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

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.

Download Full-text

Splittings and Symbolic Powers of Square-free Monomial Ideals

International Mathematics Research Notices ◽

10.1093/imrn/rnz138 ◽

2019 ◽

Author(s):

Jonathan Montaño ◽

Luis Núñez-Betancourt

Keyword(s):

Linear Optimization ◽

Monomial Ideals ◽

Sufficient Condition ◽

Graded Algebra ◽

Prime Characteristic ◽

Rees Algebra ◽

Rees Algebras ◽

Symbolic Powers ◽

Frobenius Map

Abstract We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with respect to a morphism that resembles the Frobenius map and that exists in all characteristics. Using these methods, we recover a result by Hoa and Trung that states that the normalized $a$-invariants and the Castelnuovo–Mumford regularity of the symbolic powers converge. In addition, we give a sufficient condition for the equality of the ordinary and symbolic powers of this family of ideals and relate it to Conforti–Cornuéjols conjecture. Finally, we interpret this condition in the context of linear optimization.

Download Full-text

Symbolic powers of monomial ideals and vertex cover algebras

Advances in Mathematics ◽

10.1016/j.aim.2006.06.007 ◽

2007 ◽

Vol 210 (1) ◽

pp. 304-322 ◽

Cited By ~ 46

Author(s):

Jürgen Herzog ◽

Takayuki Hibi ◽

Ngô Viêt Trung

Keyword(s):

Vertex Cover ◽

Monomial Ideals ◽

Symbolic Powers

Download Full-text

Symbolic powers, Serre conditions and Cohen-Macaulay Rees algebras

manuscripta mathematica ◽

10.1007/bf02567981 ◽

1995 ◽

Vol 86 (1) ◽

pp. 113-124 ◽

Cited By ~ 4

Author(s):

Susan Morey ◽

Sunsook Noh ◽

Wolmer V. Vasconcelos

Keyword(s):

Rees Algebras ◽

Symbolic Powers

Download Full-text

Symbolic powers of vertex cover ideals

International Journal of Algebra and Computation ◽

10.1142/s0218196720500368 ◽

2020 ◽

Vol 30 (06) ◽

pp. 1167-1183

Author(s):

S. Selvaraja

Keyword(s):

Polynomial Ring ◽

Vertex Cover ◽

Simple Graph ◽

Symbolic Powers ◽

Vertex Decomposable ◽

Decomposable Graphs ◽

Linear Quotients

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.

Download Full-text