Semi-equivelar toroidal maps and their vertex covers

TL;DR

This paper studies semi-equivelar toroidal maps, proving the existence of minimal covers with specific orbital properties and classifying their n-sheeted covers, advancing understanding of their symmetry and covering structures.

Contribution

It establishes that k-orbital semi-equivelar toroidal maps have finite index m-orbital minimal covers with m ≤ k and classifies n-sheeted covers for all natural numbers n.

Findings

Existence of finite index m-orbital minimal covers for k-orbital semi-equivelar maps.

Classification of n-sheeted covers for semi-equivelar toroidal maps.

Sharp bounds on the number of vertex orbits are confirmed.

Abstract

If the face\mbox{-}cycles at all the vertices in a map are of same type then the map is called semi\mbox{-}equivelar. A map is called minimal if the number of vertices is minimal. We know the bounds of number of vertex orbits of semi-equivelar toroidal maps. These bounds are sharp. Datta \cite{BD2020} has proved that every semi-equivelar toroidal map has a vertex-transitive cover. In this article, we prove that if a semi-equivelar map is $k$ orbital then it has a finite index $m$-orbital minimal cover for $m \le k$. We also show the existence and classification of $n$-sheeted covers of semi-equivelar toroidal maps for each $n \in \mathbb{N}$.