C60 and extended C168 are unique because neighbored nonhexagons in them are separated by exactly one carbon-carbon bond. Is there any other graphitic structure with the similar property? The answer is yes. Chern and I have written an article on the carbon nanotori and nanohelices with this property a few years ago.
Chuang, C; Jin, B.-Y.* “Hypothetical Toroidal, Cylindrical, Helical Analogs of C60.” J. Mol. Graph. Model. 2009, 28, 220-225.
Of course, it is easy to see that there are another four Archimedean solids with this property if we allow nonhexagons to be squares or triangles. They are truncated octahedron (see the following photo), truncated cube, truncated tetrahedron, and truncated dodecahedron. Note that C60 is the truncated icosahedron. So all five truncated Platonic solids belong to this class.