Hadwiger's Conjecture asserts that every $K_h$-minor-free graph is properly $(h-1)$-colourable. We prove the following improper analogue of Hadwiger's Conjecture: for fixed $h$, every $K_h$-minor-free graph is $(h-1)$-colourable with monochromatic components of bounded size. The number of colours is best possible regardless of the size of monochromatic components. It solves an open problem of Edwards, Kang, Kim, Oum and Seymour [\emph{SIAM J. Disc. Math.} 2015], and concludes a line of research initiated in 2007. Similarly, for fixed $t\geq s$, we show that every $K_{s,t}$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is best possible, solving an open problem of van de Heuvel and Wood [\emph{J.~London Math.\ Soc.} 2018]. We actually prove a single theorem from which both of the above results are immediate corollaries. For an excluded apex minor, we strengthen the result as follows: for fixed $t\geq s\geq 3$, and for any fixed apex graph $X$, every $K_{s,t}$-subgraph-free $X$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is again best possible.

翻译：Hadwiger猜想断言，每个不含$K_h$子式（$K_h$-minor-free）的图都是正常$(h-1)$-可着色的。我们证明了以下Hadwiger猜想的非正常类比：对于固定的$h$，每个不含$K_h$子式的图都是$(h-1)$-可着色的，且其单色分量具有有界大小。无论单色分量的大小如何，颜色数都是最优的。这解决了Edwards、Kang、Kim、Oum和Seymour [\emph{SIAM J. Disc. Math.} 2015] 提出的一个开放问题，并总结了自2007年发起的一系列研究。类似地，对于固定的$t\geq s$，我们证明每个不含$K_{s,t}$子式的图都是$(s+1)$-可着色的，且其单色分量具有有界大小。颜色数是最优的，解决了van de Heuvel和Wood [\emph{J.~London Math.\ Soc.} 2018] 提出的一个开放问题。实际上，我们证明了一个单一定理，上述两个结果都是其直接推论。对于排除的顶点子式（apex minor），我们加强了结果如下：对于固定的$t\geq s\geq 3$以及任何固定的顶点子式图$X$，每个不含$K_{s,t}$子图且不含$X$子式的图都是$(s+1)$-可着色的，且其单色分量具有有界大小。颜色数同样是最优的。