Odd coloring graphs with linear neighborhood complexity
James Davies, Meike Hatzel, Kolja Knauer, Rose McCarty, Torsten Ueckerdt
TL;DR
This work shows that linear neighborhood complexity imposes strong structural control, ensuring a bounded improper odd chromatic number and, consequently, χ_o-boundedness for several natural graph classes (e.g., circle graphs, bounded-twin-width/merge-width, and forbidden vertex-minors). The authors develop and deploy Haussler's Shallow Packing Lemma in conjunction with matroid fundamentals and the Growth Rate Theorem to bound neighborhood variation and produce constructive colorings. Central contributions include the main theorem linking η_G(m) ≤ r·m to a bounded-improper-odd coloring, two corollaries to transfer the result to broader classes, and a concrete 98-color bound for bipartite circle graphs via a fundamental-cut strategy. Additionally, they establish a minor-exclusion framework that yields explicit bounds on odd colorings of fundamental graphs, with implications for circle graphs and related families, and pose open questions on the limits of fundamental-cut colorings. These results advance our understanding of χ_o-boundedness beyond traditional sparsity notions and provide practical coloring bounds for several graph classes.
Abstract
We prove that any class of graphs with linear neighborhood complexity has bounded improper odd chromatic number. As a result, if $\mathcal{G}$ is the class of all circle graphs, or if $\mathcal{G}$ is any class with bounded twin-width, bounded merge-width, or a forbidden vertex-minor, then $\mathcal{G}$ is $χ_{\mathrm{o}}$-bounded.