Published online by Cambridge University Press: 05 July 2022
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 determine the clustered chromatic number of any minor-closed class with bounded treedepth, and prove a best possible upper bound on the clustered chromatic number of any minor-closed class with bounded pathwidth. As a consequence, we determine the fractional clustered chromatic number of every minor-closed class.