# Graphs and Regular Maps

See also Named Graphs

This page lists some symmetric graphs, with the regular maps that they
can form.

Many symmetric graphs can be embedded
in some compact manifold to give a regular map. Normally, when a symmetric
graph can be embedded in one compact manifold to give one regular map, it can
also be embedded in another compact manifold so as to give a different regular
map, the Petrie dual of the first.
These Petrie dual pairs are seen as the third and fourth columns of the table.
Some symmetric graphs can be embedded in only one compact manifold, giving a
regular map which is self-Petrie dual.

The column headed "m" is the multiplicity of the graph. A graph with a
multiplicity *n* has one copy of each vertex but *n* copies
of each edge.

Graph | m | 1st regular map | 2nd regular map |
---|

1-cycle | 1 | the hemi-2-hosohedron |
the dimonogon |

2-cycle | the hemi-di-square |
the 2-hosohedron |

3-cycle | the hemi-di-hexagon |
the di-triangle |

4-cycle | the hemi-di-octagon |
the di-square |

4-cube | {4,4}_{(4,0)} |
self-Petrie dual |

5-cycle | the hemi-di-decagon |
the di-pentagon |

6-cycle | the hemi-di-dodecagon |
the di-hexagon |

7-cycle | the hemi-di-14gon |
the di-heptagon |

Clebsch graph | S5:{5,5} |
C6:{4,5} |

cube | the cube |
{6,3}_{(2,2)} |

Desargues graph | C6:{10,3}_{10} |
self-Petrie dual |

Dyck graph | {6,3}_{(4,4)} |
The Dyck map |

F26A graph | {6,3}_{(2,4)} |
(a non-regular map) |

Heawood graph | {6,3}_{(1,3)} |
S3:{14,3}a |

K_{1} | the edgeless map |
self-Petrie dual |

K_{2} | the monodigon |
self-Petrie dual |

K_{3} | the di-triangle |
the hemi-di-hexagon |

K_{4} | the hemicube |
the tetrahedron |

K_{5} | {4,4}_{(2,1)} |
C5:{10,4} |

K_{5} × K_{2} | {4,4}_{(3,1)} |
S4:{10,4}a |

K_{6} | the hemi-icosahedron |
C5:{5,5} |

K_{7} | {3,6}_{(1,3)} |
(a non-regular map) |

K_{8} | {7,7}_{4} and
{7,7}_{4} |
(two non-regular maps) |

K_{3,3} | {6,3}_{(0,2)} |
self-Petrie dual |

K_{4,4} | {4,4}_{(2,2)} |
S3:{8,4|4} |

K_{2,2,2} |
C4:{6,4}_{3} |
the octahedron |

K_{3,3,3} |
{3,6}_{(3,3)} |
the regular map with C&D number N11.2 |

K_{4,4,4} |
S3:{3,8} |
the regular map with C&D number N22.4 |

Klein graph | the Klein map, S3:{7,3} |
C9:{8,3}_{7} |

Ljubljana graph |
the regular map with C&D number R17.2p |
the regular map with C&D number N104.1 |

Möbius-Kantor graph | S2:{8,3} |
S3:{12,3} |

Nauru graph | {6,3}_{(0,4)} |
S4:{12,3} |

Paley order-13 graph | {3,6}_{(2,4)} |
(a non-regular map) |

Pappus graph | {6,3}_{(3,3)} |
self-Petrie dual |

Petersen graph | the hemidodecahedron |
self-Petrie dual |

1-cycle | 2 | the hemi-4-hosohedron |
{4,4}_{(1,0)} |

4-cycle | {4,4}_{(2,0)} |
S2:{8,4} |

6-cycle | S2:{6,4} | S3:{12,4} |

C4:{6,4}_{6} |
self-Petrie dual |

8-cycle | S3:{8,4|2} |
self-Petrie dual |

cube | S3:{4,6} |
S5:{6,6} |

K_{4} |
S3:{6,6} |
C4:{4,6}_{6} |

{3,6}_{(2,2)} |
C4:{4,6}_{3} |

K_{6} |
C6:{3,10}_{5} |
the regular map with C&D number N14.3 |

K_{2,2,2} |
S2:{3,8} |
the regular map with C&D number N16.7p |

K_{3,3} | C5:{4,6} |
self-Petrie dual |

1-cycle | 3 | the hemi-6-hosohedron |
{3,6}_{(1,1)} |

3-cycle | C5:{6,6} |
{3,6}_{(0,2)} |

4-cycle | S2:{4,6} |
S4:{12,6} |

K_{2} | the 3-hosohedron |
{6,3}_{(1,1)} |

1-cycle | 4 | the hemi-8-hosohedron |
S2:{8,8} |

4-cycle |
S3:{4,8|4} |
R5.12 **or** R5.13 |

S3:{4,8|2} |
R5.13 **or** R5.12 |

K_{2} |
the 4-hosohedron |
{4,4}_{(1,1)} |

K_{4} |
S3:{3,12} |
the regular map with C&D number N16.7 |

1-cycle | 5 | the hemi-10-hosohedron |
S2:{5,10} |

K_{2} | the 5-hosohedron |
S2:{10,5} |

1-cycle | 6 | the hemi-12-hosohedron |
S3{12,12} |

K_{2} | the 6-hosohedron |
S2:{6,6} |

1-cycle | 7 | the hemi-14-hosohedron |
S3{7,14} |

K_{2} | the 7-hosohedron |
S3:{14,7} |

K_{2} | 8 |
the 8-hosohedron |
S3:{8,8}_{2} |

K_{2} |
S2:{4,8} |
S3:{8,8}_{4} |

K_{2} | 9 | the 9-hosohedron |
S4:{18,9} |

K_{2} | 10 | the 10-hosohedron |
S4:{10,10} |

K_{2} | 11 | the 11-hosohedron |
S5:{22,11} |

K_{2} | 12 |
the 12-hosohedron |
S5:{12,12} |

K_{2} | 12 |
S3:{4,12} |
S4:{6,12} |

K_{2} | 13 | the 13-hosohedron |
S6:{26,13} |

K_{2} | 14 | the 14-hosohedron |
S6:{14,14} |

## Regular Maps

General Index