Semi-equivelar toroidal maps and their k-semiregular covers

TL;DR

This paper investigates semi-equivelar toroidal maps, establishing bounds on flag orbits, showing none are semiregular among Archimedean types, and classifying their minimal covers and semiregular maps.

Contribution

It provides bounds on flag orbits, proves the non-existence of semiregularity in Archimedean types, and classifies minimal covers and semiregular maps for semi-equivelar toroidal maps.

Findings

Existence of bounds for flag orbits in semi-equivelar toroidal maps

No Archimedean type on the torus is semiregular

Existence of finite index minimal covers for semi-equivelar maps

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. In particular, it is called equivelar if the face-cycles contain same type of faces. A map is semiregular (or almost regular) if it has as few flag orbits as possible for its type. A map is $k$-regular if it is equivelar and the number of flag orbits of the map $k$ under the automorphism group. In particular, if $k =1$, its called regular. A map is $k$-semiregular if it contains more number of flags as compared to its type with the number of flags orbits $k$. Drach et al. \cite{drach:2019} have proved that every semi-equivelar toroidal map has a finite unique minimal semiregular cover. In this article, we show the bounds of flag orbits of semi-equivelar toroidal maps, i.e., there exists $k$ for each type such that every semi-equivelar map is $\ell$-uniform for some $\ell \le k$. We show that none of the Archimedean types on the torus is semiregular, i.e., for each type, there exists a map whose number of flag orbits is more than its type. We also prove that if a semi-equivelar map is $m$-semiregular then it has a finite index $t$-semiregular minimal cover for $t \le m$. We also show the existence and classification of $n$ sheeted $k$-semiregular maps for some $k$ of semi-equivelar toroidal maps for each $n \in \mathbb{N}$.