Projective Maps Last update=7 June, 2018 |
The table following is a list of projective plane embeddings of some small graphs, most of which are vertex transitive. Only 2-cell embeddings are considered. Most vertex transitive graphs are not projective planar. Except for the cycles C_{n}, which are always projective, vertex transitive graphs with at most 16 vertices which do not appear in the table are not projective planar. The naming of the graphs is described here. The embeddings were generated by software. n, ε, f mean the numbers of vertices, edges, and faces.
All projective embeddings are non-orientable, because the
projective plane is non-orientable. The order of the graph's automorphism group is
in square brackets, eg. [24].
The embeddings have subgroups as automorphism groups. Their orders are divisors
of the group order, and are listed after the group order. Download a text file containing these embeddings. [The graph named C_{3}%C_{4} is a kind of "Möbius product" of C_{3} and C_{4}. Its projective embedding looks nearly identical to the torus embedding of C_{3}xC_{4}. It can also be derived as the antipodal graph of trunc(K_{4}).] Click for a list of obstructions for the projective plane. |
graph | n | ε | f | emb. | groups | duals |
---|---|---|---|---|---|---|
K_{4} | 4 | 6 | 3 | 1 | [24] 24^{1} | |
K_{5} | 5 | 10 | 6 | 2 | [120] 10^{1},8^{1} | Wheel_{5} |
K_{3,3} | 6 | 9 | 4 | 1 | [72] 12^{1} | |
3-Prism | 6 | 9 | 4 | 3 | [12] 6^{1},4^{1},2^{1} | |
Octahedron | 6 | 12 | 7 | 4 | [48]24^{1},6^{1},4^{2} | K_{3,4} |
K_{6} | 6 | 15 | 10 | 1 | [720] 60^{1} | Petersen |
K_{3,4} | 7 | 12 | 6 | 1 | [144] 24^{1} | Octahedron |
~C_{7}=C_{7}(2) | 7 | 14 | 8 | 3 | [14] 14^{1},2^{2} | Wheel_{7} |
Q_{3} (=Cube) | 8 | 12 | 5 | 1 | [48] 4^{1} | |
C_{8}^{+} | 8 | 12 | 5 | 2 | [16] 16^{1},4^{1} | |
C_{8}(2)=~C_{8}^{+} | 8 | 16 | 9 | 5 | [16] 2^{5} | |
C_{9}(2) | 9 | 18 | 10 | 3 | [18] 18^{1}, 2^{2} | Wheel_{9} |
K_{3}x K_{3}=Paley | 9 | 18 | 10 | 1 | [72] 12^{1} | |
Petersen | 10 | 15 | 6 | 1 | [120] 60^{1} | K_{6} |
C_{10}^{+} | 10 | 15 | 6 | 2 | [20] 20^{1}, 4^{1} | |
5-Prism | 10 | 15 | 6 | 2 | [20] 4^{1}, 2^{1} | |
C_{10}(2) | 10 | 20 | 11 | 5 | [20] 2^{5} | |
C_{11}(2) | 11 | 22 | 12 | 3 | [22] 22^{1}, 2^{2} | Wheel_{11} |
C_{12}(5^{+}) | 12 | 18 | 7 | 1 | [48] 24^{1} | ~(K_{3}∪(4K_{1})) |
6-Prism | 12 | 18 | 7 | 2 | [24] 4^{1},2^{1} | |
C_{12}^{+} | 12 | 18 | 7 | 2 | [24] 24^{1},4^{1} | |
trunc(K_{4}) | 12 | 18 | 7 | 4 | [24] 24^{1},6^{1},4^{1},2^{1} | |
C_{12}(2) | 12 | 24 | 13 | 5 | [24] 2^{5} | |
L(Cube) | 12 | 24 | 13 | 4 | [48] 6^{1}, 4^{2}, 2^{1} | |
C_{3}%C_{4} | 12 | 24 | 13 | 1 | [24] 24^{1} | |
Icosahedron | 12 | 30 | 19 | 3 | [120] 6^{1},4^{1}, 2^{1} | |
C_{13}(2) | 13 | 26 | 14 | 3 | [26] 26^{1}, 2^{2} | Wheel_{13} |
7-Prism | 14 | 21 | 8 | 2 | [28] 4^{1},2^{1} | |
C_{14}^{+} | 14 | 21 | 8 | 2 | [28] 28^{1},4^{1} | |
C_{14}(2) | 14 | 28 | 15 | 5 | [28] 2^{5} | |
C_{15}(2) | 15 | 30 | 16 | 3 | [30] 30^{1}, 2^{2} | Wheel_{15} |
L(Petersen) | 15 | 30 | 16 | 1 | [120] 60^{1} | |
8-Prism | 16 | 24 | 9 | 2 | [32] 4^{1},2^{1} | |
C_{16}^{+} | 16 | 24 | 9 | 2 | [32] 32^{1},4^{1} | |
C_{16}(2) | 16 | 32 | 17 | 5 | [32] 2^{5} |