Abstract: | 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 M in 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. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 168–173, 2001 |