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Cohen-Macaulayness and Limit Behavior of Depth for Powers of Cover Ideals
, Article Communications in Algebra ; Volume 43, Issue 1 , Aug , 2015 , Pages 143-157 ; 00927872 (ISSN) ; Pournaki, M. R ; Seyed Fakhari, S. A ; Terai, N ; Yassemi, S ; Sharif University of Technology
Taylor and Francis Inc
2015
Abstract
Let K{double-struck} be a field, and let R = K{double-struck}[x1,.., xn] be the polynomial ring over K{double-struck} in n indeterminates x1,.., xn. Let G be a graph with vertex-set {x1,.., xn}, and let J be the cover ideal of G in R. For a given positive integer k, we denote the kth symbolic power and the kth bracket power of J by J(k) and J[k], respectively. In this paper, we give necessary and sufficient conditions for R/Jk, R/J (k), and R/J [k] to be Cohen-Macaulay. We also study the limit behavior of the depths of these rings
A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebra
, Article Research in Mathematical Sciences ; Volume 9, Issue 2 , 2022 ; 25220144 (ISSN) ; Pournaki, M. R ; Seyed Fakhari, S. A ; Terai, N ; Yassemi, S ; Sharif University of Technology
Springer Science and Business Media Deutschland GmbH
2022
Abstract
A very well-covered graph is a well-covered graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. In this survey article, we gather together most of the old and new results on the edge and cover ideals of these graphs. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG