The following table lists the currently known values for the number of vertices in a cubic cage. For certain small values of g the cages themselves are all known, and are given here explicitly. For larger values of g a range is given - the lower value is either the trivial Moore bound n(3,g) or a bound that has been increased by extensive computation, while the upper bound is simply the size of the smallest known cubic graph of that girth, which is given explicitly.

Under the column labeled *Number *is listed the number of graphs
known that meet the upper bound. Numbers not known to be exact are followed
by a + symbol (so 1+ means that one example is known, but there may be
more).

Girth | Moore Bound |
Range | Number | Reference |
---|---|---|---|---|

3 | 4 | 4 | 1 | K_4 |

4 | 6 | 6 | 1 | K_3,3 |

5 | 10 | 10 | 1 | Petersen |

6 | 14 | 14 | 1 | Heawood |

7 | 22 | 24 | 1 | McGee graph |

8 | 30 | 30 | 1 | Tutte's 8-cage |

9 | 46 | 58 | 18 | Brinkmann/McKay/Saager |

10 | 62 | 70 | 3 | O'Keefe/Wong |

11 | 94 | 112 | 1+ | McKay/Myrvold |

12 | 126 | 126 | 1 | Generalized hexagon |

13 | 190 | 202-272 | 1+ | McKay/Myrvold - Hoare |

14 | 254 | 258-406 | 1+ | McKay - Biggs |

15 | 382 | 384-620 | 1+ | Biggs |

16 | 510 | 512-990 | 1+ | Biggs |

17 | 766 | 768-2530 | 1+ | Parker/Rowley
Omega-cover |

18 | 1022 | 1024-3024 | 2+ | Bray/Parker/Rowley Cayley Graph |

19 | 1534 | 1536-4324 | 1+ | H(47) |

20 | 2046 | 2048-6072 | 1+ | Parker/Rowley
Subgroup Chain Graph in PSL(2,23) |

21 | 3070 | 3072-16112 | 1+ | Biggs/Whitehead |

22 | 4094 | 4096-16206 | 1+ | Whitehead S(73) |

23 | 6142 | 6144-44100 | 1+ | Parker/Rowley G_2(3) excised |

24 | 8190 | 8192-44226 | 1+ | Parker/Rowley G_2(3) |

25 | 12286 | 12288-109010 | 1+ | Bray/Parker/Rowley B-type graph |

26 | 16382 | 16384-109200 | 1+ | Bray/Parker/Rowley B-type graph |

27 | 24574 | 24576-285852 | 1+ | Bray/Parker/Rowley B-type graph |

28 | 32766 | 32768-415104 | 1+ | Bray/Parker/Rowley B-type graph |

29 | 49150 | 49152-1143408 | 1+ | Bray/Parker/Rowley B-type graph |

30 | 65534 | 65536-1227666 | 1+ | Hoare S(313) |

31 | 98302 | 98304-3649794 | 1+ | Bray/Parker/Rowley B-type graph |

32 | 131070 | 131072-3650304 | 1+ | Bray/Parker/Rowley B-type graph |