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<p>The <i>perfect matching complex</i> of a simple graph <i>G</i> is a simplicial complex having facets (maximal faces) as the <i>perfect matchings</i> of <i>G</i>. This article discusses the perfect matching complex of polygonal line tilings and the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="26_2025_755_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left( 2 \times n\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mn>2</mn> <mo>×</mo> <mi>n</mi> </mfenced> </math></EquationSource> </InlineEquation>-grid graph in particular. We use tools from discrete Morse theory to show that the perfect matching complex of any polygonal line tiling is either contractible or homotopy equivalent to a wedge of spheres. While proving our results, we also characterize all the matchings of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="26_2025_755_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left( 2 \times n\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mn>2</mn> <mo>×</mo> <mi>n</mi> </mfenced> </math></EquationSource> </InlineEquation>-grid graph that cannot be extended to form a perfect matching.</p>

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Perfect Matching Complexes of Polygonal Line Tilings

  • 期刊论文

  • 发表日期：

    2025年4月5日

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  • Himanshu Chandrakar,
  • Anurag Singh

摘要

The perfect matching complex of a simple graph G is a simplicial complex having facets (maximal faces) as the perfect matchings of G. This article discusses the perfect matching complex of polygonal line tilings and the \(\left( 2 \times n\right) \) 2 × n -grid graph in particular. We use tools from discrete Morse theory to show that the perfect matching complex of any polygonal line tiling is either contractible or homotopy equivalent to a wedge of spheres. While proving our results, we also characterize all the matchings of \(\left( 2 \times n\right) \) 2 × n -grid graph that cannot be extended to form a perfect matching.