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),where χ (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 sphere s0 for 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 plane n1 for which clar numbers attain (|v|/6) - χ (n1) in hartung's method. © 2014 yang gao and heping zhang.