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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).

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
Full Text: DOI
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