Improper colourings inspired by Hadwiger's conjecture

Hadwiger's conjecture asserts that every Kt-minor-free graph has a proper (t-1)-colouring. We relax the conclusion in Hadwiger's conjecture via improper colourings. We prove that every Kt-minor-free graph is (2t-2)-colourable with monochromatic components of order at most remvoescriptlevel=0 (t-2). This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every Kt-minor-free graph is (t-1)-colourable with monochromatic degree at most t-2. This is the best known degree bound for such a result. Both these theorems are based on a decomposition method of independent interest. We give analogous results for Ks,t-minor-free graphs, which lead to improved bounds on generalised colouring numbers for these classes. Finally, we prove that graphs containing no Kt-immersion are 2-colourable with bounded monochromatic degree.