On the twin-width of smooth manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911046078300160 |
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| author | Bonnet, Édouard Huszár, Kristóf |
| author_facet | Bonnet, Édouard Huszár, Kristóf |
| contents | Building on Whitney's classical method of triangulating smooth manifolds, we show that every compact $d$-dimensional smooth manifold admits a triangulation with dual graph of twin-width at most $d^{O(d)}$. In particular, it follows that every compact 3-manifold has a triangulation with dual graph of bounded twin-width. This is in sharp contrast to the case of treewidth, where for any natural number $n$ there exists a closed 3-manifold such that every triangulation thereof has dual graph with treewidth at least $n$. To establish this result, we bound the twin-width of the incidence graph of the $d$-skeleton of the second barycentric subdivision of the $2d$-dimensional hypercubic honeycomb. We also show that every compact, piecewise-linear (hence smooth) $d$-dimensional manifold has triangulations where the dual graph has an arbitrarily large twin-width. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10174 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the twin-width of smooth manifolds Bonnet, Édouard Huszár, Kristóf Geometric Topology Computational Geometry Discrete Mathematics 57Q15 (Primary) 57R05, 05C75, 57M15 (Secondary) F.2.2; G.2.2 Building on Whitney's classical method of triangulating smooth manifolds, we show that every compact $d$-dimensional smooth manifold admits a triangulation with dual graph of twin-width at most $d^{O(d)}$. In particular, it follows that every compact 3-manifold has a triangulation with dual graph of bounded twin-width. This is in sharp contrast to the case of treewidth, where for any natural number $n$ there exists a closed 3-manifold such that every triangulation thereof has dual graph with treewidth at least $n$. To establish this result, we bound the twin-width of the incidence graph of the $d$-skeleton of the second barycentric subdivision of the $2d$-dimensional hypercubic honeycomb. We also show that every compact, piecewise-linear (hence smooth) $d$-dimensional manifold has triangulations where the dual graph has an arbitrarily large twin-width. |
| title | On the twin-width of smooth manifolds |
| topic | Geometric Topology Computational Geometry Discrete Mathematics 57Q15 (Primary) 57R05, 05C75, 57M15 (Secondary) F.2.2; G.2.2 |
| url | https://arxiv.org/abs/2407.10174 |