Clustered Colouring in Minor-Closed Classes
Sergey Norin, Alex Scott, Paul Seymour, David R. Wood

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 such that for some integer every graph in the class is -colourable with monochromatic components of size at most . We prove that for every graph , the clustered chromatic number of the class of -minor-free graphs is tied to the tree-depth of . In particular, if is connected with tree-depth then every -minor-free graph is -colourable with monochromatic components of size at most . This provides the first evidence for a conjecture of Ossona de Mendez, Oum and Wood (2016) about defective colouring of -minor-free graphs. If 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…
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