On Oriented Colourings of Graphs on Surfaces
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929507762438144 |
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| author | Clow, Alexander |
| author_facet | Clow, Alexander |
| contents | For an oriented graph $G$, the least number of colours required to oriented colour $G$ is called the oriented chromatic number of $G$ and denoted $χ_o(G)$.For a non-negative integer $g$ let $χ_o(g)$ be the least integer such that $χ_o(G) \leq χ_o(g)$ for every oriented graph $G$ with Euler genus at most $g$. We will prove that $χ_o(g)$ is nearly linear in the sense that $Ω(\frac{g}{\log(g)}) \leq χ_o(g) \leq O(g \log(g))$. This resolves a question of the author, Bradshaw, and Xu, by improving their bounds of the form $Ω((\frac{g^2}{\log(g)})^{1/3}) \leq χ_o(g)$ and $χ_o(g) \leq O(g^{6400})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13076 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Oriented Colourings of Graphs on Surfaces Clow, Alexander Combinatorics 05C15, 05C60 For an oriented graph $G$, the least number of colours required to oriented colour $G$ is called the oriented chromatic number of $G$ and denoted $χ_o(G)$.For a non-negative integer $g$ let $χ_o(g)$ be the least integer such that $χ_o(G) \leq χ_o(g)$ for every oriented graph $G$ with Euler genus at most $g$. We will prove that $χ_o(g)$ is nearly linear in the sense that $Ω(\frac{g}{\log(g)}) \leq χ_o(g) \leq O(g \log(g))$. This resolves a question of the author, Bradshaw, and Xu, by improving their bounds of the form $Ω((\frac{g^2}{\log(g)})^{1/3}) \leq χ_o(g)$ and $χ_o(g) \leq O(g^{6400})$. |
| title | On Oriented Colourings of Graphs on Surfaces |
| topic | Combinatorics 05C15, 05C60 |
| url | https://arxiv.org/abs/2409.13076 |