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 $H$ is connected with tree-depth $t$ then every $H$-minor-free graph is $(2^{t+1}-4)$-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 $t=3$ then we prove that 4 colours suffice, which is best possible. We also determine those minor-closed graph classes with clustered chromatic number 2. Finally, we develop a conjecture for the clustered chromatic number of an arbitrary minor-closed class.

中文翻译：

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小封闭类中的聚类着色

一类图的“聚类色数”是最小整数 $k$，这样对于某个整数 $c$，该类中的每个图都是 $k$-可着色的，具有最多为 $c$ 大小的单色分量。我们证明，对于每个图 $H$，$H$-minor-free 图类的聚类色数与 $H$ 的树深度相关联。特别地，如果 $H$ 与树深度 $t$ 相关，那么每个 $H$-minor-free 图都是 $(2^{t+1}-4)$-可着色的，单色分量最多为 $ c(H)$。这为 Ossona de Mendez、Oum 和 Wood (2016) 的猜想提供了证据。如果 $t=3$ 那么我们证明 4 种颜色就足够了，这是最好的。我们还确定了那些具有聚集色数 2 的次要闭合图类。 最后，