Strong odd colorings in graph classes of bounded expansion
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912385513553920 |
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| author | Pilipczuk, Michał |
| author_facet | Pilipczuk, Michał |
| contents | We prove that for every $d\in \mathbb{N}$ and a graph class of bounded expansion $\mathscr{C}$, there exists some $c\in \mathbb{N}$ so that every graph from $\mathscr{C}$ admits a proper coloring with at most $c$ colors satisfying the following condition: in every ball of radius $d$, every color appears either zero times or an odd number of times. For $d=1$, this provides a positive answer to a question raised by Goetze, Klute, Knauer, Parada, Peña, and Ueckerdt [ArXiv 2505.02736] about the boundedness of the strong odd chromatic number in graph classes of bounded expansion. The key technical ingredient towards the result is a proof that the strong odd coloring number of a sets system can be bounded in terms of its semi-ladder index, 2VC dimension, and the maximum subchromatic number among induced subsystems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15288 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strong odd colorings in graph classes of bounded expansion Pilipczuk, Michał Combinatorics Discrete Mathematics We prove that for every $d\in \mathbb{N}$ and a graph class of bounded expansion $\mathscr{C}$, there exists some $c\in \mathbb{N}$ so that every graph from $\mathscr{C}$ admits a proper coloring with at most $c$ colors satisfying the following condition: in every ball of radius $d$, every color appears either zero times or an odd number of times. For $d=1$, this provides a positive answer to a question raised by Goetze, Klute, Knauer, Parada, Peña, and Ueckerdt [ArXiv 2505.02736] about the boundedness of the strong odd chromatic number in graph classes of bounded expansion. The key technical ingredient towards the result is a proof that the strong odd coloring number of a sets system can be bounded in terms of its semi-ladder index, 2VC dimension, and the maximum subchromatic number among induced subsystems. |
| title | Strong odd colorings in graph classes of bounded expansion |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2505.15288 |