This page lists all complete graphs that can be embedded as regular maps in manifolds with Euler characteristic not less than -200. (This limit is chosen because it is the limit for the data published by Professor Marston Conder in his paper Regular maps and hypermaps of Euler characteristic -1 to -200.)

If a regular map has mirror symmetry, then its Petrie dual exists, and embeds the same graph as a regular map in a generally different manifold. So some of the entries in the table below occur as Petrie-dual pairs. If a regular map is chiral, then its Petrie dual is not a regular map, and is not listed below.

Complete graph | manifold | Regular map | Rotational symmetry group of regular map | Comments |
---|---|---|---|---|

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

K_{2} | sphere | the monodigon | C2 | self-Petrie dual |

K_{3} | sphere | the ditriangle | S3, ≅ C3⋊C2 | |

projective plane | the hemi-di-hexagon | S3×C2 | ||

K_{4} | sphere | the tetrahedron | A4, ≅ C_{2}^{2}⋊C3 | |

projective plane | the hemicube | S4, ≅ C_{2}^{2}⋊S3 | ||

K_{5} | torus | {4,4}_{(2,1)} | C5⋊C4 | chiral, self-dual |

K_{6} | projective plane | the hemi-icosahedron | A5 | Exceptional; see paragraphs below. |

C5 | C5:{5,5} | |||

K_{7} | torus | {3,6}_{(1,3)} | C7⋊C6 | chiral |

K_{8} | S7 | S7:{7,7} | C_{2}^{3}⋊C7 | chiral dual pair |

S7:{7,7} | ||||

K_{9} | S10 | S10:{8,8} | C_{3}^{2}⋊C8 | chiral, self-dual |

K_{10} | (none) | |||

K_{11} | S12 | S12:{5,10} | C11⋊C10 | two different, chiral |

S12:{5,10} | ||||

K_{12} | (none) | |||

K_{13} | S27 | S27:{12,12} | C13⋊C12 | chiral dual pair |

S27:{12,12} | ||||

K_{14}, K_{15} | (none) | |||

K_{16} | S45 | S45:{15,15} | C_{2}^{4}⋊C15 | chiral dual pair |

S45:{15,15} | ||||

K_{17} | S52 | S52:{16,16} | C17⋊C16 | two chiral dual pairs |

S52:{16,16} | ||||

S52:{16,16} | ||||

S52:{16,16} | ||||

K_{18} | (none) | |||

K_{19} | S58 | S58:{9,18} | C19⋊C18 | three different, chiral |

S58:{9,18} | ||||

S58:{9,18} | ||||

K_{20}-K_{22} | (none) | |||

K_{23} | S93 | S93:{11,22} | C23⋊C22 | five different, chiral |

S93:{11,22} | ||||

S93:{11,22} | ||||

S93:{11,22} | ||||

S93:{11,22} |

With the exception of n=6, K_{n} can be embedded to form a regular
map iff n is a prime or prime power.

Wherever K_{n} can be embedded to form a regular map, except for n=6 (and,
trivially, n≦3), the rotational symmetry group of the regular map is a
Frobenius group acting
on its vertices. The Frobenius kernel can be considered as the additive group that
acts on the elements of the field **F**_{n}, and the Frobenius
complement as fixing one vertex and rotating the rest of the map about it.

For every prime *p* greater than 4 listed above, K_{p} can be
embedded as a regular map in one and only one orientable manifold, and the genus
of that manifold is 1 modulo *p*.

All such embeddings are explained and classified by

Norman Biggs

**Automorphisms of Imbedded Graphs**

Journal of Combinatorial Theory, 11, 132-138 (1971)

and

Lynne James and Gareth Jones

**Regular Orientable Imbeddings of Complete Graphs**

Journal of Combinatorial Theory, series B 39, 353-367 (1985)