Cubic Transitive Graphs

Lists of cubic vertex-transitive graphs have been prepared by Brendan McKay and myself for as many orders up to and including 256 vertices as we could manage.

The graphs were constructed in a variety of different ways depending on the prime factorization of the number of vertices n. Essentially a large prime in the prime factorization helps things a lot. If the number is such that all the groups are known then all the Cayley graphs can be calculated, and hence the number of non-Cayley graphs found. Unfortunately for some of the values above 100 the groups are not readily accessible so that it is easier to actually find all the graphs than to tell whether they are Cayley graphs.

The table

Each entry in the table is a number that is linked to a file containing the named graphs in graph6 format.

The columns labelled Cayley and non-Cayley include the symmetric graphs.

#Vertices Prime Factorization Symmetric Cayley Non-Cayley Total
4 2, 2 1 1 0 1
6 2, 3 1 2 0 2
8 2, 2, 2 1 2 0 2
10 2, 5 1 2 1 3
12 2, 2, 3 0 4 0 4
14 2, 7 1 3 0 3
16 2, 2, 2, 2 1 4 0 4
18 2, 3, 3 1 5 0 5
20 2, 2, 5 2 5 2 7
22 2, 11 0 3 0 3
24 2, 2, 2, 3 1 11 0 11
26 2, 13 1 4 1 5
28 2, 2, 7 1 5 1 6
30 2, 3, 5 1 7 3 10
32 2, 2, 2, 2, 2 1 10 0 10
34 2, 17 0 4 1 5
36 2, 2, 3, 3 0 12 0 12
38 2, 19 1 5 0 5
40 2, 2, 2, 5 1 10 2 12
42 2, 3, 7 1 10 0 10
44 2, 2, 11 0 7 0 7
46 2, 23 0 5 0 5
48 2, 2, 2, 2, 3 1 32 0 32
50 2, 5, 5 1 8 1 9
52 2, 2, 13 0 9 1 10
54 2, 3, 3, 3 1 13 0 13
56 2, 2, 2, 7 3 13 3 16
58 2, 29 0 6 1 7
60 2, 2, 3, 5 1 30 8 38
62 2, 31 1 7 0 7
64 2, 2, 2, 2, 2, 2 1 26 0 26
66 2, 3, 11 0 11 0 11
68 2, 2, 17 0 11 1 12
70 2, 5, 7 0 11 0 11
72 2, 2, 2, 3, 3 1 36 1 37
74 2, 37 1 8 1 9
76 2, 2, 19 0 11 0 11
78 2, 3, 13 1 14 0 14
80 2, 2, 2, 2, 5 1 29 4 33
82 2, 41 0 8 1 9
84 2, 2, 3, 7 1 27 3 30
86 2, 43 1 9 0 9
88 2, 2, 2, 11 0 16 1 17
90 2, 3, 3, 5 1 18 3 21
92 2, 2, 23 0 13 0 13
94 2, 47 0 9 0 9
96 2, 2, 2, 2, 2, 3 2 90 ? ?
98 2, 7, 7 2 13 0 13
100 2, 2, 5, 5 0 23 2 25
102 2, 3, 17 1 15 1 16
104 2, 2, 2, 13 1 ? ? 22
106 2, 53 0 10 1 11
108 2, 2, 3, 3, 3 1 ? ? 42
110 2, 5, 11 1 19 0 19
112 2, 2, 2, 2, 7 3 ? ? 38
114 2, 3, 19 1 18 0 18
116 2, 2, 29 0 17 1 18
118 2, 59 0 11 0 11
120 2, 2, 2, 3, 5 2 ? ? ?
122 2, 61 1 12 1 13
124 2, 2, 31 0 17 0 17
126 2, 3, 3, 7 1 ? ? 26
128 2, 2, 2, 2, 2, 2, 2 2 82 1 83
130 2, 5, 13 0 17 2 19
132 2, 2, 3, 11 0 ? ? 35
134 2, 67 1 13 0 13
136 2, 2, 2, 17 0 ? ? 28
138 2, 3, 23 0 19 0 19
140 2, 2, 5, 7 0 ? ? ?
142 2, 71 0 13 0 13
144 2, 2, 2, 2, 3, 3 2 ? ? ?
146 2, 73 1 14 1 15
148 2, 2, 37 0 21 1 22
150 2, 3, 5, 5 1 ? ? 28
152 2, 2, 2, 19 1 ? ? 27
154 2, 7, 11 0 19 0 19
156 2, 2, 3, 13 0 ? ? 46
158 2, 79 1 15 0 15
160 2, 2, 2, 2, 2, 5 0 ? ? ?
162 2, 3, 3, 3, 3 3 ? ? ?
164 2, 2, 41 0 23 1 24
166 2, 83 0 15 0 15
168 2, 2, 2, 3, 7 6 ? ? ?
170 2, 5, 17 0 21 2 23
172 2, 2, 43 0 23 0 23
174 2, 3, 29 0 23 0 23
176 2, 2, 2, 2, 11 0 ? ? ?
178 2, 89 0 16 1 17
180 2, 2, 3, 3, 5 0 ? ? ?
182 2, 7, 13 4 23 2 25
184 2, 2, 2, 23 0 ? ? 31
186 2, 3, 31 1 26 0 26
188 2, 2, 47 0 25 0 25
190 2, 5, 19 0 23 0 23
192 2, 2, 2, 2, 2, 2, 3 3 ? ? ?
194 2, 97 1 18 1 19
196 2, 2, 7, 7 0 ? ? 35
198 2, 3, 3, 11 0 ? ? ?
200 2, 2, 2, 5, 5 1 ? ? 61
202 2, 101 0 18 1 19
204 2, 2, 3, 17 1 ? ? 53
206 2, 103 1 19 0 19
208 2, 2, 2, 2, 13 1 ? ? ?
210 2, 3, 5, 7 0 43 ? ?
212 2, 2, 53 0 29 ? 30
214 2, 107 0 19 0 19
216 2, 2, 2, 3, 3, 3 3 ? ? ?
218 2, 109 1 20 1 21
220 2, 2, 5, 11 3 ? ? ?
222 2, 3, 37 1 30 0 30
224 2, 2, 2, 2, 2, 7 3 ? ? ?
226 2, 113 0 20 1 21
228 2, 2, 3, 19 0 ? ? 57
230 2, 5, 23 0 27 0 27
232 2, 2, 2, 29 0 ? ? 40
234 2, 3, 3, 13 2 ? ? ?
236 2, 2, 59 0 31 0 31
238 2, 7, 17 0 27 ? ?
240 2, 2, 2, 2, 3, 5 3 ? ? ?
242 2, 11, 11 1 25 0 25
244 2, 2, 61 0 33 1 34
246 2, 3, 41 0 31 0 31
248 2, 2, 2, 31 1 ? ? 41
250 2, 5, 5, 5 1 ? ? 37
252 2, 2, 3, 3, 7 0 ? ? ?
254 2, 127 1 23 0 23
256 2, 2, 2, 2, 2, 2, 2, 2 4 ? ? 284
258 2, 3, 43 1 34 0 34


[Home] Gordon Royle, gordon@cs.uwa.edu.au, September 1996