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Hadwiger’s conjecture for $$K_ 6$$-free graphs. (English) Zbl 0830.05028
Hadwiger conjectured in 1943 that every graph of chromatic number $$n$$ is contractible to the complete graph on $$n$$ vertices. A theorem of Wagner from 1937 implied that Hadwiger’s conjecture for $$n= 5$$ is equivalent to the four-colour conjecture. The authors now reduce the case $$n= 6$$ also to the four-colour theorem. They consider a minimal counterexample $$H$$ to Hadwiger’s conjecture for $$n= 6$$, and show in a complicated proof that there must exist a vertex $$z$$ in $$H$$ such that $$H- z$$ is planar. So the four-colour theorem implies that $$H$$ cannot exist.