Strong odd colorings in graph classes of bounded expansion

TL;DR

This work resolves the boundedness of the strong odd chromatic number for graph classes of bounded expansion by proving a radius-$d$ strengthening: every graph in such a class admits a proper coloring using a uniform number of colors where, in every radius-$d$ ball, each color appears either zero or an odd number of times. The key strategy reduces the problem to bounding the strong odd chromatic number of set systems in terms of the semi-ladder index, the 2VC dimension, and the maximum subchromatic number of induced subsystems, and then transferring this bound back to balls via Gaifman graphs and logical transductions. A central technical contribution is the bound on the subchromatic number for hereditary closures of ball set systems, established through a multi-tier analysis of cographs of bounded depth, shrubdepth-based decompositions, and structurally bounded expansion, culminating in a transduction-based transfer to the original graph classes. The results advance the Sparsity framework by connecting local coloring constraints to global sparsity parameters, with implications for related logical and combinatorial characterizations and potential modulus-based extensions.

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, 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.

Figures (1)

• Figure 1: A semi-ladder of order $4$, a comatching of order $4$, a ladder of order $4$, and a $2$-shattered set of size $4$. Solid edges represent membership, dashed gray edges represent non-membership, lack of an edge represents no specification.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Theorem 2.3: DingSW94, reformulated through duality
• Lemma 2.4
• proof
• Lemma 2.5
• proof