The projective dimension of the edge ideal of a very well-covered graph

Kyouko Kimura; Naoki Terai; Siamak Yassemi

doi:10.1017/nmj.2017.7

The projective dimension of the edge ideal of a very well-covered graph

Kyouko Kimura, Naoki Terai, Siamak Yassemi

Research output: Contribution to journal › Article › peer-review

9 Citations (Scopus)

Abstract

A very well-covered graph is an unmixed graph whose covering number is half of the number of vertices. We construct an explicit minimal free resolution of the cover ideal of a Cohen-Macaulay very well-covered graph. Using this resolution, we characterize the projective dimension of the edge ideal of a very well-covered graph in terms of a pairwise -disjoint set of complete bipartite subgraphs of the graph. We also show nondecreasing property of the projective dimension of symbolic powers of the edge ideal of a very well-covered graph with respect to the exponents.

Original language	English
Pages (from-to)	160-179
Number of pages	20
Journal	Nagoya Mathematical Journal
Volume	230
DOIs	https://doi.org/10.1017/nmj.2017.7
Publication status	Published - Jun 1 2018
Externally published	Yes

ASJC Scopus subject areas

Mathematics(all)

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10.1017/nmj.2017.7

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title = "The projective dimension of the edge ideal of a very well-covered graph",

abstract = "A very well-covered graph is an unmixed graph whose covering number is half of the number of vertices. We construct an explicit minimal free resolution of the cover ideal of a Cohen-Macaulay very well-covered graph. Using this resolution, we characterize the projective dimension of the edge ideal of a very well-covered graph in terms of a pairwise -disjoint set of complete bipartite subgraphs of the graph. We also show nondecreasing property of the projective dimension of symbolic powers of the edge ideal of a very well-covered graph with respect to the exponents.",

author = "Kyouko Kimura and Naoki Terai and Siamak Yassemi",

note = "Funding Information: Kimura was partially supported by JSPS Grant-in-Aid for Young Scientists (B) 24740008/15K17507. Terai was partially supported by JSPS Grant-in-Aid (C) 26400049 and thanks the American Institute of Mathematics for giving the chance to participate in SQuaRE Ordinary powers and symbolic powers. Yassemi was partially supported by a grant from University of Tehran. The authors thank the referee for many comments Publisher Copyright: {\textcopyright} 2017 by The Editorial Board of the Nagoya Mathematical Journal.",

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AU - Terai, Naoki

AU - Yassemi, Siamak

N1 - Funding Information: Kimura was partially supported by JSPS Grant-in-Aid for Young Scientists (B) 24740008/15K17507. Terai was partially supported by JSPS Grant-in-Aid (C) 26400049 and thanks the American Institute of Mathematics for giving the chance to participate in SQuaRE Ordinary powers and symbolic powers. Yassemi was partially supported by a grant from University of Tehran. The authors thank the referee for many comments Publisher Copyright: © 2017 by The Editorial Board of the Nagoya Mathematical Journal.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - A very well-covered graph is an unmixed graph whose covering number is half of the number of vertices. We construct an explicit minimal free resolution of the cover ideal of a Cohen-Macaulay very well-covered graph. Using this resolution, we characterize the projective dimension of the edge ideal of a very well-covered graph in terms of a pairwise -disjoint set of complete bipartite subgraphs of the graph. We also show nondecreasing property of the projective dimension of symbolic powers of the edge ideal of a very well-covered graph with respect to the exponents.

AB - A very well-covered graph is an unmixed graph whose covering number is half of the number of vertices. We construct an explicit minimal free resolution of the cover ideal of a Cohen-Macaulay very well-covered graph. Using this resolution, we characterize the projective dimension of the edge ideal of a very well-covered graph in terms of a pairwise -disjoint set of complete bipartite subgraphs of the graph. We also show nondecreasing property of the projective dimension of symbolic powers of the edge ideal of a very well-covered graph with respect to the exponents.

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The projective dimension of the edge ideal of a very well-covered graph

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