Fullerenes with complete Clar structure

It is known that the Fries number for a fullerene Γ=(V,E,F)Γ=(V,E,F) is bounded above by |V|3 and the fullerenes that attain this bound have been characterized. The corresponding bound for the Clar number is |V|6−2. In this paper, we characterize the fullerenes for which this bound is attained, and also characterize the fullerenes for which the upper bounds for the Clar and the Fries number are simultaneously achieved.