Given integers <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1421_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="104" /> </InlineMediaObject> <EquationSource Format="TEX">$2 \le p \le c \le q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>c</mi> <mo>≤</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation>, we construct a finite simple graph G with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1421_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu _1(G) = p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ν</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1421_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu (G) = q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation> for which the squarefree power <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1421_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$I(G)^{[k]}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> of the edge ideal I(G) of G has linear quotients for each <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1421_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$c \le k \le q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation> and is not linearly related for each <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1421_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$1 \le k < c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mo><</mo> <mi>c</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1421_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu _1(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ν</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the induced matching number of G and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1421_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the matching number of G.

Density of linearity index in the interval of matching numbers

期刊论文
发表日期：

2025年6月19日

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Nursel Erey,
Takayuki Hibi

摘要

Given integers $2 \le p \le c \le q$ $2 \leq p \leq c \leq q$ , we construct a finite simple graph G with $\nu _1(G) = p$ $ν_{1} (G) = p$ and $\nu (G) = q$ $ν (G) = q$ for which the squarefree power $I(G)^{[k]}$ $I {(G)}^{[k]}$ of the edge ideal I(G) of G has linear quotients for each $c \le k \le q$ $c \leq k \leq q$ and is not linearly related for each $1 \le k < c$ $1 \leq k < c$ , where $\nu _1(G)$ $ν_{1} (G)$ is the induced matching number of G and $\nu (G)$ $ν (G)$ is the matching number of G.