Clar Structure and Fries Set of Fullerenes and (4,6)-Fullerenes on Surfaces

Abstract

Fowler and Pisanski showed that the Fries number for a fullerene on surface Σ is bounded above by $|V|/3$, and fullerenes which attain this bound are exactly the class of leapfrog fullerenes on surface Σ. We showed that the Clar number of a fullerene on surface Σ is bounded above by $(|V|/6)-\chi (\mathrm{\Sigma })$, where $\chi (\mathrm{\Sigma })$ stands for the Euler characteristic of Σ. By establishing a relation between the extremal fullerenes and the extremal (4,6)-fullerenes on the sphere, Hartung characterized the fullerenes on the sphere $S_0$ for which Clar numbers attain $(|V|/6)-\chi (S_0)$. We prove that, for a (4,6)-fullerene on surface Σ, its Clar number is bounded above by $(|V|/6)+\chi (\mathrm{\Sigma })$ and its Fries number is bounded above by $(|V|/3)+\chi (\mathrm{\Sigma })$, and we characterize the (4,6)-fullerenes on surface Σ attaining these two bounds in terms of perfect Clar structure. Moreover, we characterize the fullerenes on the projective plane $N_1$ for which Clar numbers attain $(|V|/6)-\chi (N_1)$ in Hartung’s method.