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 |
|---|---|---|---|---|
| K1 | sphere | the edgeless map | 1 | self-Petrie dual |
| K2 | sphere | the monodigon | C2 | self-Petrie dual |
| K3 | sphere | the ditriangle | S3, ≅ C3⋊C2 | |
| projective plane | the hemi-di-hexagon | S3×C2 | ||
| K4 | sphere | the tetrahedron | A4, ≅ C22⋊C3 | |
| projective plane | the hemicube | S4, ≅ C22⋊S3 | ||
| K5 | torus | {4,4}(2,1) | C5⋊C4 | chiral, self-dual |
| K6 | projective plane | the hemi-icosahedron | A5 | Exceptional; see paragraphs below. |
| C5 | C5:{5,5} | |||
| K7 | torus | {3,6}(1,3) | C7⋊C6 | chiral |
| K8 | S7 | S7:{7,7} | C23⋊C7 | chiral dual pair |
| S7:{7,7} | ||||
| K9 | S10 | S10:{8,8} | C32⋊C8 | chiral, self-dual |
| K10 | (none) | |||
| K11 | S12 | S12:{5,10} | C11⋊C10 | two different, chiral |
| S12:{5,10} | ||||
| K12 | (none) | |||
| K13 | S27 | S27:{12,12} | C13⋊C12 | chiral dual pair |
| S27:{12,12} | ||||
| K14, K15 | (none) | |||
| K16 | S45 | S45:{15,15} | C24⋊C15 | chiral dual pair |
| S45:{15,15} | ||||
| K17 | S52 | S52:{16,16} | C17⋊C16 | two chiral dual pairs |
| S52:{16,16} | ||||
| S52:{16,16} | ||||
| S52:{16,16} | ||||
| K18 | (none) | |||
| K19 | S58 | S58:{9,18} | C19⋊C18 | three different, chiral |
| S58:{9,18} | ||||
| S58:{9,18} | ||||
| K20-K22 | (none) | |||
| K23 | 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, Kn can be embedded to form a regular map iff n is a prime or prime power.
Wherever Kn 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 Fn, 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, Kp 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)