Splitting

Splitting is a non-symmetric relationship between some pairs of regular maps. It "splits" a regular map into a version with n times as many vertices, and faces n times as large.

2-splitting is slightly simpler than n-splitting, and is described first.

2-splitting

If a regular map is described by

{p,q} V F E

(meaning, each face has p edges, each vertex has q edges, it has V vertices, V faces and E edges), and p is odd, then it can be 2-split. This yields a regular map described by

{2p,q} 2V F 2E.

(If we apply the procedure with p even, we get a two disconnected graphs, instead of one connected graph.)

For example, if we 2-split the tetrahedron we get {6,3}_(2,2).

If you have a regular map and want to 2-split it,

Replace each vertex by a pair of vertices, one "red" and one "green".
Replace each edge by a pair of edges, each joining new vertices of different colours.
Retain each face as a face, but with twice as many edges

It may not be obvious how to embed the resulting structure in the appropriate surface.

The vertex multiplicity of the result is the same as that of the original map.
The face multiplicity of the result is double that of the original map.
If the Petrie polygons of the original map are even in size, those of the result are the same size; if odd, twice the size.
Iff the original map is chiral, the result is chiral. Iff the original map is non-orientable, the result is non-orientable.

If the underlying graph of a regular map is G, and we 2-split it, the underlying graph of the result is G × K₂. If the face adjacency graph of a regular map is F, and we 2-split it, the underlying graph of the result is F with its multiplicity doubled.

For any non-orientable regular map with odd-sized faces and even-sized Petrie polygons, there are at least two candidate non-orientable regular maps with the right statistics to be the result of 2-splitting it.

For any non-orientable regular map with odd-sized faces and odd-sized Petrie polygons, there is always, as well as a non-orientable map that is the result of 2-splitting it, another with the same statistics except that its Petrie polygons are the same size as those of the original; with the following exceptions:

A single map in each genus of order 1 modulo 6, starting with the hemi-octahedron
{7,6}₇ in genus 34

(There may be other exceptions out of the range of Professor Conder's published data.)

If we start with a regular map {p,q} with p,q both odd, the dual of the 2-split of the dual of the 2-split of it is the same as the 2-split of the dual of the 2-split of the dual of it.

n-splitting

n-Splitting extends the idea of 2-splitting to multiples greater than 2.

n-Splitting can only be applied when the underlying graph of the dual of the regular map is a bipartite graph. Also, it only works if the face size of the original regular map is prime to n; otherwise we end up with several disconnected graphs, the number of them being the highest common factor of the face size and n.

If you have a regular map and want to n-split it,

Colour its faces blue and yellow, with each face bordered only by faces of the other colour, like a chess board. This is possible because we have specified that the graph of its dual is a bipartite graph. (Necessary, but not sufficient, conditions for the dual graph to be bipartite are that the faces, and the Petrie polygons, of the original regular map are even.)
Replace each vertex by a set of n vertices, labelled with the integers from 0 to n-1.
Replace each edge by a set of n edges, each joining a new vertex labelled m on the left to one labelled m+1 modulo n on the right, with left and right as seen from the incident blue face.
Retain each face as a face, but with n times as many edges

It may not be obvious how to embed the resulting structure in the appropriate surface.

Other relationships between regular maps
General Index