On the nonembeddability and crossing numbers of some Kleinical polyhedral maps on the torus (Q1777236)

The graphical Cartesian product \(C_n \times C_m\) embeds nicely on the torus and is known as the toroidal grid. The author defines similar grids on the Klein bottle. He then gives a property of Kleinical graphs that ensures the graphs are not toroidal; namely, that the graphs have four disjoint cycles such that deleting each from the embedding in the Klein bottle gives a cylinder. Using this property, he finds the crossing number on the torus of two infinite classes of the Kleinical grids.