Homological face-width condition forcing $K_6$-minors in graphs on   surfaces
  It is proved that every graph embedded on a (non-spherical) surface with non-separating 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. 
Roi Krakovski and Bojan Mohar
math.CO