See also Graphs and Regular Maps

identified with regular maps | not identified with regular maps | known not to be regular

When I first became interested in regular maps, I did not know what they were called. But as each one involves a symmetric graph, I realised that I could try to find them by choosing some symmetric graph, maybe from Wikipedia's "gallery of named graphs", and trying to find a way to embed it regularly in a compact 2-manifold.

The page lists some symmetric graphs, and for each which I know how to embed regularly in some manifold, gives a link to a page where I show it as a regular map.

Some named graphs can be embedded nicely in a manifold, but not in a fully regular way. For example, the Franklin graph can be embedded in the projective plane, where it is seen to be a truncated hemi-octahedron, which is not face-transitive. A few of these are listed in the main table of this page, but I have not tried to list irregular named graphs systematically.

Name of graph (link to wikipedia) |
Picture of graph | Valency | Vertices | Edges | Full symmetry group of graph |
Rotational symmetry group of regular map |
Schläfli symbol (link to polyhedron) |
Picture of regular map | Faces of regular map |
---|---|---|---|---|---|---|---|---|---|

Cube | 3 | 8 | 12 | 48 S4×C2 |
24 S4 |
{4,3} | 6 squares | ||

Cycle graph | 2 | n |
n |
2nD2 |
nC |
{5,2} | 2 pentagons | ||

Desargues graph | 3 | 20 | 30 | 240 S5×C2 |
60 A5 |
C6:{10,3}_{10} |
6 decagons | ||

Dodecahedron | 3 | 12 | 30 | 120 S5 |
60 A5 |
{3,5} | 12 pentagons | ||

Dyck graph | 3 | 32 | 48 | 192 | 96 | {6,3}_{(4,4)} |
16 hexagons | ||

96 | S3{8,3} | 12 octagons | |||||||

F26A graph | 3 | 26 | 39 | 78 | 78 | {6,3}_{(2,4)} |
13 hexagons | ||

Franklin graph | 3 | 12 | 18 | 48 S4×C2 |
26 S4 |
Not regular. Can be embedded in the projective plane, as a truncated hemi-octahedron, having .. | .. 3 squares and 4 hexagons | ||

Heawood graph | 3 | 14 | 21 | 336 PGL(2,7) |
21 C7⋊C3 |
S3:{14,3} | Not quite regular. 3 14-gons | ||

Icosahedron | 3 | 20 | 30 | 120 A5×C2 |
60 A5 |
{3,5} | 20 triangles | ||

K_{1} |
0 | 0 | 0 | 1 S1 |
1 S1 |
{0,0} | 1 point-bounded face | ||

K_{2} |
1 | 2 | 1 | 2 S2 |
2 S2 |
{2,1} | 1 digon | ||

K_{3} |
2 | 3 | 3 | 6 S3 |
6 S3 |
{3,2} | 2 triangles | ||

K_{4} |
3 | 4 | 6 | 24 S4 |
12 A4 |
{3,3} | 4 triangles | ||

K_{5} |
4 | 5 | 10 | 120 S5 |
20 C5⋊C4 |
{4,4}_{(2,1)} |
5 squares | ||

K_{6} |
5 | 6 | 15 | 720 S6 |
60 A5 |
C1:{3,5}, the hemi-icosahedron | 10 triangles | ||

K_{7} |
6 | 7 | 21 | 5,040 S7 |
42 C7⋊C6 |
{3,6}_{(1,3)} |
14 triangles | ||

K_{8} |
7 | 8 | 28 | 40,320 S8 |
56 (C2×C2×C2)⋊C7 |
S7:{7,7} and its dual S7:{7,7} | 8 heptagons | ||

K_{n} | See a list of complete graphs embedded as regular maps.
| ||||||||

K_{3,3}Utility |
3 | 6 | 9 | 72 (S3×S3)⋊C2 |
18 D6×C3 |
{6,3}_{(0,2)} |
3 hexagons | ||

K_{4,4} |
4 | 4 | 8 | 1,152 (S4×S4)⋊C2 |
16 C |
{4,4}_{(2,2)} |
8 squares | ||

16 | S3{8,4|4} | 4 octagons | |||||||

K_{5,5} |
4 | 10 | 25 | 28,800 | 50 | S6:{10,5} | 5 hexagons | ||

K_{6,6} |
5 | 12 | 36 | 1,036,000 | 72 | S4:{4,6} | 18 squares | ||

K_{7,7} |
6 | 14 | 49 | 50,803,200 | 98 | S15:{14,7} | 7 14-gons | ||

K_{2,2,2} |
4 | 6 | 12 | 96 (S2×S2×S2)⋊S3 |
12 A4 |
{3,4} | 8 triangles | ||

K_{3,3,3} |
6 | 9 | 27 | 1,296 (S3×S3×S3)⋊S3 |
54 | {3,6}_{(3,3)} |
18 triangles | ||

K_{4,4,4} |
8 | 12 | 48 | 82,944 (S4×S4×S4)⋊S3 |
96 | S3:{3,8}, the dual of the Dyck map | 12 octagons | ||

K_{5,5,5} |
10 | 15 | 75 | 10,368,000 | 96 | S6:{3,10} | 50 triangles | ||

Klein graph | 3 | 56 | 84 | 336 PGL(2,7) |
168 PSL(2,7) |
S3:{7,3}, the Klein map | 24 heptagons | ||

Möbius-Kantor graph | 3 | 16 | 24 | 96 | 48 | S2:{8,3} | 6 octagons | ||

Nauru graph | 3 | 24 | 36 | 144 S4×D6 |
72 | {6,3}_{(0,4)} |
12 hexagons | ||

Octahedron | See K_{2,2,2} above | ||||||||

Paley order-13 graph | 6 | 13 | 39 | 78 | 78 | {3,6}_{(2,4)} |
26 triangles | ||

Pappus graph | 3 | 18 | 27 | 216 | 54 | {6,3}_{(3,3)} |
9 hexagons | ||

Petersen graph | 3 | 10 | 15 | 120 S5 |
60 A5 |
C1:{5,3}, the hemidodecahedron | 6 pentagons | ||

Shrikhande graph | 6 | 16 | 48 | 192 | 96 | {3,6}_{(4,4)} |
32 triangles | ||

Tetrahedron | See K4 above |

These either don't exist as regular maps, or do but I don't know how.

Biggs-Smith graph | 3 | 102 | 153 | 2448 PSL(2,17) |
? | ||

Clebsch graph also called the Greenwood-Gleason graph |
5 | 16 | 40 | 1920 | (20 squares) | ||

Coxeter graph | 3 | 28 | 42 | 336 PGL(2,7) |
(12 heptagons) | ||

10-crown | 4 | 10 | 20 | 80 | (8 pentagons, but this graph is bipartite) | ||

Double-star snark | 3 | 30 | 45 | ? | 3 30-gons | ||

Foster graph | 3 | 90 | 135 | 4320 | ? | ||

Hall-Janko graph | ? | 100 | ? | ? | ? | ||

Higman-Sims graph | 22 | 100 | 1100 | 88,704,000 HS⋊C2) |
? | ||

Hoffman-Singleton graph | 7 | 50 | 175 | 252,000 PSU(3,52)⋊C2 |
? | ||

Horton graph | 3 | 96 | 144 | 96 S4×C2×C2 |
Not the same as {6,3}_{(0,8)} |
||

Tutte-Coxeter graph | 3 | 30 | 45 | 1,440 Aut(S6) |
? |

- Balaban 10-cage
- Balaban 11-cage
- Bidiakis cube
- Blanuša snarks
- Brinkmann graph
- Bull graph
- Chvátal graph
- Double-star snark
- Doyle graph
- Dürer graph
- Errera graph
- Flower snark J3
- Flower snark J5
- Flower snark J7
- Folkman graph
- Frucht graph
- Goldner-Harary graph
- Gray graph
- Grötzsch graph
- Harries graph
- Harries-Wong graph
- Herschel graph
- Hoffman graph
- Holt graph
- Ljubljana graph
- Markström graph
- McGee graph
- Meredith graph
- Robertson graph
- Szekeres graph
- Tietze's graph
- Tutte graph
- Wagner graph
- Watkins snark

More on Regular Maps

.Copyright N.S.Wedd 2009

All the images on this page were taken from Wikipedia.