On the nonembeddability and crossing numbers of some toroidal graphs on the Klein bottle Original Research Article

We show that toroidal polyhedral maps with four or more disjoint homotopic noncontractible circuits are not embeddable on the projective plane and that toroidal polyhedral maps with five or more disjoint homotopic noncontractible circuits are not embeddable on the Klein bottle. We also show that the Klein bottle crossing numbers of Cm×Cn (m⩽n) for m=3,4,5,6 are 1,2,4, and 6, respectively, and give upper bounds for all other values of n. These crossing numbers display atypical behavior in that the value depends only on m instead of on both m and n as is the case for the plane and projective plane.