Enumeration of r-regular Maps on the Torus. Part II: Enumeration of Unsensed Maps

2017/09/11 by Evgeniy Krasko, Krasko, Evgeniy, Alexander Omelchenko +1 · 1 citation
Mathematics · #05C30 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.CO #msc:05C30

paper · pdf · doi:10.48550/arxiv.1709.03230

arxiv created 2017/09/11 · openalex publication_date 2017/09/11 · arxiv updated 2017/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The second part of the paper is devoted to enumeration of r-regular toroidal maps up to all homeomorphisms of the torus (unsensed maps). We describe in detail the periodic orientation reversing homeomorphisms of the torus which turn out to be representable as glide reflections. We show that considering quotients of the torus with respect to these homeomorphisms leads to maps on the Klein bottle, annulus and the Möbius band. Using 3- and 4-regular maps as an example we describe the technique of enumerating quotient maps on surfaces with a boundary. Obtained recurrence relations are used to enumerate unsensed r-regular maps on the torus for various r.

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