At least half of the leapfrog fullerene graphs have exponentially many Hamilton cycles

A fullerene graph is a 3-connected cubic planar graph with pentagonal and hexagonal faces. The leapfrog transformation of a planar graph produces the trucation of the dual of the given graph. A fullerene graph is leapfrog if it can be obtained from another fullerene graph by the leapfrog transformation. We prove that leapfrog fullerene graphs on $n=12k-6$ vertices have at least $2^{k}$ Hamilton cycles.