  The defective chromatic number of a graph class is the infimum $k$ such that
there exists an integer $d$ such that every graph in this class can be
partitioned into at most $k$ induced subgraphs with maximum degree at most $d$.
Finding the defective chromatic number is a fundamental graph partitioning
problem and received attention recently partially due to Hadwiger's conjecture
about coloring minor-closed families. In this paper, we prove that the
defective chromatic number of any minor-closed family equals the simple lower
bound obtained by the standard construction, confirming a conjecture of Ossona
de Mendez, Oum, and Wood. This result provides the optimal list of unavoidable
finite minors for infinite graphs that cannot be partitioned into a fixed
finite number of induced subgraphs with uniformly bounded maximum degree. As
corollaries about clustered coloring, we obtain a linear relation between the
clustered chromatic number of any minor-closed family and the tree-depth of its
forbidden minors, improving an earlier exponential bound proved by Norin,
Scott, Seymour, and Wood and confirming the planar case of their conjecture.
