In order to obtain perfect fullerenes, we start from any 5/6 fullerene given in geometrical form (i.e., by a 3-valent graph). Then we use mid-edge-truncation and vertex bifurcation in all vertices of the triangular faces obtained, transforming them into hexagons with alternating digonal edges. Let some fullerene (e.g., C20) be given in its geometrical form. By connecting the midpoints of all adjacent edges we obtain the 3/5 fullerene covered by connected triangular net and pentagonal faces preserved from C20. After that,we introduce digons in all vertices of the truncated polyhedron, to turn all triangles into hexagonal faces. This way, C60 can be derived (in its chemical form) from C20.
Mid-edge-truncation can be applied to any 5/6 (geometrical) fullerene, giving a new perfect (chemical) fullerene formed by carbon rings. Similarly, from a 5/6 fullerene with v vertices we can always derive new perfect 5/6 fullerenes with 3v vertices. Moreover, symmetry of the generating fullerene is preserved. According to the theorem by Grünbaum and Motzkin (1963), for every non-negative n6 ¹1, there exists a 3-valent convex 5/6 polyhedron having n5 = 12 pentagonal and n6 hexagonal faces. Hence, from the infinite class of 3-valent 5/6 polyhedra with v = 20+n6 vertices, we obtain the infinite class of perfect fullerenes with v = 60+3n6 vertices. Perfect fullerenes satisfy two important chemical conditions:
The IPR rule means that there are no adjacent pentagons, and HPR means that all pentagons are "holes", i.e., that every pentagon may have only external double bonds. The first 5/6 fullerene satisfying IPR is C60, and it also satisfies HPR. The IPR is well known as the stability criterion: all fullerenes of lower order (less than 60) are unstable, because they don't satisfy IPR. On the other hand, C70 satisfies IPR, but cannot satisfy HPR.
The same holds for C80, which has the same icosahedral geometrical symmetry as C60, but since HPR cannot be satisfied, its symmetry will be reduced due to edge-coloring. Therefore we conclude that only perfect fullerenes with G = G' = [3,5] = Ih = S5, satisfying both IPR and HPR, are C60, C180, C240, etc. We need also to notice that for n6 = 0,2,3 there are always exactly one 3-valent 5/6 polyhedron (i.e., the geometrical form of C20, C24, C26), but for some larger values (e.g. n6 = 4,5,7,9) there are several geometrical isomers of generating fullerenes, and consequently, the same number of derived perfect fullerenes. Hence, considering fullerene isomers, we can distinguish geometrical isomers, that is, different geometrical forms of some fullerene treated as 3-valent 5/6 polyhedra, and chemical isomers- different arrangements of double bonds, obtained from the same 3-valent graph by its edge-coloring.