Improper Colourings inspired by Hadwiger's Conjecture

Hadwiger's Conjecture asserts that every $K_t$-minor-free graph has a proper $(t-1)$-colouring. We relax the conclusion in Hadwiger's Conjecture via improper colourings. We prove that every $K_t$-minor-free graph is $(2t-2)$-colourable with monochromatic components of order at most $\lceil{\frac12(t-2)}\rceil$. This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every $K_t$-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 $K_{s,t}$-minor-free graphs, which lead to improved bounds on generalised colouring numbers for these classes. Finally, we prove that graphs containing no $K_t$-immersion are $2$-colourable with bounded monochromatic degree.