On reducible partition of graphs and its application to Hadwiger conjecture

An undirected graph $H$ is called a minor of the graph $G$ if $H$ can be formed from $G$ by deleting edges and vertices and by contracting edges. If $G$ does not have a graph $H$ as a minor, then we say that $G$ is $H$-free. Hadwiger conjecture claim that the chromatic number of $G$ may be closely r