Face colorings and chiral face colorings of icosahedral giant fullerenes: C80 to C240

Abstract

Combinatorial techniques based on Sheehan’s modification of Pόlya’s theorem and Mȍbius inversion technique together with character cycle indices are applied to face colorings of giant fullerenes. These techniques are applied to icosahedral fullerenes, C₈₀ with a chamfered dodecahedron structure, a chiral fullerene C₁₄₀, icosahedral C₁₈₀ and C₂₄₀ with a chamfered truncated icosahedron geometry. The techniques are shown to provide both achiral and chiral face colorings of giant fullerenes for (a) coloring only pentagons with all hexagons painted white, (b) coloring hexagonal faces with all pentagons kept white, (c) both pentagons and hexagons painted with variable colors from a single set, and (d) both pentagons and hexagons painted with variable colors but chosen from two different sets. We have shown that in order to produce chiral colorings for the I_h fullerenes at least 2 black colors and remaining being white are needed when both pentagons and hexagons are varied. Results provide new insights into facial labeling and facial dynamics of fullerenes, while therefore being fully prepared to be inscribed in the new so called neutrosophical science (of carbon) nano-structural physical-chemistry.