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

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)-χ(Σ), whereχ(Σ)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 sphereS0for which Clar numbers attain(|V|/6)-χ(S0). We prove that, for a (4,6)-fullerene on surface Σ, its Clar number is bounded above by(|V|/6)+χ(Σ)and its Fries number is bounded above by(|V|/3)+χ(Σ), 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 planeN1for which Clar numbers attain(|V|/6)-χ(N1)in Hartung’s method.