  A graph coloring has bounded clustering if each monochromatic component has
bounded size. Equivalently, it is a partition of the vertices into induced
subgraphs with bounded size components. This paper studies clustered colorings
of graphs, where the number of colors depends on an excluded minor and/or an
excluded subgraph. We prove the following results (for fixed integers $s,t$ and
a fixed graph $H$). First we show that graphs with no $K_{s,t}$ subgraph and
with no $H$-minor are $(s+2)$-colorable with bounded clustering. The number of
colors here is best possible. This result implies that graphs with no
$K_{s+1}$-minor are $(s+2)$-colorable with bounded clustering, which is within
two colors of the clustered coloring version of Hadwiger's conjecture. For
graphs of bounded treewidth (or equivalently, excluding a planar minor) and
with no $K_{s,t}$ subgraph, we prove $(s+1)$-choosability with bounded
clustering, which is best possible. We then consider excluding an odd minor. We
prove that graphs with no $K_{s,t}$ subgraph and with no odd $H$-minor are
$(2s+1)$-colorable with bounded clustering, generalizing a result of the first
author and Oum who proved the case $s=1$. Moreover, at least $s-1$ color
classes are stable sets. Finally, we consider the clustered coloring version of
a conjecture of Gerards and Seymour and prove that graphs with no odd
$K_{s+1}$-minor are $(8s-4)$-colorable with bounded clustering, which improves
on previous such bounds.

Combinatorics