Triangulated spheres with holes in triangulated surfaces
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arXiv
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| Format: | Preprint |
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2024
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| author | Clinch, Katie Dewar, Sean Fuladi, Niloufar Gorsky, Maximilian Huynh, Tony Kastis, Eleftherios Nakamoto, Atsuhiro Nixon, Anthony Servatius, Brigitte |
| author_facet | Clinch, Katie Dewar, Sean Fuladi, Niloufar Gorsky, Maximilian Huynh, Tony Kastis, Eleftherios Nakamoto, Atsuhiro Nixon, Anthony Servatius, Brigitte |
| contents | Let $\mathbb{S}_h$ denote a sphere with $h$ holes. Given a triangulation $G$ of a surface $\mathbb{M}$, we consider the question of when $G$ contains a spanning subgraph $H$ such that $H$ is a triangulated $\mathbb{S}_h$. We give a new short proof of a theorem of Nevo and Tarabykin 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 $\mathbb{S}_{2h}$. We also prove that for every $0 \leq g' < g$ and $w \in \mathbb{N}$, 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 $\mathbb{S}_{g'}$. Our results are motivated by, and have applications for, rigidity questions in the plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04450 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Triangulated spheres with holes in triangulated surfaces Clinch, Katie Dewar, Sean Fuladi, Niloufar Gorsky, Maximilian Huynh, Tony Kastis, Eleftherios Nakamoto, Atsuhiro Nixon, Anthony Servatius, Brigitte Combinatorics Discrete Mathematics 57Kxx, 05C10 G.2.2 Let $\mathbb{S}_h$ denote a sphere with $h$ holes. Given a triangulation $G$ of a surface $\mathbb{M}$, we consider the question of when $G$ contains a spanning subgraph $H$ such that $H$ is a triangulated $\mathbb{S}_h$. We give a new short proof of a theorem of Nevo and Tarabykin 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 $\mathbb{S}_{2h}$. We also prove that for every $0 \leq g' < g$ and $w \in \mathbb{N}$, 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 $\mathbb{S}_{g'}$. Our results are motivated by, and have applications for, rigidity questions in the plane. |
| title | Triangulated spheres with holes in triangulated surfaces |
| topic | Combinatorics Discrete Mathematics 57Kxx, 05C10 G.2.2 |
| url | https://arxiv.org/abs/2410.04450 |