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 G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen-Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo-Mumford regularity of R/I(G) can be determined from the invariants of G.
We focus in this paper on edge ideals associated to bipartite graphs and give a combinatorial characterization of those having regularity 3. When the regularity is strictly bigger than 3, we determine the first step $i$ in the minimal graded free resolution where there exists a minimal generator of degree $>i+3$, show that at this step the highest degree of a minimal generator is $i+4$, and determine the value of the corresponding graded Betti number $β_{i,i+4}$ in terms of the combinatorics of the associated bipartite graph. The results can then be easily extended to the non-squarefree case t…
Let $G$ be a bipartite graph and $I=I(G)$ be its edge ideal. The aim of this note is to investigate different aspects of the Rees algebra $\mathcal{R}(I)$ of $I$. We compute its regularity and the universal Gröbner basis of its defining equations; interestingly, both of them are described in terms of the combinatorics of $G$. We apply these ideas to study the regularity of the powers of $I$. For any $s \ge \text{match}(G)+\lvert E(G) \rvert +1$ we prove that $\text{reg}(I^{s+1})=\text{reg}(I^s)+2$.
We prove that the projective dimension of any (hyper)graph can be bounded from above by the (Castelnuovo-Mumford) regularity of its Levi graph (or incidence bipartite graph). This in particular brings the use of regularity's upper bounds on the calculation of projective dimension of (hyper)graphs. When G is just a (simple) graph, we prove that there exists an induced subgraph H of G such that prod-dim(G)=reg(S(H)), where S(H) is the subdivision graph of H. Moreover, we show that known upper bounds on prod-dim(G) involving domination parameters are in fact upper bounds to reg(S(G)).
In this paper we prove that if $I(G)$ is a bipartite edge ideal with regularity three then for all $s\geq 2$ the regularity of $I(G)^s$ is exactly $2s+1$.
We give the Castelnuovo-Mumford regularity of arrangements of (n-2)-planes in P^n whose incidence graph is a sufficiently large complete bipartite graph, and determine when such arrangements are arithmetically Cohen-Macaulay.
We study the Castelnuovo-Mumford regularity of the vanishing ideal over a bipartite graph endowed with a decomposition of its edge set. We prove that, under certain conditions, the regularity of the vanishing ideal over a bipartite graph obtained from a graph by attaching a path of length $\ell$ increases by $\lfloor \frac{\ell}{2}\rfloor (q-2)$, where $q$ is the order of the field of coefficients. We use this result to show that the regularity of the vanishing ideal over a bipartite graph, $G$, endowed with a weak nested ear decomposition is equal to $$\textstyle \frac{|V_G|+ ε-3}{2}(q-2),$$ …
In this paper we provide some precise formulas for regularity of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs. For the projective dimension of such an edge ideal, we give its exact formula. Meanwhile, we also give the upper and lower bounds of projective dimension of higher power of such edge ideals. Some examples show that these formulas are related to direction selection.
Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. For all $s \geq 1$, we obtain upper bounds for reg$(I(G)^s)$ for bipartite graphs. We then compare the properties of $G$ and $G'$, where $G'$ is the graph associated with the polarization of the ideal $(I(G)^{s+1} : e_1\cdots e_s)$, where $e_1,\ldots e_s$ are edges of $G$. Using these results, we explicitly compute reg$(I(G)^s)$ for several subclasses of bipartite graphs.
Let $G$ be a finite simple graph. We give a lower bound for the Castelnuovo-Mumford regularity of the toric ideal $I_G$ associated to $G$ in terms of the sizes and number of induced complete bipartite graphs in $G$. When $G$ is a chordal bipartite graph, we find an upper bound for the regularity of $I_G$ in terms of the size of the bipartition of $G$. We also give a new proof for the graded Betti numbers of the toric ideal associated to the complete bipartite graph $K_{2,n}$.
Let $G$ be a finite simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in $S = K[x_1, \ldots, x_n, y_1, \ldots, y_n].$ In this article, we prove that if $G$ is a fan graph of a complete graph, then $reg(S/J_G) \leq c(G)$, where $c(G)$ denote the number of maximal cliques in $G$. Further, we show that if $G$ is a $k$-pure fan graph, then $reg(S/J_G) = k+1$. We then compute a precise expression for the regularity of Cohen-Macaulay bipartite graphs.
Let $S$ be a polynomial ring in $n$ variables over a field. Let $I$ be a homogeneous ideal in $S$ generated by forms of degree at most $d$ with $\text{dim}(S/I)=r$. In the first part of this paper, we show how to derive from a result of Hoa an upper bound for the regularity of $\sqrt{I}$. More specifically we show that $\text{reg}(\sqrt{I})\leq d^{(n-1)2^{r-1}}$. In the second part, we show that the $r$-th arithmetic degree of $I$ is bounded above by $2\cdot d^{2^{n-r-1}}$. This is done by proving upper bounds for arithmetic degrees of strongly stable ideals and ideals of Borel type.
We study the Castelnuovo-Mumford regularity of powers of edge ideals. We prove that if G is a bipartite graph, then reg(I(G)^s) \leq 2s + reg I(G) - 2 for all s \geq 2, which is the best possible upper bound for any s. Suspension plays a key role in proof of the base case s =2.
Let $\operatorname{pd}(I(G))$ and $\operatorname{reg}(I(G))$ respectively denote the projective dimension and the regularity of the edge ideal $I(G)$ of a graph $G$. For any positive integer $n$, we determine all pairs $(\operatorname{pd}(I(G)),\, \operatorname{reg}(I(G)))$ as $G$ ranges over all connected bipartite graphs on $n$ vertices.
Let $K[G]$ denote the edge ring of a finite connected simple graph $G$ on $[d]$ and $\mat(G)$ the matching number of $G$. It is shown that $\reg(K[G]) \leq \mat(G)$ if $G$ is non-bipartite and $K[G]$ is normal, and that $\reg(K[G]) \leq \mat(G) - 1$ if $G$ is bipartite.
In this paper, we obtain a combinatorial formula for computing the Betti numbers in the linear strand of edge ideals of bipartite Kneser graphs. We deduce lower and upper bounds for regularity of powers of edge ideals of these graphs in terms of associated combinatorial data and show that the lower bound is attained in some cases. Also, we obtain bounds on the projective dimension of edge ideals of these graphs in terms of combinatorial data.
A graph $G$ is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore two times this cardinality is equal to $|V(G)|$, the graph $G$ is called very well-covered. The class of very well-covered graphs contains bipartite well-covered graphs. Recently in \cite{CRT} it is shown that a very well-covered graph $G$ is Cohen-Macaulay if and only if it is pure shellable. In this article we improve this result by showing that $G$ is Cohen-Macaulay if and only if it is pure vertex decomposable. In addition, if $I(G)$ denotes the edge ide…
Let X* be a subset of an affine space A^s, over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x --> [x] and x --> [(x,1)] respectively, where [x] and [(x,1)] are points in the projective spaces P^{s-1} and P^s respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud-Goto regularity conjecture. We …
A classical theorem by Hartshorne states that the dual graph of any arithmetically Cohen--Macaulay projective scheme is connected. We give a quantitative version of Hartshorne's result, in terms of Castelnuovo--Mumford regularity. If $X \subset \mathbb{P}^n$ is an arithmetically Gorenstein projective scheme of regularity $r+1$, and if every irreducible component of $X$ has regularity $\le r'$, we show that the dual graph of $X$ is $\lfloor{\frac{r+r'-1}{r'}}\rfloor$-connected. The bound is sharp. We also provide a strong converse to Hartshorne's result: Every connected graph is the dual graph …
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 graph with $n$ vertices and $S=\mathbb{K}[x_1,\dots,x_n]$ 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 if $G$ is a very well-covered graph such that $J(G)$ has linear resolution, then $J(G)^{(k)}$ has linear resolution, for every integer $k\geq 1$. We also prove that for a every very well-covered graph $G$, the depth of symbolic powers of $J(G)$ forms a non-increasing sequence. Finally, we determine a linear upper bound for the regularity of powers of cov…
Let $I_{G} \subset K[x_{1},...,x_{m}]$ be the toric ideal associated to a finite graph $G$. In this paper we study the binomial arithmetical rank and the $G$-homogeneous arithmetical rank of $I_G$ in 2 cases: $G$ is bipartite, $I_G$ is generated by quadratic binomials. In both cases we prove that the binomial arithmetical rank and the $G$-arithmetical rank coincide with the minimal number of generators of $I_G$.
Let G be a graph obtained by taking r>=2 paths and identifying all first vertices and identifying all the last vertices. We compute the Castelnuovo--Mumford regularity of the quotient S/I(X), where S is the polynomial ring on the edges of G and I(X) is the vanishing ideal of the projective toric subset parameterized by G. The case we consider is the first case where the regularity was unknown, following earlier computations (by several authors) of the regularity when G is a tree, cycle, complete graph or complete bipartite graph, but specially in light of the reduction of the computation of th…
In this paper, we explain the regularity, projective dimension and depth of edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a $C_5$-free vertex decomposable graph $G$, $\T{reg}(R/I(G))= c_G$, where $c_G$ is the maximum number of 3-disjoint edges in $G$. Moreover for this class of graphs we characterize $\T{pd}(R/I(G))$ and $\T{depth}(R/I(G))$. As a corollary we describe these invariants in forests and sequentially Cohen-Macaulay bipartite graphs.
We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the $h^\ast$-polynomial in case of complete bipartite graphs. In particular, we show that the $h^\ast$-polynomial is $γ$-positive and real-rooted. This proves Gal's conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthing due to Nevo and Petersen (20…
Bounds for the Castelnuovo-Mumford regularity and Hilbert coefficients are given in terms of the arithmetic degree (if the ring is reduced) or in terms of the defining degrees. From this it follows that there exists only a finite number of Hilbert functions associated to reduced algebras over an algebraically closed field with a given arithmetic degree and dimension. A good bound is also given for the Castelnuovo-Mumford regularity of initial ideals which depends neither on term orders nor on the coordinates, and holds for any field.
Generalized Andrásfai graphs and special Betti diagrams of edge ideals
Instituto de Investigación en Matemáticas (IMUVa), Universidad de Valladolid（瓦拉多利德大学数学研究所）；Instituto de Matemáticas y Aplicaciones (IMAULL), Sección de Matemáticas, Facultad de Ciencias, Universidad de La Laguna（拉帕鲁纳大学理学院数学系数学与应用研究所）
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 $S=\mathbb{K}[x_1,\dots,x_n]$ 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…
We call a (simple) graph G codismantlable if either it has no edges or else it has a codominated vertex x, meaning that the closed neighborhood of x contains that of one of its neighbor, such that G-x codismantlable. We prove that if G is well-covered and it lacks induced cycles of length four, five and seven, than the vertex decomposability, codismantlability and Cohen-Macaulayness for G are all equivalent. The rest deals with the computation of Castelnuovo-Mumford regularity of codismantlable graphs. Note that our approach complements and unifies many of the earlier results on bipartite, cho…
Let $J_G$ denote the binomial edge ideal of a connected undirected graph on $n$ vertices. This is the ideal generated by the binomials $x_iy_j - x_jy_i, 1\leq i < j \leq n,$ in the polynomial ring $S= K[x_1,...,x_n,y_1,...,y_n]$ where $\{i,j\}$ is an edge of $G$. We study the arithmetic properties of $S/J_G$ for $G$, the complete bipartite graph. In particular we compute dimensions, depths, Castelnuovo-Mumford regularities, Hilbert functions and multiplicities of them. As main results we give an explicit description of the modules of deficiencies, the duals of local cohomology modules, and pro…
This work concerns the study of properties of a group of Koszul algebras coming from the toric ideals of a chordal bipartite infinite family of graphs (alternately, these rings may be interpreted as coming from determinants of certain ladder-like structures). We determine a linear system of parameters for each ring and explicitly determine the Hilbert series for the resulting Artinian reduction. As corollaries, we obtain the multiplicity and regularity of the original rings. This work extends results easily derived from lattice theory for a subfamily coming from a two-sided ladder to a family …
We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones, and we show the extremal Betti numbers of some classes of Cohen-Macaulay binomial edge ideals: Cohen-Macaulay bipartite and fan graphs. In addition, we compute the Hilbert-Poincaré series of the binomial edge ideals of some Cohen-Macaulay bipartite graphs.
We study the basic $k$-covers of a bipartite graph $G$; the algebra $\AG$ they span, first studied by Herzog, is the fiber cone of the Alexander dual of the edge ideal. We characterize when $\AG$ is a domain in terms of the combinatorics of $G$; if follows from a result of Hochster that when $\AG$ is a domain, it is also Cohen-Macaulay. We then study the dimension of $\AG$ by introducing a geometric invariant of bipartite graphs, the "graphical dimension". We show that the graphical dimension of $G$ is not larger than $\dim(\AG)$, and equality holds in many cases (e.g. when $G$ is a tree, or a…
Previously, Ohsugi and Hibi gave a combinatorial description of bipartite graphs $G$ whose toric edge ideal $I_G$ is generated by quadrics, showing that every cycle of $G$ of length at least $6$ must have a chord. This corresponds to the Green-Lazarsfeld condition $\mathbf{N}_1$. In this paper, we investigate the higher syzygies of $I_G$ and give combinatorial descriptions of the Green-Lazarsfeld conditions $\mathbf{N}_p$ of toric edge ideals of bipartite graphs for all $p \ge 1$. In particular, we show that $I_G$ is linearly presented (i.e. satisfies condition $\mathbf{N}_2$) if and only if t…
In this paper we introduce $p-$Ferrer diagram, note that $1-$ Ferrer diagram are the usual Ferrer diagrams or Ferrer board, and corresponds to planar partitions. To any $p-$Ferrer diagram we associate a $p-$Ferrer ideal. We prove that $p-$Ferrer ideal have Castelnuovo mumford regularity $p+1$. We also study Betti numbers, minimal resolutions of $p-$Ferrer ideals. Every $p-$Ferrer ideal is $p-$joined ideals in a sense defined in a fortcoming paper \cite{m2}, which extends the notion of linearly joined ideals introduced and developped in the papers \cite{bm2}, \cite{bm4},\cite{eghp} and \cite{m1…
Let X be an algebraic toric set in a projective space over a finite field. We study the vanishing ideal, I(X), of X and show some useful degree bounds for a minimal set of generators of I(X). We give an explicit description of a set of generators of I(X), when X is the algebraic toric set associated to an even cycle or to a connected bipartite graph with pairwise disjoint even cycles. In this case, a fomula for the regularity of I(X) is given. We show an upper bound for this invariant, when X is associated to a (not necessarily connected) bipartite graph. The upper bound is sharp if the graph …
Each partition $λ= (λ_1, λ_2, ..., λ_n)$ determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution, i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution: This is obtained by associating a suitable polyhedra…
The algebra of basic covers of a graph G, denoted by \A(G), was introduced by Juergen Herzog as a suitable quotient of the vertex cover algebra. In this paper we show that if the graph is bipartite then \A(G) is a homogeneous algebra with straightening laws and thus is Koszul. Furthermore, we compute the Krull dimension of \A(G) in terms of the combinatorics of G. As a consequence we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Finally, we characterize the Cohen-Macaulay property and the Castelnuovo-Mumford regularity of the edge ideal of a certain cl…
We introduce a new class of polynomial ideals associated to a simple graph, $G$. Let $K[E_G]$ be the polynomial ring on the edges of $G$ and $K[V_G]$ the polynomial ring on the vertices of $G$. We associate to $G$ an ideal, $I(X_G)$, defined as the preimage of $(x_i^2-x_j^2 : i,j\in V_G)\subseteq K[V_G]$ by the map $K[E_G]\to K[V_G]$ which sends a variable, $t_e$, associated to an edge $e=\{i,j\}$, to the product $x_ix_j$ of the variables associated to its vertices. We show that $K[E_G]/I(X_G)$ is a one-dimensional, Cohen-Macaulay, graded ring, that $I(X_G)$ is a binomial ideal and that, with …
We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global $F$-regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita's conjecture to mixed characteristic.
