Semi-equivelar toroidal maps and their k-edge covers

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.