Clustered Colouring in Minor-Closed Classes

TL;DR

This paper establishes a relationship between the clustered chromatic number of $H$-minor-free graphs and the tree-depth of $H$, providing bounds and confirming conjectures for specific cases.

Contribution

It proves that the clustered chromatic number of $H$-minor-free graphs depends on the tree-depth of $H$, and confirms the conjecture for certain cases, including when the tree-depth is 3.

Findings

Clustered chromatic number tied to tree-depth of $H$.

For connected $H$ with tree-depth $t$, $(2^{t+1}-4)$-colourability with bounded monochromatic components.

Confirmed the conjecture for $t=3$, showing 4 colours suffice.

Abstract

The "clustered chromatic number" of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$-colourable with monochromatic components of size at most $c$. We prove that for every graph $H$, the clustered chromatic number of the class of $H$-minor-free graphs is tied to the tree-depth of $H$. In particular, if $H$ is connected with tree-depth $t$ then every $H$-minor-free graph is $(2^{t+1}-4)$-colourable with monochromatic components of size at most $c(H)$. This provides the first evidence for a conjecture of Ossona de Mendez, Oum and Wood (2016) about defective colouring of $H$-minor-free graphs. If $t=3$ then we prove that 4 colours suffice, which is best possible. We also determine those minor-closed graph classes with clustered chromatic number 2. Finally, we develop a conjecture for the clustered chromatic number of an arbitrary minor-closed class.