91Ó°ÊÓ

Let \(G\) be a graph of average degree at least \(2^{r-2}\). By considering the neighbourhood of a vertex in a minimal minor \(H \preccurlyeq G\) with \(\varepsilon(H) \geqslant \varepsilon(G)\), prove Mader's (1967) theorem that \(G \succcurlyeq K^{T}\).

Show that Hadwiger's conjecture for \(r+1\) implies the conjecture for \(r\).