The arc chromatic number for Galois projective planes, affine planes and Euclidean grids
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915757501186048 |
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| author | Araujo-Pardo, Gabriela Martínez-Sandoval, Leonardo |
| author_facet | Araujo-Pardo, Gabriela Martínez-Sandoval, Leonardo |
| contents | We establish that the minimum number of arcs required to partition the Galois projective plane $\text{PG}(2,q)$ is $q+1$. Furthermore, we determine the exact value for a fractional variant of this problem.
We extend our analysis to affine planes $\text{AG}(2,q)$, proving that they can be partitioned into $q$ arcs. In particular, we show that this partition is tight when $q$ is an odd prime power, and that a $(q-1)$-partition is attainable for $q=2^k$ with $k \in \{1,2,3\}$. For $q=2^k$ with $k \geq 4$, we provide bounds between two possible values.
Finally, we apply these results to Euclidean grids, demonstrating that a partition into $(1+ε)n$ sets in general position exists for any $ε> 0$ and sufficiently large $n$. We also present exact minimal partitions for small Euclidean grids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19043 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The arc chromatic number for Galois projective planes, affine planes and Euclidean grids Araujo-Pardo, Gabriela Martínez-Sandoval, Leonardo Combinatorics Discrete Mathematics 05B25, 51E15, 51E21, 05C15, 68V05 We establish that the minimum number of arcs required to partition the Galois projective plane $\text{PG}(2,q)$ is $q+1$. Furthermore, we determine the exact value for a fractional variant of this problem. We extend our analysis to affine planes $\text{AG}(2,q)$, proving that they can be partitioned into $q$ arcs. In particular, we show that this partition is tight when $q$ is an odd prime power, and that a $(q-1)$-partition is attainable for $q=2^k$ with $k \in \{1,2,3\}$. For $q=2^k$ with $k \geq 4$, we provide bounds between two possible values. Finally, we apply these results to Euclidean grids, demonstrating that a partition into $(1+ε)n$ sets in general position exists for any $ε> 0$ and sufficiently large $n$. We also present exact minimal partitions for small Euclidean grids. |
| title | The arc chromatic number for Galois projective planes, affine planes and Euclidean grids |
| topic | Combinatorics Discrete Mathematics 05B25, 51E15, 51E21, 05C15, 68V05 |
| url | https://arxiv.org/abs/2601.19043 |