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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.23057 |
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| _version_ | 1866917971771785216 |
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| author | Kaiser, Tomáš Lo, On-Hei Solomon Nakamoto, Atsuhiro Nozaki, Yuta Ozeki, Kenta |
| author_facet | Kaiser, Tomáš Lo, On-Hei Solomon Nakamoto, Atsuhiro Nozaki, Yuta Ozeki, Kenta |
| contents | Youngs proved that every non-bipartite quadrangulation of the projective plane $\mathbb{R}\mathrm{P}^2$ is 4-chromatic. Kaiser and Stehl\'ık [J. Combin. Theory Ser. B 113 (2015), 1-17] generalised the notion of a quadrangulation to higher dimensions and extended Youngs' theorem by proving that every non-bipartite quadrangulation of the $d$-dimensional projective space $\mathbb{R}\mathrm{P}^d$ with $d \geq 2$ has chromatic number at least $d+2$. On the other hand, Hachimori et al. [European. J. Combin. 125 (2025), 104089] defined another kind of high-dimensional quadrangulation, called a normal quadrangulation. They proved that if a non-bipartite normal quadrangulation $G$ of $\mathbb{R}\mathrm{P}^d$ with any $d \geq 2$ satisfies a certain geometric condition, then $G$ is $4$-chromatic, and asked whether the geometric condition can be removed from the result. In this paper, we give a negative solution to their problem for the case $d=3$, proving that there exist 3-dimensional normal quadrangulations of $\mathbb{R}\mathrm{P}^3$ whose chromatic number is arbitrarily large. Moreover, we prove that no normal quadrangulation of $\mathbb{R}\mathrm{P}^d$ with any $d \geq 2$ has chromatic number $3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_23057 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Colouring normal quadrangulations of projective spaces Kaiser, Tomáš Lo, On-Hei Solomon Nakamoto, Atsuhiro Nozaki, Yuta Ozeki, Kenta Combinatorics Youngs proved that every non-bipartite quadrangulation of the projective plane $\mathbb{R}\mathrm{P}^2$ is 4-chromatic. Kaiser and Stehl\'ık [J. Combin. Theory Ser. B 113 (2015), 1-17] generalised the notion of a quadrangulation to higher dimensions and extended Youngs' theorem by proving that every non-bipartite quadrangulation of the $d$-dimensional projective space $\mathbb{R}\mathrm{P}^d$ with $d \geq 2$ has chromatic number at least $d+2$. On the other hand, Hachimori et al. [European. J. Combin. 125 (2025), 104089] defined another kind of high-dimensional quadrangulation, called a normal quadrangulation. They proved that if a non-bipartite normal quadrangulation $G$ of $\mathbb{R}\mathrm{P}^d$ with any $d \geq 2$ satisfies a certain geometric condition, then $G$ is $4$-chromatic, and asked whether the geometric condition can be removed from the result. In this paper, we give a negative solution to their problem for the case $d=3$, proving that there exist 3-dimensional normal quadrangulations of $\mathbb{R}\mathrm{P}^3$ whose chromatic number is arbitrarily large. Moreover, we prove that no normal quadrangulation of $\mathbb{R}\mathrm{P}^d$ with any $d \geq 2$ has chromatic number $3$. |
| title | Colouring normal quadrangulations of projective spaces |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.23057 |