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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.13195 |
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| _version_ | 1866909083324383232 |
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| author | Kelly, Samuel |
| author_facet | Kelly, Samuel |
| contents | We prove that for any finite tree $T$ with $n$ vertices and maximal degree $3$, there is a topological embedding of $T$ into the integer grid $Z^2$ which maps vertices to vertices and whose image meets at most $\frac{7}{3}n$ vertices. This recovers a weaker form of a result due to Valiant 10.5555/1963635.1963641 with stronger constants. We address question $7.7$ of arXiv:2112.05305, giving the first example of a pair of graphs $X,Y$ such that there is no regular map $X\to Y$ but the coarse wiring profile of $X$ into $Y$ grows linearly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_13195 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Topological Embedding of the Binary Tree into the Square Lattice Kelly, Samuel Metric Geometry Group Theory We prove that for any finite tree $T$ with $n$ vertices and maximal degree $3$, there is a topological embedding of $T$ into the integer grid $Z^2$ which maps vertices to vertices and whose image meets at most $\frac{7}{3}n$ vertices. This recovers a weaker form of a result due to Valiant 10.5555/1963635.1963641 with stronger constants. We address question $7.7$ of arXiv:2112.05305, giving the first example of a pair of graphs $X,Y$ such that there is no regular map $X\to Y$ but the coarse wiring profile of $X$ into $Y$ grows linearly. |
| title | A Topological Embedding of the Binary Tree into the Square Lattice |
| topic | Metric Geometry Group Theory |
| url | https://arxiv.org/abs/2311.13195 |