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

摘要

We show that if G has a minor M with maximum degree at most 4, then the crossing number of G in a surface Σ is at least one fourth the crossing number of Min Σ. We use this result to show that every graph embedded on the torus with representativity r ≥ 6 has Klein bottle crossing number at least 「2r/3」~2/64.