A fullerene is a cubic polyhedral graph having all faces 5- or 6-cycles. Examples include the 20-vertex dodecahedral
graph, 24-vertex generalized Petersen
graph 
,
graph on 26 vertices given by Gosil and Royle (2001, p. 208), truncated
icosahedral graph, and stable molecule 
(Babić et al. 2002), illustrated above.
Every fullerene has exactly twelve 5-cycles. The complement of a fullerene on 
vertices is 
-regular, and it has precisely 12 odd chordless cycles,
all of them of order 5.
The numbers of fullerenes on 
,
22, 24, ... vertices (counting enantiomers as equivalent) are given by 1, 0, 1, 1,
2, 3, 6, 6, 15, 17, 40, 45, 89, ... (OEIS A007894).
Brinkmann and McKay have written programs for the enumeration and generation of fullerenes.
Canonical polyhedra corresponding to fullerenes on 20 to 34 vertices are illustrated above.
Fullerenes of type I (isomorphic to the skeletons of 
-Goldberg
polyhedra) and type II (isomorphic to the skeletons
of 
-Goldberg
polyhedra) are implemented in the Wolfram
Language as BuckyballGraph[n,
"I"] and BuckyballGraph[n,
"II"], respectively.
While almost all small fullerenes have fractional chromatic number 5/2, those listed in the following table (indexed according to Brinkmann and McKay) do not.
| fullerene |  |
| (24, 1) | 8/3 |
| (28, 1) | 68/27 |
| (28, 2) | 28/11 |
| (30, 2) | 28/11 |
