In this paper, we study the componentwise linearity of edge ideals of weighted oriented graphs. We show that if D is a weighted oriented graph whose edge ideal I(D) is componentwise linear, then the underlying simple graph G of D is co-chordal. We give combinatorial characterizations of componentwise linearity of I(D) if the vertices in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1401_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$V^+$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>V</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation> are sinks or <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1401_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\vert V^+ \vert \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msup> <mi>V</mi> <mo>+</mo> </msup> <mrow> <mo stretchy="false">|</mo> <mo>≤</mo> <mn>1</mn> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, if G belongs to certain chordal graphs or G is bipartite or the vertices in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1401_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$V^+$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>V</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation> are sinks or <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1401_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\vert V^+ \vert \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msup> <mi>V</mi> <mo>+</mo> </msup> <mrow> <mo stretchy="false">|</mo> <mo>≤</mo> <mn>1</mn> </mrow> </mrow> </math></EquationSource> </InlineEquation>, then we show the following equivalence for I(D): Vertex splittable <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1401_Article_IEq5.gif" Format="GIF" Height="6" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$\Longleftrightarrow $</EquationSource> <EquationSource Format="MATHML"><math> <mo stretchy="false">⟺</mo> </math></EquationSource> </InlineEquation> Linear quotient <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1401_Article_IEq5.gif" Format="GIF" Height="6" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$\Longleftrightarrow $</EquationSource> <EquationSource Format="MATHML"><math> <mo stretchy="false">⟺</mo> </math></EquationSource> </InlineEquation> Componentwise linear.

Componentwise linearity of edge ideals of weighted oriented graphs

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发表日期：

2025年3月26日

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Manohar Kumar,
Ramakrishna Nanduri,
Kamalesh Saha

摘要

In this paper, we study the componentwise linearity of edge ideals of weighted oriented graphs. We show that if D is a weighted oriented graph whose edge ideal I(D) is componentwise linear, then the underlying simple graph G of D is co-chordal. We give combinatorial characterizations of componentwise linearity of I(D) if the vertices in $V^+$ $V^{+}$ are sinks or $\vert V^+ \vert \le 1$ $| V^{+} | \leq 1$ . Furthermore, if G belongs to certain chordal graphs or G is bipartite or the vertices in $V^+$ $V^{+}$ are sinks or $\vert V^+ \vert \le 1$ $| V^{+} | \leq 1$ , then we show the following equivalence for I(D): Vertex splittable $\Longleftrightarrow $ $⟺$ Linear quotient $\Longleftrightarrow $ $⟺$ Componentwise linear.