The Generation of Fullerenes

TL;DR

This paper introduces a highly efficient algorithm for generating fullerenes, significantly surpassing previous tools in speed and capacity, and provides extensive data and verification of related conjectures for fullerenes up to 400 vertices.

Contribution

The authors present a new, faster algorithm for fullerene generation, correcting previous software errors and extending the generation capacity beyond 100 vertices.

Findings

Algorithm is 3.5 times faster than previous methods.

Generated fullerenes and IPR fullerenes up to 400 vertices.

Verified conjectures related to cubic planar graphs and spiral structures.

Abstract

We describe an efficient new algorithm for the generation of fullerenes. Our implementation of this algorithm is more than 3.5 times faster than the previously fastest generator for fullerenes -- fullgen -- and the first program since fullgen to be useful for more than 100 vertices. We also note a programming error in fullgen that caused problems for 136 or more vertices. We tabulate the numbers of fullerenes and IPR fullerenes up to 400 vertices. We also check up to 316 vertices a conjecture of Barnette that cubic planar graphs with maximum face size 6 are hamiltonian and verify that the smallest counterexample to the spiral conjecture has 380 vertices.