Semi-equivelar toroidal maps and their k-edge covers

TL;DR

The paper addresses edge-orbit structure in semi-equivelar toroidal maps, focusing on edge-homogeneous cases and their coverings. It adopts a plane-tiling perspective, realizing toroidal maps as quotients of plane tilings by discrete automorphism subgroups, and employs covering-space theory to construct and classify edge covers. The authors establish sharp bounds on the number of edge orbits for toroidal edge-homogeneous maps, extend these bounds to non-edge-homogeneous semi-equivelar toroidal maps, and show that every edge-homogeneous toroidal map has finite-index minimal covers with $m\le k$, along with a complete classification of $n$-sheeted covers via Hermite normal forms. These results deepen the understanding of symmetry, covering structures, and the interplay between edge-orbits and toroidal topology in semi-equivelar tilings.

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 tiling is edge-homogeneous if any two edges with vertices of congruent face-cycles. In general, edge-homogeneous maps on a surface form a bigger class than edge-transitive maps. There are edge-homogeneous toroidal maps which are not edge\mbox{-}transitive. An edge-homogeneous map is called $k$-edge-homogeneous if it contains $k$ number of edge orbits. In particular, if $k=1$ then it is called edge-transitive map. In general, a map is called $k$-edge orbital or $k$-orbital if it contains $k$ number of edge orbits. A map is called minimal if the number of edges is minimal. A surjective mapping $η\colon M \to K$ from a map $M$ to a map $K$ is called a covering if it preserves adjacency and sends vertices, edges, faces of $M$ to vertices, edges, faces of $K$ respectively. Orbani{\' c} et al. and {\v S}ir{á}{\v n} et al. have shown that every edge-homogeneous toroidal map has edge-transitive cover. In this article, we show the bounds of edge orbits of edge-homogeneous toroidal maps. Using these bounds, we show the bounds of edge orbits of non-edge-homogeneous semi-equivelar toroidal maps. We also prove that if a edge-homogeneous map is $k$ edge orbital then it has a finite index $m$-edge orbital minimal cover for $m \le k$. We also show the existence and classification of $n$ sheeted covers of edge-homogeneous toroidal maps for each $n \in \mathbb{N}$. We extend this to non-edge-homogeneous semi-equivelar toroidal maps and prove the same results, i.e., if a non-edge-homogeneous map is $k$ edge orbital then it has a finite index $m$-edge orbital minimal cover (non-edge-homogeneous) for $m \le k$ and then classify them for each sheet.

Figures (10)

• Figure 1: Edge-homogeneous tilings on the plane
• Figure 2: $E_9$ ($[3^1,4^1,6^1,4^1]$)
• Figure 3: $E_8$ ($[3^1,12^2]$)
• Figure 4: $E_5$ ($[3^2,4^1,3^1,4^1]$)
• Figure 5: $E_4$ ($[3^3,4^2]$)

Theorems & Definitions (20)

• proof : Proof of Theorem \ref{['thm:no-of-orbits']}
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• proof : Proof of Theorem \ref{['thm-main1']}
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