关于非混合二部图的极小点覆盖半群环的维数

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 $μ(I^2)=\ell(I)μ(I)-{\ell(I)\choose 2}$ 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.

二部边理想的正则性、深度和算术秩

Regularity, Depth and Arithmetic Rank of Bipartite Edge Ideals

研究了二部图的边理想的极小自由分解。将有向图与边理想为纯的二部图联系起来，利用有向图的不变量给出了边理想的正则性和深度的表达式.对于某些类的非混合边理想，我们证明了理想的算术秩等于其商的投射维数。

二部图的二项式边理想

Binomial edge ideals of bipartite graphs

对二项式边理想$J_G$为Cohen-Macaulay的二部图$G$进行了分类.这类图的连通分支可以通过用两个操作粘合有限个基本块来获得。在这种情况下，我们证明了一个众所周知的结果，由于Hartshorne的逆，表明这些理想的科恩-麦考利性是等价于其对偶图的连通性。我们还研究了非二部图的有趣性质，并在未混合的情况下，构建类的二部图与$J_G$未混合，而不是科恩-麦考利。

属于Domain的非混合图

Unmixed Graphs that are Domains

给定任意图G，研究其基本覆盖代数，即G的边理想的Alexander对偶的符号纤维锥。推广了Villarreal和Benedetti-Constantinescu-Varbaro的结果，该结果仅在G是二分的情况下有效，我们用组合的方式刻画了以下情形：1）基本覆盖代数是整环; 2）它是整环且G（的边理想）是非混合的.最后一个结果给出了边理想的任何符号幂都由同度单项式生成的图的一个完全刻画。

边理想和完全二部子图的贝蒂数的非零性

Non-vanishing of Betti numbers of edge ideals and complete bipartite subgraphs

给定一个有限的单图，我们可以把边理想联系起来。本文证明了在一定条件下，若原图含有一组完全二部子图，则边理想的分次贝蒂数不为零。并给出了非混合二部图的边理想的投射维数的组合刻画。

Almost Cohen-Macaulay bipartite graphs and connected in codimension two

In this paper we study almost Cohen-Macaulay bipartite graphs. Furthermore, we prove that if $G$ is almost Cohen-Macaulay bipartite graph with at least one vertex of positive degree, then there is a vertex of $°(v) \leq 2$. In particular, if $G$ is an almost Cohen-Macaulay bipartite graph and $u$ is a vertex of degree one of $G$ and $v$ its adjacent vertex, then $G\setminus\{v\}$ is almost Cohen-Macaulay. Also, we show that an unmixed Ferrers graph is almost Cohen-Macaulay if and only if it is connected in codimension two. Moreover, we give some examples.

Gorenstein homogeneous subrings of graphs

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.

加权定向图的单式理想

Monomial ideals of weighted oriented graphs

设I=I（D）是加权定向图D的边理想.我们确定了I的不可冗余不可约分解。此外，我们还刻画了I的相伴素数和非混合性质。此外，当D是二部的、D是须状的或D是圈的时，我们给出了I的非混合性的一个组合刻画。最后，我们研究了I.

图的覆盖理想的符号幂的正则性

Regularity of symbolic powers of cover ideals of graphs

设$G$是一个图，它属于下列任一类：（i）二部图，（ii）纯图，或（iii）无爪图.假设$J(G)$是覆盖理想$G$，而$J(G)^{(k)}$是它的$k$次符号幂。我们证明了$$k{\rm deg}(J(G))\leq {\rm reg}(J(G)^{(k)})\leq (k-1){\rm deg}(J(G))+|V(G)|-1.$$。我们还确定了图族的上述不等式是平等的。

一维单纯复形的环面环

The toric ring of one dimensional simplicial complexes

本文研究一维单纯复形（即有限简单图）的环面环，分类正规情形并刻画其 Gorenstein 与伪 Gorenstein 性质。

非常好覆盖图的点可分解性和正则性

Vertex decomposability and regularity of very well-covered graphs

一个图$G$是好覆盖的，如果它没有孤立点，并且所有的极大独立集具有相同的基数。如果这个基数的另外两倍等于$|V(G)|$，则图$G$被称为非常好覆盖的。非常好覆盖图类包含二部好覆盖图。最近在\cite{CRT}中证明了一个非常好覆盖的图$G$是Cohen-Macaulay当且仅当它是纯可壳的。本文改进了这一结果，证明了$G$是Cohen-Macaulay的当且仅当它是纯顶点可分解的.另外，如果$I(G)$表示$G$的边理想，我们证明了$R/I(G)$的Castelnuovo-Mumford正则性等于$G$的最大3-不相交边数.这改进了Kummini关于非混合二部图的结果。

Footprint and minimum distance functions

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, …

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…