Bounding the crossing number of a graph in terms of the crossing number ofa minor with small maximum degree

We show that if G has a minor M with maximum degree at most 4, then the cro ssing number of G in a surface Sigma is at least one fourth the crossing nu mber of M in Sigma. We use this result to show that every graph embedded on the torus with representativity r greater than or equal to 6 has Klein bot tle crossing number at least (2r/3](2)/64. (C) 2001 John Wiley & Sons, Inc.