Abstract
A fullerene graph F is a 3-connected plane cubic graph with exactly 12 pentagons and the remaining faces as 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 we 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.
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