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 FnFn be a fullerene graph with nn vertices. A set HH of mutually disjoint hexagons of FnFn is a sextet pattern if FnFn has a perfect matching which alternates on and off every hexagon in HH. The maximum cardinality of sextet patterns of FnFn is the Clar number of FnFn. It was shown that the Clar number is no more than ⌊n−126⌋. Many fullerenes with experimental evidence attain the upper bound, for instance, C60C60 and C70C70. In this paper, we characterize extremal fullerene graphs whose Clar numbers equal n−126. By the characterization, we show that there are precisely 18 fullerene graphs with 60 vertices, including C60C60, achieving the maximum Clar number 8 and we construct all these extremal fullerene graphs.