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Glenn Tesler,
The Kekulé Covalent Bond Arrangements of C60,
Advances in Applied Mathematics 14 (1993), 217-225.

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We present an algorithm that enumerates all 12,500 arrangements of double bonds on C60 in which each carbon atom forms a single bond with two of its neighbors and a double bond with the third. We then present statistics concerning the symmetry groups of the arrangements and the number of double bonds along pentagonal faces. We exhibit a generating function for the allowed arrangements whose factorization suggests there should be a direct combinatorial solution to the enumeration. We conclude by exhibiting a complete list of double bond arrangements, up to symmetry.