Trivalent cages
Table 1: known trivalent cages
| g |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12
|
| n(3,g) |
10 |
14 |
24 |
30 |
58 |
70 |
112 |
126
|
| number of cages |
1 |
1 |
1 |
1 |
18 |
3 |
1 |
1
|
Table 2: bounds for trivalent cages
Optimal graphs are marked in bold
| Girth g |
Lower bound |
Upper bound |
Number of cages |
Author or description
|
| 5 |
10 |
10 |
1 |
Petersen
|
| 6 |
14 |
14 |
1 |
Heawood
|
| 7 |
24 |
24 |
1 |
McGee
|
| 8 |
30 |
30 |
1 |
Tutte
|
| 9 |
58 |
58 |
18 |
Brinkmann-McKay-Saager
|
| 10 |
70 |
70 |
3 |
O’Keefe-Wong
|
| 11 |
112 |
112 |
1 |
McKay-Myrvold; Balaban
|
| 12 |
126 |
126 |
1 |
Benson
|
| 13 |
202 |
272 |
|
McKay-Myrvold; Hoare
|
| 14 |
258 |
384 |
|
McKay; Exoo
|
| 15 |
384 |
620 |
|
Biggs
|
| 16 |
512 |
960 |
|
Exoo
|
| 17 |
768 |
2176 |
|
Exoo
|
| 18 |
1024 |
2560 |
|
Exoo
|
| 19 |
1536 |
4324 |
|
Hoare, H(47)
|
| 20 |
2048 |
5376 |
|
Exoo
|
| 21 |
3072 |
16028 |
|
Exoo
|
| 22 |
4096 |
16206 |
|
Biggs-Hoare, S(73)
|
| 23 |
6144 |
49326 |
|
Exoo
|
| 24 |
8192 |
49608 |
|
Bray-Parker-Rowley
|
| 25 |
12288 |
108906 |
|
Exoo
|
| 26 |
16384 |
109200 |
|
Bray-Parker-Rowley
|
| 27 |
24576 |
285852 |
|
Bray-Parker-Rowley
|
| 28 |
32768 |
415104 |
|
Bray-Parker-Rowley
|
| 29 |
49152 |
1143408 |
|
Bray-Parker-Rowley
|
| 30 |
65536 |
1227666 |
|
Biggs-Hoare, S(313)
|
| 31 |
98304 |
3649794 |
|
Bray-Parker-Rowley
|
| 32 |
131072 |
3650304 |
|
Bray-Parker-Rowley
|
Cages of girth 5 and 6
Optimal graphs are marked in bold.
Table 1: known cages of Girth 5
| k |
3 |
4 |
5 |
6 |
7
|
| n(k,5) |
10 |
19 |
30 |
40 |
50
|
| number of cages |
1 |
1 |
4 |
1 |
1
|
Table 2: bounds for cages of girth 5
| Degree k |
Lower bound |
Upper bound |
Author or description
|
| 3 |
10 |
10 |
Petersen
|
| 4 |
19 |
19 |
Robertson
|
| 5 |
30 |
30 |
Robertson-Wegner-Wong
|
| 6 |
40 |
40 |
Wong
|
| 7 |
50 |
50 |
Hoffman-Singleton
|
| 8 |
67 |
80 |
Royle
|
| 9 |
86 |
96 |
Jørgensen
|
| 10 |
103 |
126 |
Exoo
|
| 11 |
124 |
156 |
Jørgensen
|
| 12 |
147 |
203 |
Exoo
|
| 13 |
174 |
240 |
Exoo
|
| 14 |
199 |
288 |
Jørgensen
|
| 15 |
230 |
312 |
Jørgensen
|
| 16 |
259 |
336 |
Jørgensen
|
| 17 |
294 |
448 |
Schwenk
|
| 18 |
327 |
480 |
Schwenk
|
| 19 |
364 |
512 |
Schwenk
|
| 20 |
403 |
576 |
Jørgensen
|
Table 3: bounds for cages of girth 6
| Degree k |
Lower bound |
Upper bound |
Author or description
|
| 3 |
14 |
14 |
Projective Plane
|
| 4 |
26 |
26 |
Projective Plane
|
| 5 |
42 |
42 |
Projective Plane
|
| 6 |
62 |
62 |
Projective Plane
|
| 7 |
90 |
90 |
O’Keefe-Wong
|
| 8 |
114 |
114 |
Projective Plane
|
| 9 |
146 |
146 |
Projective Plane
|
| 10 |
182 |
182 |
Projective Plane
|
| 11 |
224 |
240 |
Wong
|
| 12 |
266 |
266 |
Projective Plane
|
| 13 |
314 |
336 |
Abreu-Funk-Labbate-Napolitano
|
| 14 |
366 |
366 |
Projective Plane
|
| 15 |
422 |
462 |
Abreu-Funk-Labbate-Napolitano
|
| 16 |
482 |
504 |
Abreu-Funk-Labbate-Napolitano
|
| 17 |
546 |
546 |
Projective Plane
|
| 18 |
614 |
614 |
Projective Plane
|
| 19 |
686 |
720 |
Abreu-Funk-Labbate-Napolitano
|
| 20 |
762 |
762 |
Projective Plane
|