Tucker, T,; Watkins, M. and Širáň, J.
(2001).
Realizing finite edgetransitive orientable maps.
Journal of Graph Theory, 37(1) pp. 1–34.
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Abstract
J.E. Graver and M.E. Watkins, Memoirs Am. Math. Soc. 126 (601) (1997) established that the automorphism group of an edgetransitive, locally finite map manifests one of exactly 14 algebraically consistent combinations (called types) of the kinds of stabilizers of its edges, its vertices, its faces, and its Petrie walks. Exactly eight of these types are realized by infinite, locally finite maps in the plane. H.S.M. Coxeter (Regular Polytopes, 2nd ed., McMillan, New York, 1963) had previously observed that the nine finite edgetransitive planar maps realize three of the eight planar types. In the present work, we show that for each of the 14 types and each integer n ≥ 11 such that n ≡ 3, 11 (mod 12), there exist finite, orientable, edgetransitive maps whose various stabilizers conform to the given type and whose automorphism groups are (abstractly) isomorphic to the symmetric group Sym(n). Exactly seven of these types (not a subset of the planar eight) are shown to admit infinite families of finite, edgetransitive maps on the torus, and their automorphism groups are determined explicitly. Thus all finite, edgetransitive toroidal maps are classified according to this schema. Finally, it is shown that exactly one of the 14 types can be realized as an abelian group of an edgetransitive map, namely, as ℤ_{n} × ℤ_{2} where n ≡ 2 (mod 4).
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