A Relative of Hadwiger's Conjecture

Hadwiger’s conjecture asserts that if a simple graph G has no Kt+1 minor, then its vertex set V (G) can be partitioned into t stable sets. This is still open, but we prove under the same hypotheses that V (G) can be partitioned into t sets X1, . . . ,Xt, such that for 1 ≤ i ≤ t, the subgraph induced on Xi has maximum degree at most a function of t. This is sharp, in that the conclusion becomes false if we ask for a partition into t − 1 sets with the same property.