Complete graphs as regular maps

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
manifoldRegular mapRotational symmetry
group of regular map
Comments
K1spherethe edgeless map1self-Petrie dual
K2spherethe monodigonC2self-Petrie dual
K3spherethe ditriangleS3, ≅ C3⋊C2
projective planethe hemi-di-hexagonS3×C2
K4spherethe tetrahedronA4, ≅ C22⋊C3
projective planethe hemicubeS4, ≅ C22⋊S3
K5torus{4,4}(2,1)C5⋊C4chiral, self-dual
K6projective planethe hemi-icosahedronA5Exceptional; see paragraphs below.
C5C5:{5,5}
K7torus{3,6}(1,3)C7⋊C6chiral
K8S7S7:{7,7}C23⋊C7chiral dual pair
S7:{7,7}
K9S10S10:{8,8}C32⋊C8chiral, self-dual
K10(none)
K11S12S12:{5,10}C11⋊C10two different, chiral
S12:{5,10}
K12(none)
K13S27S27:{12,12}C13⋊C12chiral dual pair
S27:{12,12}
K14, K15(none)
K16S45S45:{15,15}C24⋊C15chiral dual pair
S45:{15,15}
K17S52S52:{16,16}C17⋊C16two chiral dual pairs
S52:{16,16}
S52:{16,16}
S52:{16,16}
K18(none)
K19S58S58:{9,18}C19⋊C18three different, chiral
S58:{9,18}
S58:{9,18}
K20-K22(none)
K23S93S93:{11,22}C23⋊C22five 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.

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)

Index to Regular Maps
Glossary for Regular Maps