K6 and icosahedron minors in 5-connected projective planar graphs

We show that every 5-connected graph admitting an embedding into the projective plane with face-width at least 3 contains K6 as a minor. Further we find a simple proof that every 5-connected planar graph contracts to icosahedron. We adapt the proof and find a simple face-width type condition such that every 5-connected graph embedded in an arbitrary surface satisfying the condition contracts to icosahedron.