  We prove that for every graph $H$, if a graph $G$ has no (odd) $H$ minor,
then its vertex set $V(G)$ can be partitioned into three sets $X_1$, $X_2$,
$X_3$ such that for each~$i$, the subgraph induced on $X_i$ has no component of
size larger than a function of~$H$ and the maximum degree of~$G$. This improves
a previous result of Alon, Ding, Oporowski and Vertigan~(2003) stating that
$V(G)$ can be partitioned into four such sets if $G$ has no $H$ minor. Our
theorem generalizes a result of Esperet and Joret~(2014), who proved it for
graphs embeddable on a fixed surface and asked whether it is true for graphs
with no $H$ minor.
  As a corollary, we prove that for every positive integer $t$, if a graph $G$
has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $3t$
sets $X_1,\ldots,X_{3t}$ such that for each~$i$, the subgraph induced on $X_i$
has no component of size larger than a function of~$t$. This corollary improves
a result of Wood~(2010), which states that $V(G)$ can be partitioned into
$\lceil 3.5t+2\rceil$ such sets.
