Homological Face-Width Condition Forcing $K_6$-Minors in Graphs on Surfaces
Roi Krakovski, Bojan Mohar · SIAM Journal on Discrete Mathematics · 2014
It is proved that every graph embedded on a (nonspherical) surface with nonseparating face-width at least 7 contains a minor isomorphic to $K_6$. It is also shown that face-width four yields the same conclusion for graphs on the projective plane.