A regular map is a cellular embedding of a connected graph in a closed surface whose automorphism group acts transitively on the flags of the embedding, where a flag is an incident vertex-edge-face triple. On an orientable surface, an orientably regular map is one whose orientation-preserving automorphism group acts transitively on the graph arcs (Jones and Singerman 1978).
A regular map has Schläfli symbol 
, or type 
, if every face has 
edges
and every vertex has vertex
degree 
.
If it has 
vertices, 
edges, and 
faces, double counting incidences
gives
 |
The dual map has type 
.
The five Platonic solids give familiar regular maps on the sphere. Regular maps also occur on surfaces
of positive genus. Orientably regular maps of type 
are Hurwitz
maps.
