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We describe all regular tiings of the torus and the Klein bottle. We apply this to describe, for each orientable (respectively nonorientable) surface S, all (but finitely many) vertex-transitive graphs which can be drawn on S but not on any surface of smaller genus (respectively crosscap number). In particular, we prove the conjecture of Babai that, for each g > 3, there are only finitely many vertex-transitive graphs of genus g. In fact, they all have order 2, there are only finitely many groups that act on the surface of genus g . We also derive a nonorientable version of Hurwitz' theorem.

Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface