On k-resonant fullerene graphs

A fullerene graph $F$ is a 3-connected plane cubic graph with exactly 12 pentagons and the remaining hexagons. Let $M$ be a perfect matching of $F$. A cycle $C$ of $F$ is $M$-alternating if the edges of $C$ appear alternately in and off $M$. 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 all hexagons in $\mathcal H$ are $M$-alternating. A fullerene graph $F$ is $k$-resonant if any $i$ ($0\leq i \leq k$) disjoint hexagons of $F$ form a resonant pattern. In this paper, we prove that every hexagon of a fullerene graph is resonant and all leapfrog fullerene graphs are 2-resonant. Further, we show that a 3-resonant fullerene graph has at most 60 vertices and construct all nine 3-resonant fullerene graphs, which are also $k$-resonant for every integer $k>3$. Finally, sextet polynomials of the 3-resonant fullerene graphs are computed.