Torus Maps

Last update=6 June, 2018

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A table of torus embeddings of some small graphs appears in
A. Gagarin, W. Kocay, and D. Neilson, "Embeddings of Small Graphs on the Torus" (2001), (260K) CUBO 5(2003), pp. 251-171.

The software that generated the original embeddings missed several embeddings. The condition that controls the loop was incorrect, causing the loop to sometimes stop too soon. The corrected numbers of embeddings are shown in the tables following. The graphs with corrected entries are marked with [*] in the last column.

The naming of the graphs is described here.
n, ε, f mean the numbers of vertices, edges, and faces.
Only 2-cell embeddings are considered.
Some embeddings are orientable and some are non-orientable.
An embedding is orientable if flipping it over results in a non-isomorphic embedding. Otherwise it is non-orientable. Orientability is indicated in the graph name, eg., [12or] following the name means an orientable embedding with a group of order 12. The non-orientable embeddings have no special indication. Vertex transitive graphs with at most 12 vertices which do not appear in the table are not embeddable on the torus.

A group order like [24] in the table means that the graph has an automorphism group of order 24. The embeddings have subgroups as automorphism groups. Their orders are divisors of the group order, and are listed after the group order. Δ means a triangulation of the torus. Some diagrams follow the tables.

Individual files of distinct embeddings can be downloaded by clicking on the hilighted graph names. The unhilighted files are not yet available.

Download a text file containing all the embeddings available.

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graphnεfemb.or. non.groupsduals
K4462202[24] 41,31
K551056 24[120] 201,41,23,11self(1)
K3,36932 02[72] 181,21
3-Prism6935 03[12] 61,22,12
Octahedron612617 413[48] 121,61,43,31,26,15self(1)
K661594 22[720] 62,21,11
K3,471253 03[144] 41,31,21
~C7=C7(2)714728 235[14] 141,214,113self(1) [*]
K7   [Δ]721141 10[5040] 421Heawood
K3,581571 01[720] 31
Cube=C4xK281245 05[48] 241,82,31,21Dbl(K4)
C8+81245 14[16] 24,11 [*]
K4,481682 02[1152] 321,161self(2)
~C8+=C8(2)816837 2017[16] 161,44,215,117self(1) [*]
~Cube81688 44[48] 42,25,11
~C88201210 100[16] 27,13 [*]
~(2C4)820126 24[128] 82,42,22 [*]
~(4K2)   [Δ]824161 01[384] 161DblCvr(Q3)
K3,691891 01[4320] 181 Paley(9)
C9(2)9189 37 343[18] 181,61,218,117self(1) [*]
C9(3)91896 42[18] 181,31, 23,11self(1) [*]
K3x K3=Paley91897 34[72] 361,181,41,23,11K3,6, self(1)
~(3K3)   [Δ]927181 01[1296] 541Pappus
~C9   [Δ]927181 10[18] 181(9,3) config
Petersen101551 01[120] 31
C10+101556 15[20] 101, 24, 11
C5x K2=5-Prism 101555 05[20] 23, 12
C10(2)102010 59 4118[20] 43, 223, 133 [*]
C10(4)1020101 01[20] 201self(1)
~(K5x K2)1020101 10[240] 401self(1)
~C10(2)1025151 01[20] 101
~C10(4)1025154 40[20] 101,23
~(C5x K2)   [Δ]1030201 01[20] 201(10,3) config
C11(2)112211 77 743[22] 221, 236, 140self(1) [*]
C11(3)1122111 10[22] 221self(1)
~C11(3)   [Δ]1133221 10[22] 221(11,3) config
C12(5+)121863 12[48] 121, 61, 21 [*]
C6x K2=6-Prism121869 09[24] 121,42,24,12
C12+121867 16[24] 61,24,12
trunc(K4)121869 09[24] 41,31,22,15
C12(3+,6)122412 1 0 1[48] 241self(1)
C12(2)122412 137 109 28[24] 81, 62, 44, 245, 185 [*]
C12(3)122412 1 1 0[24] 241self(1)
C12(4)122412 2 1 1[24] 241, 41self(1)
C12(5)122412 2 0 2[768] 242self(2) [*]
C12(5+,6)122412 10 10 0[48] 62, 26,12
L(Cube)122412 16 1 15[48] 241, 82, 41, 31, 23, 18 [*]
C4x C3122412 3 0 3[48] 241, 41, 21self(1) [*]
antip(trunc(K4))122412 1 0 1[24] 31
Icosahedron123018 12 5 7[120] 31,24, 17
Octahedron x K2123018 1 0 1[96] 121
C12(5,6)123018 8 5 3[768] 121, 61, 41, 24, 11
C12(2,5+)123018 1 1 0[12] 121
C12(4,5+)123018 1 1 0[12] 121
C12(2,3)   [Δ]123624 1 1 0[24] 241(12,3) config
C12(2,5)   [Δ]123624 1 0 1[144] 721(12,3) config
C12(3,4)   [Δ]123624 1 1 0[24] 241(12,3) config
C12(4,5)   [Δ]123624 1 0 1[48] 241(12,3) config
Q4=C4xC4163216 1 0 1[384] 641self(1)

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Petersen C4xC3
C4xC3 C4xC3
C5xK2 C5xK2
C5xK2 C5xK2
C5xK2 C5xK2
C5xK2 C5xK2

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