Symbolic powers of cover ideal of very well-covered and bipartite graphs

Let G be a graph with n vertices and S=\mathbb{K}[x_1,\dots ,x_n] be the polynomial ring in n variables over a field \mathbb{K}. Assume that J(G) is the cover ideal of G and J(G)^{(k)} is its k-th symbolic power. We show that if G is a very well-covered graph such that J(G) has a linear resolution,...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2018-01, Vol.146 (1), p.97-110
Main Author: FAKHARI, S. A. SEYED
Format: Article
Language:English
Subjects:
A. ALGEBRA, NUMBER THEORY, AND COMBINATORICS
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Summary:Let G be a graph with n vertices and S=\mathbb{K}[x_1,\dots ,x_n] be the polynomial ring in n variables over a field \mathbb{K}. Assume that J(G) is the cover ideal of G and J(G)^{(k)} is its k-th symbolic power. We show that if G is a very well-covered graph such that J(G) has a linear resolution, then for every integer k\geq 1, the ideal J(G)^{(k)} has a linear resolution and moreover, the modules J(G)^{(k)} and S/J(G)^{(k)} satisfy Stanley's inequality, i.e., their Stanley depth is an upper bound for their depth. Finally, we determine a linear upper bound for the Castelnuovo-Mumford regularity of powers of cover ideals of bipartite graphs.
ISSN:0002-9939
1088-6826
DOI:10.1090/proc/13721