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

Abstract:

We describe all regular tilings 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 $g \geqslant 3$, there are only finitely many vertex-transitive graphs of genus $g$. In fact, they all have order $< {10^{10}}g$. The weaker conjecture for Cayley graphs was made by Gross and Tucker and extends Hurwitz’ theorem that, for each $g \geqslant 2$, there are only finitely many groups that act on the surface of genus $g$. We also derive a nonorientable version of Hurwitz’ theorem.