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 Cn, 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.
Some diagrams follow the tables.

Download a text file containing these embeddings.

[The graph named C3%C4 is a kind of "Möbius product" of C3 and C4. Its projective embedding looks nearly identical to the torus embedding of C3xC4. It can also be derived as the antipodal graph of trunc(K4).]

Click for a list of obstructions for the projective plane.


graphnεfemb.groups    duals
K44631[24] 241
K551062[120] 101,81Wheel5
K3,36941[72] 121
3-Prism6943[12] 61,41,21
Octahedron61274[48]241,61,42 K3,4
K6615101[720] 601 Petersen
K3,471261[144] 241Octahedron
~C7=C7(2)71483[14] 141,22Wheel7
Q3 (=Cube)81251[48] 41
C8+81252[16] 161,41
C8(2)=~C8+81695[16] 25
C9(2)91810 3[18] 181, 22Wheel9
K3x K3=Paley918101[72] 121
Petersen101561[120] 601 K6
C10+101562[20] 201, 41
5-Prism101562[20] 41, 21
C10(2)102011 5[20] 25
C11(2)112212 3[22] 221, 22Wheel11
C12(5+)121871[48] 241~(K3∪(4K1))
6-Prism121872[24] 41,21
C12+121872[24] 241,41
trunc(K4)121874[24] 241,61,41,21
C12(2)122413 5[24] 25
L(Cube)122413 4[48] 61, 42, 21
C3%C4122413 1[24] 241
Icosahedron123019 3[120] 61,41, 21
C13(2)132614 3[26] 261, 22Wheel13
7-Prism142182[28] 41,21
C14+142182[28] 281,41
C14(2)142815 5[28] 25
C15(2)153016 3[30] 301, 22Wheel15
L(Petersen)1530161[120] 601
8-Prism162492[32] 41,21
C16+162492[32] 321,41
C16(2)1632175[32] 25


Cube K6
Octahedron K(3,4)
~C8+ paley
Icosahedron C12(5+)
L(Cube)1 L(Cube)2
C3%C4 C13(2)
Petersen L(Petersen)
C15(2) C4xC3


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