Bounding the Crossing Number of a Graph in terms of the Crossing Number of a Minor with Small Maximum Degree

We show that if G has a minor M with maximum degree at most 4, then the crossing number of G in a surface is at least one fourth the crossing number of M in . We use this result to show that every graph embedded on the torus with representativity r 6 has Klein bottle crossing number at least b2r=3c =64