An operation on trivalent graphs leads from the truncated cube to buckminsterfullerene, and C60 is the only fullerene with disjoint pentagons which can be obtained by this method. The construction and the proof emphasize maximal independent sets that contain two fifths of the vertices of trivalent graphs. In the case of C60, these sets define the structure of the experimentally obtained bromofullerene C60_Br24 and presumably also the fullerol C60_(OH)_24. These special independent sets seem to be related to the Golay code, and the fullerol is studied in oncology. The construction and characterization of the icosahedral C60 is a result of work on conjectures of Graffiti.
DIMACS Series in Discrete Mathematics and Theoretical Computer Science, to appear in Volume