... Regen and Bjarne Toft for stimulating discussions on critical graphs and for assistance in using certain programs, in particular, I thank Friedrich and Marco for developing programs for sorting and ... C4 + C6 , 2C5 , or C10 In each case it is easy to exhibit a K7 minor in Gx , and so G ≥ K8 − Minimum degree 10 and K8 minors In this section, we shall apply the following result of Mader − Theorem ... every − 8-chromatic graph contains a K7 minor In this paper we prove that every double-critical − 8-chromatic graph contains a K8 minor The proof of this result is surprisingly complicated and uses...