Datta, Basudeb and Maity, Dipendu (2017) Semi-equivelar and vertex-transitive maps on the torus. In: Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 58 (3). pp. 617-634. ISSN 0138-4821
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Abstract
A vertex-transitive map X is a map on a closed surface on which the automorphism group Aut (X) acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. We show that there are eleven types of semi-equivelar maps on the torus. Three of these are equivelar maps. It is known that two of these three types are always vertex-transitive. We show that this is true for the remaining one type of equivelar maps and one other type of semi-equivelar maps, namely, if X is a semi-equivelar map of type [ 6 3] or [ 3 3, 4 2] then X is vertex-transitive. We also show, by presenting examples, that this result is not true for the remaining seven types of semi-equivelar maps. There are ten types of semi-equivelar maps on the Klein bottle. We present examples in each of the ten types which are not vertex-transitive.
| Item Type: | Journal Article |
|---|---|
| Publication: | Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry |
| Publisher: | Springer Verlag |
| Additional Information: | The copyright for this article belongs to the Authors. |
| Keywords: | Archimedean tiling; Equivelar maps; Polyhedral map on torus; Vertex-transitive map |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 29 May 2022 07:39 |
| Last Modified: | 29 May 2022 07:39 |
| URI: | https://eprints.iisc.ac.in/id/eprint/72768 |
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