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 hypothesis 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.
All Science Journal Classification (ASJC) codes
- Hadwiger's conjecture
- Improper coloring