Families of Regular Graphs in Regular Maps

The question of when a given graph can be the underlying graph of a regular map has roots a hundred years old and is currently the object of several threads of research. This paper outlines this topic briefly and proves that a product of graphs which have regular embeddings also has such an embedding. We then present constructions for members of three families: (1) circulant graphs, (2) wreath graphs W(k, n), whose vertices are ordered pairs (i, j), 0⩽i<k, 0⩽j<n, and whose edges are all possible (i, j)−(i+1, j′), and (3) depleted wreath DW(k, n), the subgraph of W(k, n) left by removing all edges in which j=j′. We open the question of multiplicity of occurrence and we list the underlying graphs of rotary maps with no more than 50 edges.