On the Nonembeddability and Crossing Numbers of Some Kleinical Polyhedral Maps on the Torus

We define a family of graph bundles over cycles which embed naturally on the Klein bottle and which are analogous to the celebrated toroidal grid graphs (cartesian product of a cycle with a cycle). We give a criterion for a polyhedral map on the Klein bottle to fail to embed on the torus, and use this to calculate toroidal crossing numbers of two one-parameter infinite families of our Kleinical graphs.Mathematics Subject Classification (1991): 05C10