Odd coloring graphs with linear neighborhood complexity

TL;DR

This paper establishes that classes of graphs with linear neighborhood complexity have bounded improper odd chromatic number, extending to circle graphs and classes with bounded twin-width, merge-width, or forbidden vertex-minors.

Contribution

It proves a new bound on the odd chromatic number for graph classes with linear neighborhood complexity, unifying several graph classes under this property.

Findings

Graph classes with linear neighborhood complexity have bounded improper odd chromatic number.

Circle graphs and classes with bounded twin-width, merge-width, or forbidden vertex-minors are $ ext{chi}_o$-bounded.

The results unify and extend previous bounds for various 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 $\chi_{\mathrm{o}}$-bounded.