problem. We give a method to construct all possible $\mathbf{CCM}$-completion of a graph. We find the $\mathbf{CCM}$-completion number and the set of all minimal $\mathbf{CCM}$-completions for a l…
We classify all unmixed monomial ideals I of codimension 2 which are generically a complete intersection and which have the property that the symbolic power algebra A(I) is standard graded. We give a lower bound for the highest degree of a generator of A(I) in the case that I is a modification of the vertex cover ideal of a bipartite graph, and show that this highest degree can be any given number. We furthermore give an upper bound for the highest degree of a generator of the integral closure of A(I) in the case that I is a monomial ideal which is generically a complete intersection.
Determining whether an arbitrary subring $R$ of $k[x_1^{\pm 1},\dots, x_n^{\pm 1}]$ is a normal domain is, in general, a nontrivial problem, even in the special case of a monomial generated domain. In this paper, we provide a complete characterization of the normality and normalizations of quadratic-monomial generated domains. For a quadratic-monomial generated domain $R$, we develop a combinatorial structure that assigns, to each quadratic monomial of the ring, an edge in a mixed signed, directed graph $G$, i.e., a graph with signed edges and directed edges. We classify the normality and the …
It is well-known that the factorization properties of a domain are reflected in the structure of its group of divisibility. The main theme of this paper is to introduce a topological/graph-theoretic point of view to the current understanding of factorization in integral domains. We also show that connectedness properties in the graph and topological space give rise to a generalization of atomicity.
