Abstract
The first note of this paper determines for which g the orientable surface of genus g can be embedded in euclidean 3-space so as to have prismatic, cubical/octahedral, tetrahedral, or icosahedral/dodecahedral symmetry. The second note proves, through entirely elementary methods, that the clique number of the graph underlying a regular map is m = 2, 3, 4, 6; for m = 6 the map must be non-orientable and for m = 4, 6 the graph has a K m factorization. Here a regular map is one having maximal symmetry: reflections in all edges and full rotational symmetry about every vertex, edge and face.
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Tucker, T.W. (2014). Two Notes on Maps and Surface Symmetry. In: Connelly, R., Ivić Weiss, A., Whiteley, W. (eds) Rigidity and Symmetry. Fields Institute Communications, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0781-6_17
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DOI: https://doi.org/10.1007/978-1-4939-0781-6_17
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