Regular map (graph theory) explained

See also: Regular map (algebraic geometry).

In mathematics, a regular map is a symmetric tessellation of a closed surface. More precisely, a regular map is a decomposition of a two-dimensional manifold (such as a sphere, torus, or real projective plane) into topological disks such that every flag (an incident vertex-edge-face triple) can be transformed into any other flag by a symmetry of the decomposition. Regular maps are, in a sense, topological generalizations of Platonic solids. The theory of maps and their classification is related to the theory of Riemann surfaces, hyperbolic geometry, and Galois theory. Regular maps are classified according to either: the genus and orientability of the supporting surface, the underlying graph, or the automorphism group.

Overview

Regular maps are typically defined and studied in three ways: topologically, group-theoretically, and graph-theoretically.

Topological approach

Topologically, a map is a 2-cell decomposition of a compact connected 2-manifold.

\chi (M) = |V| - |E| + |F| which is equal to 2 -2 g if the map is orientable, and 2 - g if the map is non-orientable. It is a crucial fact that there is a finite (non-zero) number of regular maps for every orientable genus except the torus.

Group-theoretical approach

Group-theoretically, the permutation representation of a regular map M is a transitive permutation group C, on a set

\Omega of flags, generated by three fixed-point free involutions r₀, r₁, r₂ satisfying (r₀r₂)²= I. In this definition the faces are the orbits of F = <r₀, r₁>, edges are the orbits of E = <r₀, r₂>, and vertices are the orbits of V = <r₁, r₂>. More abstractly, the automorphism group of any regular map is the non-degenerate, homomorphic image of a <2,m,n>-triangle group.

Graph-theoretical approach

Graph-theoretically, a map is a cubic graph

\Gamma with edges coloured blue, yellow, red such that: \Gamma is connected, every vertex is incident to one edge of each colour, and cycles of edges not coloured yellow have length 4. Note that \Gamma is the flag graph or graph-encoded map (GEM) of the map, defined on the vertex set of flags \Omega and is not the skeleton G = (V,E) of the map. In general, |\Omega | = 4|E|.

A map M is regular if Aut(M) acts regularly on the flags. Aut(M) of a regular map is transitive on the vertices, edges, and faces of M. A map M is said to be reflexible iff Aut(M) is regular and contains an automorphism

\phi that fixes both a vertex v and a face f, but reverses the order of the edges. A map which is regular but not reflexible is said to be chiral.

Examples

• The great dodecahedron is a regular map with pentagonal faces in the orientable surface of genus 4.
• The hemicube is a regular map of type in the projective plane.
• The hemi-dodecahedron is a regular map produced by pentagonal embedding of the Petersen graph in the projective plane.
• The p-hosohedron is a regular map of type .
• The Dyck map is a regular map of 12 octagons on a genus-3 surface. Its underlying graph, the Dyck graph, can also form a regular map of 16 hexagons in a torus.

The following is a complete list of regular maps in surfaces of positive Euler characteristic, χ: the sphere and the projective plane.

χ	g	Schläfli	Vert.	Edges	Faces	Group		Graph		Notes
2	0		p	p	2	C2 × Dihp	4p	Cp		Dihedron
2	0		2	p	p	C2 × Dihp	4p	p-fold K2		Hosohedron
2	0		4	6	4	S4	24	K4		Tetrahedron
2	0		8	12	6	C2 × S4	48	K4 × K2		Cube
2	0		6	12	8	C2 × S4	48	K2,2,2		Octahedron
2	0		20	30	12	C2 × A5	120			Dodecahedron
2	0		12	30	20	C2 × A5	120	K6 × K2		Icosahedron
1	n1	/2	p	p	1	Dih2p	4p	Cp		Hemi-dihedron
1	n1	/2	2	p	p	Dih2p	4p	p-fold K2		Hemi-hosohedron
1	n1	/2	4	6	3	S4	24	K4		Hemicube
1	n1	/2	3	6	4	S4	24	2-fold K3		Hemioctahedron
1	n1	/2	10	15	6	A5	60	Petersen graph		Hemidodecahedron
1	n1	/2	6	15	10	A5	60	K6		Hemi-icosahedron

The images below show three of the 20 regular maps in the triple torus, labelled with their Schläfli types.

Toroidal polyhedra

Regular maps exist as torohedral polyhedra as finite portions of Euclidean tilings, wrapped onto the surface of a duocylinder as a flat torus. These are labeled _b,c for those related to the square tiling, .^[1] _b,c are related to the triangular tiling,, and _b,c related to the hexagonal tiling, . b and c are whole numbers.^[2] There are 2 special cases (b,0) and (b,b) with reflective symmetry, while the general cases exist in chiral pairs (b,c) and (c,b).

Regular maps of the form _m,0 can be represented as the finite regular skew polyhedron, seen as the square faces of a m×m duoprism in 4-dimensions.

Here's an example _8,0 mapped from a plane as a chessboard to a cylinder section to a torus. The projection from a cylinder to a torus distorts the geometry in 3 dimensions, but can be done without distortion in 4-dimensions.

Regular maps with zero Euler characteristic^[3]

χ	g	Schläfli	Vert.	Edges	Faces	Group		Notes
0	1	b,0 n=b^2	n	2n	n	[4,4](b,0)	8n	Flat toroidal polyhedra Same as

Notes and References

1. [Coxeter]
2. [Coxeter]
3. [Coxeter]