Every Extremal Fullerene Graph with No Less than 60 Vertices is 2-Resonant*

摘要

A fullerene graph is a 3-regular 3-connected plane graph with only pentagonal and hexagonal faccs. For a fullerene graph F, a set H of disjoint hexagons is called a resonant pattern (or sextet pattern) if it has a perfect matching such that every hexagon in H is M-alternating. F is said to be k-resonant if any i (0 ≤ i ≤ k) disjoint hexagons of F form a resonant pattern. Every fullerene graph is shown to be 1-resonant and exactly nine fullerene graphs are k-resonant (k > 3). But not all fullerene graphs are 2-resonant. For a fullerene graph Fn with n vertices, the size of a maximum resonant pattern of Fn is the Clar number, denoted by c(Fn). It is known that c(F_n) ≤ n-12/6. The fullerene graphs Fn with c(Fn) = n-12/6 are extremal In this paper, we show that every extremal fullerene graph with no less than 60 vertices is 2-resonant. However, non-2-resonant extremal fullerene graphs with less than 60 vertices exist.