Semi-equivelar toroidal maps and their k-edge covers

Abstract

A map K on a surface is called edge-transitive if the automorphism group of K acts transitively on the set of edges of K. 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-transitive. A map is called minimal if the number of edges is minimal. A map f : M → K is a covering map if f is a covering map from the vertex set of M to the vertex set of K such that f is a surjection and a local isomorphism : the neighbourhood of a vertex v in M is mapped bijectively onto the neighbourhood of f (v) in K. Orbanić et al. and Širáň et al. have shown that every edge-homogeneous toroidal map has edgetransitive cover. In this article, we show the bounds of edge orbits of edge-homogeneous toroidal maps. We also prove that if a edge-homogeneous map is k edge orbital (or k orbital) then it has a finite index m-edge orbital minimal cover for m ≤ k. We also show the existence and classification of n sheeted covers of edge-homogeneous toroidal maps for each n ∈ N.