Clustered Colouring of Odd-$H$-Minor-Free Graphs

TL;DR

This paper investigates the relationship between the structure of a graph H and the clustered chromatic number of odd-H-minor-free graphs, establishing a connection to the tree-depth of H.

Contribution

It adapts a proof method to show that the clustered chromatic number of odd-H-minor-free graphs depends on the tree-depth of H.

Findings

Clustered chromatic number linked to tree-depth of H.

Method adaptation from Norin et al. (2019).

Provides structural insights into odd-minor-free graph classes.

Abstract

The clustered chromatic number of a graph class $\mathcal{G}$ is the minimum integer $c$ such that every graph $G\in\mathcal{G}$ has a $c$-colouring where each monochromatic component in $G$ has bounded size. We study the clustered chromatic number of graph classes $\mathcal{G}_H^{\text{odd}}$ defined by excluding a graph $H$ as an odd-minor. How does the structure of $H$ relate to the clustered chromatic number of $\mathcal{G}_H^{\text{odd}}$? We adapt a proof method of Norin, Scott, Seymour and Wood (2019) to show that the clustered chromatic number of $\mathcal{G}_H^{\text{odd}}$ is tied to the tree-depth of $H$.