Colouring the 1-skeleton of $d$-dimensional triangulations
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| Format: | Preprint |
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2024
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| _version_ | 1866913577651142656 |
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| author | Planken, Tim |
| author_facet | Planken, Tim |
| contents | While every plane triangulation is colourable with three or four colours, Heawood showed that a plane triangulation is 3-colourable if and only if every vertex has even degree. In $d \geq 3$ dimensions, however, every $k \geq d+1$ may occur as the chromatic number of some triangulation of ${\mathbb S}^d$. As a first step, Joswig structurally characterised which triangulations of ${\mathbb S}^d$ have a $(d+1)$-colourable 1-skeleton. In the 20 years since Joswig's result, no characterisations have been found for any $k>d+1$.
In this paper, we structurally characterise which triangulations of ${\mathbb S}^d$ have a $(d+2)$-colourable 1-skeleton: they are precisely the triangulations that have a subdivision such that for every $(d-2)$-cell, the number of incident $(d-1)$-cells is divisible by three. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_11762 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Colouring the 1-skeleton of $d$-dimensional triangulations Planken, Tim Combinatorics Algebraic Topology 05C15 (Primary), 05E45, 05C10 (Secondary) While every plane triangulation is colourable with three or four colours, Heawood showed that a plane triangulation is 3-colourable if and only if every vertex has even degree. In $d \geq 3$ dimensions, however, every $k \geq d+1$ may occur as the chromatic number of some triangulation of ${\mathbb S}^d$. As a first step, Joswig structurally characterised which triangulations of ${\mathbb S}^d$ have a $(d+1)$-colourable 1-skeleton. In the 20 years since Joswig's result, no characterisations have been found for any $k>d+1$. In this paper, we structurally characterise which triangulations of ${\mathbb S}^d$ have a $(d+2)$-colourable 1-skeleton: they are precisely the triangulations that have a subdivision such that for every $(d-2)$-cell, the number of incident $(d-1)$-cells is divisible by three. |
| title | Colouring the 1-skeleton of $d$-dimensional triangulations |
| topic | Combinatorics Algebraic Topology 05C15 (Primary), 05E45, 05C10 (Secondary) |
| url | https://arxiv.org/abs/2409.11762 |