PentHex puzzles. A reliable and efficient top-down approach to fullerene-structure enumeration (Q1271874)

The authors give an algorithmic solution to the problem of constructing fullerene-structures up to isomorphism. The approach relies on Petrie paths which are zig-zag paths along the network of edges provided by a fullerene and guarantees complete lists of fullerenes for up to 162 vertices and lists of fullerenes with non-adjacent pentagons for up to 210 vertices. The paper contains a table with the numbers of isomorphism types of both kinds of fullerenes. For details on the implementation and its isomorphism testing the authors refer to another description. A Hamiltonian cycle checker by Brendan McKay checked that for up to 150 vertices all these structures are Hamiltonian. This method is a big step forward compared to earlier approaches which appeared mostly in the literature of computational chemistry.