On k-Resonant Fullerene Graphs

A fullerene graph F is a cubic 3-connected plane graph with exact 12 pentagons and other hexagons. Let M be a perfect matching of F . A cycle C of F is M -alternating if and only if the edges of C appear alternately in and off M . A set H of disjoint hexagons of F is called a resonant pattern if F has a perfect matching M such that all hexagons in H are M -alternating. A fullerene graph F is k-resonant if any i (0 ≤ i ≤ k) mutually 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. Finally, we show that a k-resonant (k ≥ 3) fullerene graph has at most 60 vertices and construct all nine 3-resonant fullerene graphs, which are also k-resonant (k > 3).