Difference between revisions of "Tables and Results"
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Revision as of 12:14, 16 September 2019
Contents
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 | 1141484 | Exoo-Jajcay | |
| 30 | 65536 | 1143408 | Exoo-Jajcay | |
| 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 |