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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Carsten Thomassen
Journal: Trans. Amer. Math. Soc. 323 (1991), 605-635
MSC: Primary 57Q99; Secondary 05C10, 05C25
MathSciNet review: 1040045
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1040045-3
PII: S 0002-9947(1991)1040045-3
Article copyright: © Copyright 1991 American Mathematical Society