  A fullerene graph $F$ is a planar cubic graph with exactly 12 pentagonal
faces and other hexagonal faces. A set $\mathcal{H}$ 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 $\mathcal{H}$ is $M$-alternating. $F$
is said to be $k$-resonant if any $i$ ($0\leq i\leq k$) disjoint hexagons of
$F$ form a resonant pattern. It was known that each fullerene graph is
1-resonant and all 3-resonant fullerenes are only the nine graphs. 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.
