Extremal fullerene graphs with the maximum Clar number

A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let F_n be a fullerene graph with n vertices. A set of mutually disjoint hexagons of F_n is a sextet pattern if F_n has a perfect matching which alternates on and off every hexagon in . The maximum cardinality of sextet patterns of F_n is the Clar number of F_n. It was shown that the Clar number is no more than . Many fullerenes with experimental evidence attain the upper bound, for instance, C₆₀ and C₇₀. In this paper, we characterize extremal fullerene graphs whose Clar numbers equal . By the characterization, we show that there are precisely 18 fullerene graphs with 60 vertices, including C₆₀, achieving the maximum Clar number 8 and we construct all these extremal fullerene graphs.