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arXiv · 2505.15288

Strong odd colorings in graph classes of bounded expansion

Michał Pilipczuk
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Abstract

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, Pe\~na, 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.

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Michał Pilipczuk. 2025-05-21. Strong odd colorings in graph classes of bounded expansion. https://arxiv.org/abs/2505.15288

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