کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4654435 | 1632825 | 2008 | 19 صفحه PDF | دانلود رایگان |

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 nn vertices is obtained in terms of arithmetic functions in elementary number theory, such as the number of integer divisors of nn. The asymptotic behaviour for n→∞n→∞ is also discussed, and an example is given for nn such that the number of distinct equivelar triangulations of the torus with nn vertices is larger than nn 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+q2p2+pq+q2 (or p2+q2p2+q2, resp.) can be interpreted and visualized by the hierarchy of covering maps between regular and chiral equivelar maps or type {3,6} (or {4,4}, resp.).
Journal: European Journal of Combinatorics - Volume 29, Issue 8, November 2008, Pages 1843–1861