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

TL;DR

This paper proves that leapfrog fullerene graphs with a specific number of vertices have exponentially many Hamilton cycles, revealing a rich structure in these molecular graphs.

Contribution

It establishes a lower bound of exponential growth in the number of Hamilton cycles for leapfrog fullerene graphs with n=12k-6 vertices.

Findings

Leapfrog fullerene graphs have at least 2^k Hamilton cycles.

The number of Hamilton cycles grows exponentially with the parameter k.

The result applies to a class of structurally significant molecular graphs.

Abstract

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.