# Graphs and Regular Maps

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.

Graphm1st regular map2nd regular map
1-cycle1the hemi-2-hosohedron the dimonogon
2-cyclethe hemi-di-square the 2-hosohedron
3-cyclethe hemi-di-hexagon the di-triangle
4-cyclethe hemi-di-octagon the di-square
4-cube{4,4}(4,0) self-Petrie dual
5-cyclethe hemi-di-decagon the di-pentagon
6-cyclethe hemi-di-dodecagon the di-hexagon
7-cyclethe hemi-di-14gon the di-heptagon
Clebsch graphS5:{5,5} C6:{4,5}
cubethe cube {6,3}(2,2)
Desargues graphC6:{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
K1the edgeless map self-Petrie dual
K2the monodigon self-Petrie dual
K3the di-triangle the hemi-di-hexagon
K4the hemicube the tetrahedron
K5{4,4}(2,1) C5:{10,4}
K5 × K2{4,4}(3,1) S4:{10,4}a
K6the hemi-icosahedron C5:{5,5}
K7{3,6}(1,3) (a non-regular map)
K8{7,7}4 and {7,7}4 (two non-regular maps)
K3,3{6,3}(0,2) self-Petrie dual
K4,4{4,4}(2,2) S3:{8,4|4}
K2,2,2 C4:{6,4}3 the octahedron
K3,3,3 {3,6}(3,3) the regular map with C&D number N11.2
K4,4,4 S3:{3,8} the regular map with C&D number N22.4
Klein graphthe 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 graphS2:{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 graphthe hemidodecahedron self-Petrie dual
1-cycle2the hemi-4-hosohedron {4,4}(1,0)
4-cycle{4,4}(2,0) S2:{8,4}
6-cycleS2:{6,4}S3:{12,4}
C4:{6,4}6 self-Petrie dual
8-cycleS3:{8,4|2} self-Petrie dual
cubeS3:{4,6} S5:{6,6}
K4 S3:{6,6} C4:{4,6}6
{3,6}(2,2) C4:{4,6}3
K6 C6:{3,10}5 the regular map with C&D number N14.3
K2,2,2 S2:{3,8} the regular map with C&D number N16.7p
K3,3C5:{4,6} self-Petrie dual
1-cycle3the hemi-6-hosohedron {3,6}(1,1)
3-cycleC5:{6,6} {3,6}(0,2)
4-cycleS2:{4,6} S4:{12,6}
K2the 3-hosohedron {6,3}(1,1)
1-cycle4the 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
K2 the 4-hosohedron {4,4}(1,1)
K4 S3:{3,12} the regular map with C&D number N16.7
1-cycle5the hemi-10-hosohedron S2:{5,10}
K2the 5-hosohedron S2:{10,5}
1-cycle6the hemi-12-hosohedron S3{12,12}
K2the 6-hosohedron S2:{6,6}
1-cycle7the hemi-14-hosohedron S3{7,14}
K2the 7-hosohedron S3:{14,7}
K28 the 8-hosohedron S3:{8,8}2
K2 S2:{4,8} S3:{8,8}4
K29the 9-hosohedron S4:{18,9}
K210the 10-hosohedron S4:{10,10}
K211the 11-hosohedron S5:{22,11}
K212 the 12-hosohedron S5:{12,12}
K212 S3:{4,12} S4:{6,12}
K213the 13-hosohedron S6:{26,13}
K214the 14-hosohedron S6:{14,14}