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.
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.
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