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<p>An embedding of a graph on a translation surface is said to be <i>systolic</i> if each vertex of the graph corresponds to a singular point (or marked point) and each edge corresponds to a shortest saddle connection on the translation surface. The embedding is said to be <i>cellular</i> (respectively <i>essential</i>) if each complementary region is a topological disk (respectively not a topological disk). In this article, we prove that any finite graph admits an essential-systolic embedding on a translation surface and estimate the genera of such surfaces. For a wedge <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Sigma _n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Σ</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> of <i>n</i> circles, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n\ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we investigate that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Sigma _n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Σ</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> admits cellular-systolic embedding on a translation surface and compute the minimum and maximum genera of such surfaces. Finally, we find some collection of graphs with more than one vertex that also admit cellular-systolic embedding on so called translation surfaces.</p>

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Systolic embedding of graphs on translation surfaces

  • 期刊论文

  • 发表日期：

    2025年11月8日

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  • Achintya Dey,
  • Bidyut Sanki

摘要

An embedding of a graph on a translation surface is said to be systolic if each vertex of the graph corresponds to a singular point (or marked point) and each edge corresponds to a shortest saddle connection on the translation surface. The embedding is said to be cellular (respectively essential) if each complementary region is a topological disk (respectively not a topological disk). In this article, we prove that any finite graph admits an essential-systolic embedding on a translation surface and estimate the genera of such surfaces. For a wedge \(\Sigma _n\) Σ n of n circles, \(n\ge 2\) n ≥ 2 , we investigate that \(\Sigma _n\) Σ n admits cellular-systolic embedding on a translation surface and compute the minimum and maximum genera of such surfaces. Finally, we find some collection of graphs with more than one vertex that also admit cellular-systolic embedding on so called translation surfaces.