Semi-equivelar maps on the torus and the Klein bottle are Archimedean

Datta, Basudeb and Maity, Dipendu (2018) Semi-equivelar maps on the torus and the Klein bottle are Archimedean. In: DISCRETE MATHEMATICS, 341 (12). pp. 3296-3309.

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Abstract

If the face-cycles at all the vertices in a map on a surface are of same type then the map is called semi-equivelar. There are eleven types of Archimedean tilings on the plane. All the Archimedean tilings are semi-equivelar maps. If a map X on the torus is a quotient of an Archimedean tiling on the plane then the map X is semi-equivelar. We show that each semi-equivelar map on the torus or on the Klein bottle is a quotient of an Archimedean tiling on the plane. Vertex-transitive maps are semi-equivelar maps. We know that four types of semi-equivelar maps on the torus are always vertex-transitive and there are examples of other seven types of semi-equivelar maps which are not vertex-transitive. We show that the number of Aut(Y)-orbits of vertices for any semi-equivelar map Y on the torus is at most six. In fact, the number of orbits is at most three except one type of semi-equivelar maps. Our bounds on the number of orbits are sharp. (C) 2018 Elsevier B.V. All rights reserved.

Item Type:	Journal Article
Publication:	DISCRETE MATHEMATICS
Publisher:	ELSEVIER SCIENCE BV
Additional Information:	Copy right for this article belong to ELSEVIER SCIENCE BV
Keywords:	Polyhedral maps on torus and Klein bottle; Vertex-transitive map; Equivelar map; Archimedean tiling
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	12 Nov 2018 15:43
Last Modified:	12 Nov 2018 15:43
URI:	http://eprints.iisc.ac.in/id/eprint/61012

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