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

We describe all regular tilings of the torus and the Klein bottle. We apply this to describe, for each orientable (respectively nonorientable) surface S S , all (but finitely many) vertex-transitive graphs which can be drawn on S 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 g 3 , there are only finitely many vertex-transitive graphs of genus g g . In fact, they all have order > 10 10 g > {10¹⁰}g . The weaker conjecture for Cayley graphs was made by Gross and Tucker and extends Hurwitz’ theorem that, for each g ⩾ 2 g 2 , there are only finitely many groups that act on the surface of genus g g . We also derive a nonorientable version of Hurwitz’ theorem.