Symbolic powers of vertex cover ideals

Let [Formula: see text] be a finite simple graph and [Formula: see text] denote its vertex cover ideal in a polynomial ring over a field [Formula: see text]. In this paper, we show that all symbolic powers of vertex cover ideals of certain vertex-decomposable graphs have linear quotients. Using these results, we give various conditions on a subset [Formula: see text] of the vertices of [Formula: see text] so that all symbolic powers of vertex cover ideals of [Formula: see text], obtained from [Formula: see text] by adding a whisker to each vertex in [Formula: see text], have linear quotients. For instance, if [Formula: see text] is a vertex cover of [Formula: see text], then all symbolic powers of [Formula: see text] have linear quotients. Moreover, we compute the Castelnuovo–Mumford regularity of symbolic powers of certain vertex cover ideals.

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Upper bounds for the regularity of symbolic powers of certain classes of edge ideals

Journal of Algebra and Its Applications ◽

10.1142/s0219498823500160 ◽

2021 ◽

Author(s):

Arvind Kumar ◽

S. Selvaraja

Keyword(s):

Polynomial Ring ◽

Upper Bounds ◽

Simple Graph ◽

Edge Ideal ◽

Edge Ideals ◽

Symbolic Powers

Let [Formula: see text] be a finite simple graph and [Formula: see text] denote the corresponding edge ideal in a polynomial ring over a field [Formula: see text]. In this paper, we obtain upper bounds for the Castelnuovo–Mumford regularity of symbolic powers of certain classes of edge ideals. We also prove that for several classes of graphs, the regularity of symbolic powers of their edge ideals coincides with that of their ordinary powers.

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Vertex-Decomposable Graphs, Codismantlability, Cohen-Macaulayness, and Castelnuovo-Mumford Regularity

The Electronic Journal of Combinatorics ◽

10.37236/2387 ◽

2014 ◽

Vol 21 (1) ◽

Cited By ~ 9

Author(s):

Türker Bıyıkoğlu ◽

Yusuf Civan

Keyword(s):

Vertex Cover ◽

Infinite Family ◽

Matching Number ◽

Induced Matching ◽

Free Vertex ◽

Acyclic Digraph ◽

Vertex Decomposable ◽

Decomposable Graphs

We call a vertex $x$ of a graph $G=(V,E)$ a codominated vertex if $N_G[y]\subseteq N_G[x]$ for some vertex $y\in V\backslash \{x\}$, and a graph $G$ is called codismantlable if either it is an edgeless graph or it contains a codominated vertex $x$ such that $G-x$ is codismantlable. We show that $(C_4,C_5)$-free vertex-decomposable graphs are codismantlable, and prove that if $G$ is a $(C_4,C_5,C_7)$-free well-covered graph, then vertex-decomposability, codismantlability and Cohen-Macaulayness for $G$ are all equivalent. These results complement and unify many of the earlier results on bipartite, chordal and very well-covered graphs. We also study the Castelnuovo-Mumford regularity $reg(G)$ of such graphs, and show that $reg(G)=im(G)$ whenever $G$ is a $(C_4,C_5)$-free vertex-decomposable graph, where $im(G)$ is the induced matching number of $G$. Furthermore, we prove that $H$ must be a codismantlable graph if $im(H)=reg(H)=m(H)$, where $m(H)$ is the matching number of $H$. We further describe an operation on digraphs that creates a vertex-decomposable and codismantlable graph from any acyclic digraph. By way of application, we provide an infinite family $H_n$ ($n\geq 4$) of sequentially Cohen-Macaulay graphs whose vertex cover numbers are half of their orders, while containing no vertex of degree-one such that they are vertex-decomposable, and $reg(H_n)=im(H_n)$ if $n\geq 6$. This answers a recent question of Mahmoudi et al.

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A Construction of Sequentially Cohen–Macaulay Graphs

Algebra Colloquium ◽

10.1142/s1005386721000316 ◽

2021 ◽

Vol 28 (03) ◽

pp. 399-414

Author(s):

Aming Liu ◽

Tongsuo Wu

Keyword(s):

Simplicial Complex ◽

Betti Numbers ◽

Simple Graph ◽

Ideal Formula ◽

Vertex Decomposable ◽

Linear Quotients ◽

Facet Ideal

For every simple graph [Formula: see text], a class of multiple clique cluster-whiskered graphs [Formula: see text] is introduced, and it is shown that all such graphs are vertex decomposable; thus, the independence simplicial complex [Formula: see text] is sequentially Cohen–Macaulay. The properties of the graphs [Formula: see text] and [Formula: see text] constructed by Cook and Nagel are studied, including the enumeration of facets of the complex [Formula: see text] and the calculation of Betti numbers of the cover ideal [Formula: see text]. We also prove that the complex[Formula: see text] is strongly shellable and pure for either a Boolean graph [Formula: see text] or the full clique-whiskered graph [Formula: see text] of [Formula: see text], which is obtained by adding a whisker to each vertex of [Formula: see text]. This implies that both the facet ideal [Formula: see text] and the cover ideal [Formula: see text] have linear quotients.

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On regularity of symbolic Rees algebras and symbolic powers of vertex cover ideals of graphs

10.1090/proc/15824 ◽

2021 ◽

Author(s):

Ramakrishna Nanduri

Keyword(s):

Vertex Cover ◽

Rees Algebras ◽

Symbolic Powers

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Symbolic powers of cover ideals of graphs and Koszul property

International Journal of Algebra and Computation ◽

10.1142/s0218196721500405 ◽

2021 ◽

pp. 1-17

Author(s):

Yan Gu ◽

Huy Tài Hà ◽

Joseph W. Skelton

Keyword(s):

Vertex Cover ◽

Cycle Cover ◽

Symbolic Powers ◽

Koszul Property

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

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Symbolic Powers of Regular Primes

Canadian Journal of Mathematics ◽

10.4153/cjm-1981-102-2 ◽

1981 ◽

Vol 33 (6) ◽

pp. 1331-1337 ◽

Cited By ~ 1

Author(s):

Yasunori Ishibashi

Keyword(s):

Prime Ideal ◽

Finite Type ◽

Polynomial Ring ◽

Type A ◽

Symbolic Power ◽

Perfect Field ◽

Symbolic Powers ◽

Higher Derivatives ◽

Differential Filtration ◽

Derivatives Of

In a recent paper [6], P. Seibt has obtained the following result: Let k be a field of characteristic 0, k[T1, … , Tr] the polynomial ring in r indeterminates over k, and let P be a prime ideal of k[T1, … , Tr]. Then a polynomial F belongs to the n-th symbolic power P(n) of P if and only if all higher derivatives of F from the 0-th up to the (n – l)-st order belong to P.In this work we shall naturally generalize this result so as to be valid for primes of the polynomial ring over a perfect field k. Actually, we shall get a generalization as a corollary to a theorem which asserts: For regular primes P in a k-algebra R of finite type, a certain differential filtration of R associated with P coincides with the symbolic power filtration (P(n))n≧0.

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