Equivelar maps on the torus

Abstract. We give a classification of all equivelar polyhedral maps on the torus. In particular, we classify all triangulations and quadrangulations of the torus admitting a vertex transitive automorphism group. These are precisely the ones which are quotients of the regular tessellations {3, 6}, {6, 3} or {4, 4} by a pure translation group. An explicit formula for the number of combinatorial types of equivelar maps (polyhedral and non-polyhedral) with n vertices is obtained in terms of arithmetic functions in elementary number theory, such as the number of integer divisors of n. The asymptotic behaviour for n is also discussed, and an example is given for n such that the number of distinct equivelar triangulations of the torus with n vertices is larger than n itself. The numbers of regular and chiral maps are determined separately, as well as the ones for all other kinds of symmetry. Furthermore, arithmetic properties of the integers of type p2 +pq +q2 (or p2 +q2, resp.) can be inte...

• Type: Journal
• Year: 2008
• Where: EJC
• Authors: Ulrich Brehm, Wolfgang Kühnel
• Keywords: EJC 2007, EJC 2008, Information Technology, Regular Tessellation, Torus, Vertex Transitive Automorphism