2-uniform toroidal maps, classification and asymptotic behavior

TL;DR

This paper classifies 2-uniform maps on the torus, providing explicit formulas for their enumeration and analyzing their asymptotic behavior as the number of vertices grows large.

Contribution

It offers the first classification of 2-uniform toroidal maps and derives formulas involving number theory functions, along with asymptotic bounds.

Findings

Explicit formulas for the number of 2-uniform maps based on divisor functions

Asymptotic bounds for the growth of these maps as vertices increase

Continuous functions approximating the asymptotic behavior

Abstract

If a map has k transitivity classes of vertices that are subject to the action of the automorphism group, it is said to be k-uniform. The classification of 1-uniform maps on the torus is known. In this article, we classify 2-uniform maps on the torus up to isomorphism. Explicit formulas for the number of combinatorial types of these maps on number of vertices is obtained in terms of arithmetic functions in number theory, such as the divisor function. The asymptotic behaviour of these functions as number of vertices tends to infinity is also discussed and we obtained continuous functions which asymptotically served as upper and lower bounds.