From Combinatorics Wiki
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
|