Clustered Colouring in Minor-Closed Classes

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 has tree-depth t then every H-minor-free graph is 4t-colourable with monochromatic components of size at most c(H). This provides evidence for a conjecture of Ossona de Mendez, Oum and Wood (2016). If H is connected with tree-depth 3, then we prove that 4 colours suffice. We also determine those minor-closed graph classes with clustered chromatic number 2.