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


Author: S. A. Seyed Fakhari
Journal: Proc. Amer. Math. Soc. 146 (2018), 97-110
MSC (2010): Primary 13D02, 05E99; Secondary 13C15
DOI: https://doi.org/10.1090/proc/13721
Published electronically: July 20, 2017
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Abstract: 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.


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Additional Information

S. A. Seyed Fakhari
Affiliation: School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran
Email: aminfakhari@ut.ac.ir

DOI: https://doi.org/10.1090/proc/13721
Keywords: Cover ideal, very well-covered graph, linear resolution, regularity
Received by editor(s): September 30, 2016
Received by editor(s) in revised form: February 28, 2017
Published electronically: July 20, 2017
Communicated by: Irena Peeva
Article copyright: © Copyright 2017 American Mathematical Society

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