Bounding Betti numbers of bipartite graph ideals

Dependence of Betti Numbers on Characteristic

We study the dependence of graded Betti numbers of monomial ideals on the characteristic of the base field. The examples we describe include bipartite ideals, Stanley--Reisner ideals of vertex-decomposable complexes and ideals with componentwise linear resolutions. We give a description of bipartite graphs and, using discrete Morse theory, provide a way of looking at the homology of arbitrary simplicial complexes through bipartite ideals. We also prove that the Betti table of a monomial ideal over the field of rational numbers can be obtained from the Betti table over any field by a sequence o…

Extremal Betti numbers of some classes of binomial edge ideals

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.

二部平面图的代数与几何方面

Algebraic and geometric aspects of bipartite planar graphs

本文研究具有偶数个区域的二部平面图，给出其关联商环的分次 Betti 数与投射维数的界，并探讨最小顶点覆盖和最大匹配。

关于某些平面图的Betti数与优美性

On the Betti numbers and gracefulness of some planar graphs

本文研究二部平面图 St_r 的代数与标号性质，建立了区域数 r 与边理想投射分解中分次 Betti 数、投射维数等代数不变量的关系，并证明该类图具有边优美标号。

Very well-covered graphs by Betti splittings

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…

Resolution 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.

Regularity 3 in edge ideals associated to bipartite graphs

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…

广义安德拉斯法伊图和边理想特殊贝蒂图

Generalized Andrásfai graphs and special Betti diagrams of edge ideals

研究广义安德拉斯法伊图的边理想特性，发现其贝蒂图具有特殊形状，扩展了完整图和完整双图的正则度和射影维数结果。

Betti numbers of subgraphs

Let $G$ be a simple graph on $n$ vertices. Let $H$ be either the complete graph $K_m$ or the complete bipartite graph $K_{r,s}$ on a subset of the vertices in $G$. We show that $G$ contains $H$ as a subgraph if and only if $β_{i,α}(H) \le β_{i,α}(G)$ for all $i \ge 0$ and $α\in \mathbb{Z}^n$. In fact, it suffices to consider only the first syzygy module. In particular, we prove that $β_{1,α}(H) \le β_{1,α}(G)$ for all $α\in \mathbb{Z}^n$ if and only if $G$ contains a subgraph that is isomorphic to either $H$ or a multipartite graph $K_{2,\dots,2,a,b}$.

Betti numbers of toric ideals of graphs: A case study

We compute the graded Betti numbers for the toric ideal of a family of graphs constructed by adjoining a cycle to a complete bipartite graph. The key observation is that this family admits an initial ideal which has linear quotients. As a corollary, we compute the Hilbert series and $h$-vector for all the toric ideals of graphs in this family.

Certain Homological Invariants of Bipartite Kneser Graphs

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.

Splittings of monomial ideals

We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bipartite graphs. Finally, we consider the frequency with which one can find particular splittings of monomial ideals and raise questions about ideals whose resolutions are characteristic-dependent.

Distributive Lattices, Bipartite Graphs and Alexander Duality

A certain squarefree monomial ideal $H_P$ arising from a finite partially ordered set $P$ will be studied from viewpoints of both commutative algebra and combinatorics. First, it is proved that the defining ideal of the Rees algebra of $H_P$ possesses a quadratic Gröbner basis. Thus in particular all powers of $H_P$ have linear resolutions. Second, the minimal free graded resolution of $H_P$ will be constructed explicitly and a combinatorial formula to compute the Betti numbers of $H_P$ will be presented. Third, by using the fact that the Alexander dual of the simplicial complex $Δ$ whose Stan…

On vertex decomposable simplicial complexes and their Alexander duals

In this paper we study the Alexander dual of a vertex decomposable simplicial complex. We define the concept of a vertex splittable ideal and show that a simplicial complex $Δ$ is vertex decomposable if and only if $I_{Δ^{\vee}}$ is a vertex splittable ideal. Moreover, the properties of vertex splittable ideals are studied. As the main result, it is proved that any vertex splittable ideal has a Betti splitting and the graded Betti numbers of such ideals are explained with a recursive formula. As a corollary, recursive formulas for the regularity and projective dimension of $R/I_Δ$, when $Δ$ is…

2-线性邻域复形的完全分类

A Complete Classification of 2-Linear Neighborhood Complexes

本文刻画了图G的Stanley-Reisner环k[N(G)]何时具有2-线性分解，等价于G是邻域共形且其公共邻域图是弦图，即G是仅含4-圈作为诱导圈的二分图，并由此得到Katzman下界等号成立条件及Cohen-Macaulay支配复形的分类。

Bounds on the regularity of toric ideals of 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 $K_{2,n}$.

通过组合不变量对超图边理想的代数不变量的若干解释

Some interpretations for algebraic invariants of edge ideals of hypergraphs via combinatorial invariants

本文引入并比较几类类似匹配数的超图组合不变量，用以刻画和界定边理想的分次 Betti 数、投射维数与 CM 正则度等代数不变量。

边添加下非匹配复形的同调及其在Stanley-Reisner理想中的应用

Homology of non-matching complexes under edge additions and applications to their Stanley-Reisner ideals

该研究针对二分图的$t$-非匹配复形，证明边添加诱导同调单射，结合模型得到Leray数等结果，还确定了完全二分图对应理想的相关同调性质并解决了一个猜想。

加权有向图边理想的同调不变量

Homological invariants of edge ideals of weighted oriented graphs

该研究确定了固定顶点数加权有向图边理想的深度、维数和正则性三元组，并计算了固定顶点数加权有向树及二部图边理想的贝蒂表大小。

偶环二部补的二项式边理想

Binomial edge ideals of bipartite complements of even cycles

本文研究偶环二部补的二项式边理想，通过割点性质分类描述极小素理想，并研究维数、Betti数、Hilbert级数、射影维数、深度及正则性等不变量。

闭邻域理想与Barile-Macchia resolutions

Barile-Macchia Resolutions and the closed neighborhood ideal

本文研究了图的闭邻域理想在BM resolutions框架下的极小自由分辨率，并证明了树的闭邻域理想具有桥友性，从而得到其射影维数等于独立数的结论。

单纯复形的线图

Line graphs of simplicial complexes

本文研究纯单纯复形的线图，证明可基于其线图的组合结构计算面理想的第二阶分次Betti数，并刻画了线图为完全（二部）图的纯单纯复形及线图应满足的条件。