Cubic Graphs of Large Girth


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