Symmetries of (3, 6)-Fullerenes

A (3, 6)-fullerene is a cubic planar graph whose faces all have 3 or 6 sides.

We give an exact enumeration of (3, 6)-fullerenes with V vertices and each of five possible symmetry types: *332, 332, 2*2, *222, and 222, in orbifold notation.

We use this enumeration, together with some constructions, to resolve three conjectures about the existence of (3, 6)-fullerenes with 2*2$, *222, and 222 symmetry for given numbers of vertices.