In order to obtain perfect fullerenes, we start from any 5/6 fullerene given in geometrical form (i.e., by a 3valent graph). Then we use midedgetruncation and vertex bifurcation in all vertices of the triangular faces obtained, transforming them into hexagons with alternating digonal edges. Let some fullerene (e.g., C_{20}) 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 C_{20}. After that,we introduce digons in all vertices of the truncated polyhedron, to turn all triangles into hexagonal faces. This way, C_{60} can be derived (in its chemical form) from C_{20}. Midedgetruncation 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 nonnegative n_{6} ¹1, there exists a 3valent convex 5/6 polyhedron having n_{5} = 12 pentagonal and n_{6} hexagonal faces. Hence, from the infinite class of 3valent 5/6 polyhedra with v = 20+n_{6} vertices, we obtain the infinite class of perfect fullerenes with v = 60+3n_{6} 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 C_{60}, 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, C_{70} satisfies IPR, but cannot satisfy HPR. The same holds for C_{80}, which has the same icosahedral geometrical symmetry as C_{60}, but since HPR cannot be satisfied, its symmetry will be reduced due to edgecoloring. Therefore we conclude that only perfect fullerenes with G = G' = [3,5] = I_{h} = S_{5}, satisfying both IPR and HPR, are C_{60}, C_{180}, C_{240}, etc. We need also to notice that for n_{6} = 0,2,3 there are always exactly one 3valent 5/6 polyhedron (i.e., the geometrical form of C_{20}, C_{24}, C_{26}), but for some larger values (e.g. n_{6} = 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 3valent 5/6 polyhedra, and chemical isomers different arrangements of double bonds, obtained from the same 3valent graph by its edgecoloring.
