Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {S}_h$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">S</mi> <mi>h</mi> </msub> </math></EquationSource> </InlineEquation> denote a sphere with h holes. Given a triangulation G of a surface <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {M}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">M</mi> </math></EquationSource> </InlineEquation>, we consider the question of when G contains a spanning subgraph H such that H is a triangulated <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {S}_h$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">S</mi> <mi>h</mi> </msub> </math></EquationSource> </InlineEquation>. We give a new short proof of a theorem of Nevo and Tarabykin [<CitationRef CitationID="CR15">15</CitationRef>] that every triangulation G of the torus contains a spanning subgraph which is a triangulated cylinder. For arbitrary surfaces, we prove that every high facewidth triangulation of a surface with h handles contains a spanning subgraph which is a triangulated <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {S}_{2h}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">S</mi> <mrow> <mn>2</mn> <mi>h</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. We also prove that for every <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$0 \le g' < g$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo><</mo> <mi>g</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$w \in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, there exists a triangulation of facewidth at least w of a surface of Euler genus g that does not have a spanning subgraph which is a triangulated <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {S}_{g'}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">S</mi> <msup> <mi>g</mi> <mo>′</mo> </msup> </msub> </math></EquationSource> </InlineEquation>. Our results are motivated by, and have applications for, rigidity questions in the plane.

Triangulated Spheres with Holes in Triangulated Surfaces

期刊论文
发表日期：

2026年3月12日

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Katie Clinch,
Sean Dewar,
Niloufar Fuladi,
Maximilian Gorsky,
Tony Huynh,
Eleftherios Kastis,
Atsuhiro Nakamoto,
Anthony Nixon,
Brigitte Servatius

摘要

Let $\mathbb {S}_h$ $S_{h}$ denote a sphere with h holes. Given a triangulation G of a surface $\mathbb {M}$ $M$ , we consider the question of when G contains a spanning subgraph H such that H is a triangulated $\mathbb {S}_h$ $S_{h}$ . We give a new short proof of a theorem of Nevo and Tarabykin [15] that every triangulation G of the torus contains a spanning subgraph which is a triangulated cylinder. For arbitrary surfaces, we prove that every high facewidth triangulation of a surface with h handles contains a spanning subgraph which is a triangulated $\mathbb {S}_{2h}$ $S_{2 h}$ . We also prove that for every $0 \le g' < g$ $0 \leq g^{'} < g$ and $w \in \mathbb {N}$ $w \in N$ , there exists a triangulation of facewidth at least w of a surface of Euler genus g that does not have a spanning subgraph which is a triangulated $\mathbb {S}_{g'}$ $S_{g^{'}}$ . Our results are motivated by, and have applications for, rigidity questions in the plane.