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 S 0 for which Clar numbers attain ( | V | / 6 ) - χ ( S 0 ) . 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 N 1 for which Clar numbers attain ( | V | / 6 ) - χ ( N 1 ) in Hartung’s method.