Clustered coloring of graphs excluding a subgraph and a minor

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 Ks,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 Kₛ₊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 Ks,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 Ks,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 Kₛ₊₁-minor are $(8s-4)$-colorable with bounded clustering, which improves on previous such bounds.