  The clustering of a graph coloring is the maximum size of monochromatic
components. This paper studies colorings with bounded clustering in graph
classes with bounded layered treewidth, which include planar graphs, graphs of
bounded Euler genus, graphs embeddable on a fixed surface with a bounded number
of crossings per edge, map graphs, amongst other examples. Our main theorem
says that every graph with layered treewidth at most $k$ and with maximum
degree at most $\Delta$ is $3$-colorable with clustering
$O(k^{19}\Delta^{37})$. This is the first known polynomial bound on the
clustering. This greatly improves upon a corresponding result of Esperet and
Joret for graphs of bounded genus.
