## Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface.(English)Zbl 0722.05031

This beautiful paper is concerned with the problem of constructing and enumerating all possible regular tilings on a surface. The author describes all regular tilings of the torus and the Klein bottle. As a consequence, it can be described, for each orientable (resp. nonorientable) surface S, all (but finitely many) vertex-transitive graphs which can be drawn on S but not on any surface of smaller genus (resp. crosscap number). In particular, it is proved the conjecture of Babai that, for each $$g\geq 3$$, there are only finitely many vertex- transitive graphs of genus g [see J. L. Gross and T. W. Tucker, Topological Graph Theory (1987; Zbl 0621.05013)]. Indeed, they all have order $$<10^{10}g$$. T. W. Tucker [J. Comb. Theory, Ser. B 34, 82-98 (1983; Zbl 0521.05027)] proved that, if a finite group acts on a surface S, then some Cayley graph G of the group has a 2-cell embedding on S such that every homeomorphism in the group induces an isomorphism. The results of the present paper tell what the graph G is, provided the group is large. Thus the author can obtain an alternative description of all (but finitely many) finite homeomorphism groups of a surface (compare also the Hurwitz theorem). In particular, it follows that $$S_ 0$$, $$S_ 1$$, $$N_ 1$$, $$N_ 2$$ are the only surfaces having infinitely many homeomorphism groups. Here $$S_ g$$ and $$N_ k$$ denote the orientable (resp. nonorientable) surface of genus g (resp. crosscap number k).

### MSC:

 05C10 Planar graphs; geometric and topological aspects of graph theory 57M15 Relations of low-dimensional topology with graph theory 57S25 Groups acting on specific manifolds

### Citations:

Zbl 0621.05013; Zbl 0521.05027
Full Text:

### References:

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