Induced Matchings and the v-Number of Graded Ideals

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><...

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Main Authors:	Gonzalo Grisalde, Enrique Reyes, Rafael H. Villarreal
Format:	article
Language:	EN
Published:	MDPI AG 2021
Subjects:	graded ideals v-number induced matchings edge ideals regularity very well-covered graphs Mathematics QA1-939
Online Access:	https://doaj.org/article/659ec7b0da7e42ca9d4fefed4cd2bc1e
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spelling	oai:doaj.org-article:659ec7b0da7e42ca9d4fefed4cd2bc1e2021-11-25T18:16:39ZInduced Matchings and the v-Number of Graded Ideals10.3390/math92228602227-7390https://doaj.org/article/659ec7b0da7e42ca9d4fefed4cd2bc1e2021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2860https://doaj.org/toc/2227-7390We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> of a graph <i>G</i>, the induced matching number of <i>G</i> is an upper bound for the v-number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> when <i>G</i> is very well-covered, or <i>G</i> has a simplicial partition, or <i>G</i> is well-covered connected and contains neither four, nor five cycles. In all these cases, the v-number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a lower bound for the regularity of the edge ring of <i>G</i>. We classify when the induced matching number of <i>G</i> is an upper bound for the v-number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> when <i>G</i> is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mn>2</mn></msub></semantics></math></inline-formula>-graphs.Gonzalo GrisaldeEnrique ReyesRafael H. VillarrealMDPI AGarticlegraded idealsv-numberinduced matchingsedge idealsregularityvery well-covered graphsMathematicsQA1-939ENMathematics, Vol 9, Iss 2860, p 2860 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	graded ideals v-number induced matchings edge ideals regularity very well-covered graphs Mathematics QA1-939
spellingShingle	graded ideals v-number induced matchings edge ideals regularity very well-covered graphs Mathematics QA1-939 Gonzalo Grisalde Enrique Reyes Rafael H. Villarreal Induced Matchings and the v-Number of Graded Ideals
description	We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> of a graph <i>G</i>, the induced matching number of <i>G</i> is an upper bound for the v-number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> when <i>G</i> is very well-covered, or <i>G</i> has a simplicial partition, or <i>G</i> is well-covered connected and contains neither four, nor five cycles. In all these cases, the v-number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a lower bound for the regularity of the edge ring of <i>G</i>. We classify when the induced matching number of <i>G</i> is an upper bound for the v-number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> when <i>G</i> is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mn>2</mn></msub></semantics></math></inline-formula>-graphs.
format	article
author	Gonzalo Grisalde Enrique Reyes Rafael H. Villarreal
author_facet	Gonzalo Grisalde Enrique Reyes Rafael H. Villarreal
author_sort	Gonzalo Grisalde
title	Induced Matchings and the v-Number of Graded Ideals
title_short	Induced Matchings and the v-Number of Graded Ideals
title_full	Induced Matchings and the v-Number of Graded Ideals
title_fullStr	Induced Matchings and the v-Number of Graded Ideals
title_full_unstemmed	Induced Matchings and the v-Number of Graded Ideals
title_sort	induced matchings and the v-number of graded ideals
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/659ec7b0da7e42ca9d4fefed4cd2bc1e
work_keys_str_mv	AT gonzalogrisalde inducedmatchingsandthevnumberofgradedideals AT enriquereyes inducedmatchingsandthevnumberofgradedideals AT rafaelhvillarreal inducedmatchingsandthevnumberofgradedideals
_version_	1718411385559318528

Induced Matchings and the v-Number of Graded Ideals

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