Vertex-Transitive Toroidal Maps

This article explores vertex-transitive maps on the torus, with a focus on polyhedral maps, which are embeddings of graphs on surfaces where face intersections are limited to vertices or edges. A map is said to be semi-equivelar if, up to cyclic permutations and reversed order, any two face-cycles are equivalent. There are eleven types of semi-equivelar maps on the torus. A map is termed vertex-transitive if its automorphism group acts transitively on the vertex set. This study builds on previous works that classified and enumerated various types of equivelar and semi-equivelar maps on the torus. Among these, only four types are vertex-transitive, while for the remaining seven types, both vertex-transitive and non-vertex-transitive examples are known. We extend the understanding of semi-equivelar maps by providing a detailed enumeration of vertex-transitive maps of specific types, namely \([3^2, 4^1, 3^1, 4^1], [4^1, 8^2], [3^1, 6^1, 3^1, 6^1], [3^1, 12^2], [3^1, 4^1, 6^1, 4^1], [4^1, 6^1, 12^1],\) $[3^2,4^1,3^1,4^1],[4^1,8^2],[3^1,6^1,3^1,6^1],[3^1,{12}^2],[3^1,4^1,6^1,4^1],[4^1,6^1,{12}^1],$ and \([3^4, 6^1]\) $[3^4,6^1]$ . We determine the conditions under which vertex-transitive maps exist for any given number of vertices. The results show that for each type, there exist integer values of n (the number of vertices) such that vertex-transitive maps can be constructed. This work provides a comprehensive classification of both vertex-transitive and non-vertex-transitive semi-equivelar maps on the torus, contributing significantly to the ongoing study of polyhedral map classification and symmetry on surfaces.