Strong odd colorings in graph classes of bounded expansion

TL;DR

This paper proves that graphs in classes of bounded expansion have a bounded strong odd chromatic number with a specific coloring property, answering a previously open question.

Contribution

It establishes the existence of a bounded strong odd chromatic number for graphs in classes of bounded expansion, using new bounds related to set system parameters.

Findings

Bounded strong odd chromatic number in graph classes of bounded expansion.

A new bound on the strong odd coloring number based on set system parameters.

Resolution of an open question about the boundedness of strong odd chromatic number.

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.