2-resonant fullerenes

A fullerene graph F is a planar cubic graph with exactly 12 pentagonal faces and other hexagonal faces. A set H of disjoint hexagons of F is called a resonant pattern (or sextet pattern) if F has a perfect matching M 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. It was known that each fullerene graph is 1-resonant and there are only nine fullerene graphs that are 3-resonant. In this paper, we show that the fullerene graphs which do not contain the subgraph L or R as illustrated in Fig. 1 are 2-resonant except for the specific eleven graphs. This result implies that each IPR fullerene is 2-resonant.