Definition

For the purpose of these pages, a "regular map" is defined as an of embedding a graph (a set of vertices and edges) in a compact 2-manifold such that

• the 2-manifold is partitioned into faces,
• each face has the topology of a disc,
• it is face-transitive: for any two faces, there is a symmetry operation of the whole thing that takes one face to the other,
• its faces are regular: for any face and any two edges of that face, there is a symmetry operation which maps one edge to the other and maps all edges of that face to edges of that face,
• it is vertex-transitive: for any two vertices, there is a symmetry operation of the whole thing that takes one vertex to the other,
• its vertices are regular: for any vertex and any two edges of that vertex, there is a symmetry operation which maps one edge to the other and maps all edges of that vertex to edges of that vertex.

For the reasoning behind this choice of definition, see What do we mean by "Regular" for Regular Maps?

Further, optional, criteria are listed below. Regular maps violating these criteria are listed on these pages, with red marks indicating the violations.

1.   Each face has at least three edges
2.   Each vertex has at least three edges
3.   A face may not share a vertex with itself, equivalently a vertex may not share a face with itself.
4.   A face may not share an edge with itself, equivalently an edge may not share a face with itself.
5.   An edge may not share a vertex with itself, equivalently a vertex may not share an edge with itself.
6.   It is "flag-transitive", with full symmetry including reflection, not chiral

For the sphere, this definition gives the five regular maps usually known as the five "platonic solids" or "regular polyhedra", and some other things. For manifolds of genus greater than 0, it gives some things which which have a pleasing amount of symmetry, but will be less familiar to many readers.