Castelnuovo-Mumford regularity of graphs

Castelnuovo-Mumford正则性与树的组合不变量

Castelnuovo-Mumford Regularity and Combinatorial Invariants of Trees

本文研究了树的边理想及其多须变体的Castelnuovo-Mumford正则性，用阶、直径和悬挂顶点数给出组合界，并导出了诱导匹配数和独立数的不等式。

A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements

We show that the ideal of an arrangement of d linear subspaces of projective space is d-regular in the sense of Castelnuovo and Mumford, answering a question of B. Sturmfels. In particular this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d.

Castelnuovo-Mumford regularity and Schubert geometry

We study the Castelnuovo-Mumford regularity of tangent cones of Schubert varieties. Conjectures about this statistic are presented; these are proved for the covexillary case. This builds on work of L. Li and the author on these tangent cones, as well as that of J. Rajchgot-Y. Ren-C. Robichaux-A. St. Dizier-A. Weigandt and J. Rajchgot-C. Robichaux-A. Weigandt on tableau rules for computing regularity of some matrix Schubert varieties.

Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity

We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation. We then provide a counterexample to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri on a formula for regularities of standard open patches of particular Grassmannian Schubert varieties and show that our work gives rise to an alternate explicit formula in these cases. We end with a new conjecture on the regularities of standard open patches of arbitrary Gr…

Castedelnuovo-Mumford多项式的支持

Supports of Castelnuovo-Mumford polynomials

本文定义了新的排列家族，证明其Castedelnuovo-Mumford多项式支持为M-凸，揭示了schubitope的整数点结构。

Projective dimension of (hyper)graphs and the Castelnuovo-Mumford regularity of bipartite graphs

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)).

On the Castelnuovo-Mumford regularity of products of ideal sheaves

In this paper we give bounds on the Castelnuovo-Mumford regularity of products of ideals and ideal sheaves. In particular, we show that the regularity of a product of ideals is bounded by the sum of the regularities of its factors if the corresponding schemes intersect in a finite set of points. We also show how approximations of sheaves can be used to bound the regularity of an arrangement of two-planes in projective space.

Connectivity through bounds for the Castelnuovo-Mumford regularity

We present a simple method to obtain information regarding the connectivity of the 1-skeleta of a wide family of simplicial complexes through bounds for the Castelnuovo-Mumford regularity of their Stanley-Reisner rings. In this way we generalize and unify two results on connectivity: one by Balinsky and Barnette, one by Athanasiadis. In particular, if $Δ$ is a simplicial $d$-pseudomanifold, and $s$ is the highest integer such that there is an $s$-dimensional simplex not contained in $Δ$, but such that its boundary is, then the 1-skeleton of $Δ$ is $\left\lceil \frac{(s+1)d}{s} \right\rceil$-co…

Castelnuovo-Mumford regularity of ladder determinantal varieties and patches of Grassmannian Schubert varieties

We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and $1432$-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which appeared in previous joint work of the authors with Y. Ren and A. St. Dizier. We apply our formulas to compute Castelnuovo-Mumford regularity of classes of generalized determinantal ideals. In particular, we give combinatorial formulas for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of cert…

图的广义二项式边理想Castelnuovo-Mumford正则度研究

Castelnuovo-Mumford regularity of generalized binomial edge ideals of graphs

本文研究图的广义二项式边理想的正则度，证明其范围为2到n-1，并给出P4-free图的新特征化。

Castelnuovo-Mumford regularity by approximation

The Castelnuovo-Mumford regularity of a module gives a rough measure of its complexity. We bound the regularity of a module given a system of approximating modules whose regularities are known. Such approximations can arise naturally for modules constructed by inductive combinatorial means. We apply these methods to bound the regularity of ideals constructed as combinations of linear ideals and the module of derivations of a hyperplane arrangement as well as to give degree bounds for invariants of finite groups.

$321$-avoiding Kazhdan-Lusztig 簇的 Castelnuovo-Mumford 正则度

Castelnuovo-Mumford regularity for $321$-avoiding Kazhdan-Lusztig varieties

本文证明 $321$-avoiding Kazhdan-Lusztig 簇的正则度可由 $K$-理论斜激发杨图组合计算，提出下界算法并用于精确计算双侧混合阶梯行列式簇。

An Eisenbud-Goto-type Upper Bound for the Castelnuovo-Mumford Regularity of Fake Weighted Projective Spaces

We will give an upper bound for the $k$-normality of very ample lattice simplices, and then give an Eisenbud-Goto-type bound for some special classes of projective toric varieties.

斜对称矩阵Schubert簇的Castelnuovo-Mumford正则度

Castelnuovo-Mumford regularity of skew-symmetric matrix Schubert varieties

本文通过给出辛Grothendieck多项式次数的组合公式，计算了斜对称矩阵Schubert簇的Castelnuovo-Mumford正则度，并刻画了其最高次齐次分量及最大正则度。

图的闭邻域理想的Castelnuovo-Mumford正则度

Castelnuovo-Mumford regularity of the closed neighborhood ideal of a graph

该研究围绕图的闭邻域理想的同调不变量展开，证明了森林情形下其Castelnuovo-Mumford正则度等于匹配数（证实Sharifan与Moradi的猜想），给出任意图下该正则度的匹配数下界，还证明含单纯顶点时理想有Betti分裂，且森林与单圈图的投射维数也以匹配数为下界。

On the Castelnuovo-Mumford regularity of squarefree powers of edge ideals

Assume that $G$ is a graph with edge ideal $I(G)$ and matching number ${\rm match}(G)$. For every integer $s\geq 1$, we denote the $s$-th squarefree power of $I(G)$ by $I(G)^{[s]}$. It is shown that for every positive integer $s\leq {\rm match}(G)$, the inequality ${\rm reg}(I(G)^{[s]})\leq {\rm match}(G)+s$ holds provided that $G$ belongs to either of the following classes: (i) very well-covered graphs, (ii) semi-Hamiltonian graphs, or (iii) sequentially Cohen-Macaulay graphs. Moreover, we prove that for every Cameron-Walker graph $G$ and for every positive integer $s\leq {\rm match}(G)$, we …

拟正则性与至多一个奇点的托里克品种

Castelnuovo-Mumford regularity of toric varieties with at most one singular point

研究了至多一个奇点的简单投影托里克品种的拟正则性上界，结合组合与同调方法证明了Eisenbud-Goto界在光滑和单奇点情况下均成立。

关于覆盖理想的符号幂的Castelnuovo-Mumford正则度

On the Castelnuovo-Mumford regularity of symbolic powers of cover ideals

本文研究了图覆盖理想符号幂的正则度，基于星装填数给出了精确上界，并计算了双重Cohen-Macaulay图、Cameron-Walker图及特定无爪图的正则度。

Joins, Ears and Castelnuovo-Mumford regularity

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 …

Krull dimension and regularity of binomial edge ideals of block graphs

We give a lower bound for the Castelnuovo-Mumford regularity of binomial edge ideals of block graphs by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs. Moreover, we present a linear time algorithm to compute the Castelnuovo-Mumford regularity and Krull dimension of binomial edge ideals of block graphs.

Regularity and multiplicity of toric rings of three-dimensional Ferrers diagrams

We investigate the Castelnuovo-Mumford regularity and the multiplicity of the toric ring associated with a three-dimensional Ferrers diagram. In particular, in the rectangular case, we provide direct formulas for these two important invariants. Then, we compare these invariants for an accompanying pair of Ferrers diagrams under some mild conditions and bound the Castelnuovo-Mumford regularity for more general cases.

弦二部图、双团顶点划分与1-细分图的Castelnuovo-Mumford正则性

Chordal bipartite graphs, biclique vertex partitions and Castelnuovo-Mumford regularity of $1$-subdivision graphs

本文研究1-细分图的正则性与双团顶点划分数之间的关系，证明了一般图的下界不等式，并针对弦二部图建立了等式，同时刻画了其独立复形的同伦类型。

二项式边理想的Castelnuovo-Mumford正则性的组合数学研究

Combinatorics of Castelnuovo-Mumford Regularity of Binomial Edge Ideals

该研究提出新的图不变量$ν(G)$，证明其与二项式边理想Castelnuovo-Mumford正则性的不等式关系，刻画等式成立的多类图情形，最终给出块图正则性的组合解释，解决了相关公开问题。

Some algebraic invariants of edge ideal of circulant graphs

Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq\{ 1,\ldots,\left \lfloor\frac{n}{2}\right \rfloor\}$ and let $I(G)$ be its edge ideal in the ring $K[x_0,\ldots,x_{n-1}]$. Under the hypothesis that $n$ is prime we : 1) compute the regularity index of $R/I(G)$; 2) compute the Castelnuovo-Mumford regularity when $R/I(G)$ is Cohen-Macaulay; 3) prove that the circulant graphs with $S=\{1,\ldots,s\}$ are sequentially $S_2$ . We end characterizing the Cohen-Macaulay circulant graphs of Krull dimension $2$ and computing their Cohen-Macaulay type and Castelnuovo-Mumford regularity.

容许匹配与无平方幂的Castelnuovo-Mumford正则度

Admissible matchings and the Castelnuovo-Mumford regularity of square-free powers

本文引入超图的广义$k$-容许匹配概念，给出无平方单项式理想$k$次幂正则度的紧下界，证明块图等特殊情形下的等式，并提出弦图边理想无平方幂正则度的猜想。

关于单项式理想幂的正则性的若干不等式

Some inequalities regarding the regularity of powers of monomial ideals

本文研究单项式理想幂的正则性，证明了其符号幂与幂的整闭包的 Castelnuovo–Mumford 正则性所满足的若干不等式。

On vertex decomposability and regularity of graphs

There are two motivation questions in \cite{MTS, MTS1} about Castelnuovo-Mumford regularity and vertex decomposable of simple graph $G$. In this paper, we disprove the questions by providing of two counterexamples.

Bounds on Regularity of Quadratic Monomial Ideals

Castelnuovo-Mumford regularity is a measure of algebraic complexity of an ideal. Regularity of monomial ideals can be investigated combinatorially. We use a simple graph decomposition and results from structural graph theory to prove, improve and generalize many of the known bounds on regularity of quadratic square-free monomial ideals.

Sequentially Cohen-Macaulay bipartite graphs: vertex decomposability and regularity

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.

Regularity of Edge Ideals Via Suspension

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.

Regularity of Edge Ideals and Their Powers

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{reg} I(G)$ and the asymptotic linear function $\text{reg} I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.$

Binomial edge ideals of cographs

We determine the Castelnuovo-Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). For cographs with $n$ vertices the maximum regularity grows as $2n/3$. We also bound the regularity by graph theoretic invariants and construct a family of counterexamples to a conjecture of Hibi and Matsuda.

The regularity of some families of circulant graphs

We compute the Castelnuovo-Mumford regularity of the edge ideals of two families of circulant graphs, which includes all cubic circulant graphs. A feature of our approach is to combine bounds on the regularity, the projective dimension, and the reduced Euler characteristic to derive an exact value for the regularity.