A computer-assisted proof of Barnette-Goodey conjecture: Not only fullerene graphs are Hamiltonian

TL;DR

This paper proves the Barnette-Goodey conjecture by demonstrating that all 3-connected planar graphs with faces of size at most 6, including fullerene graphs, are Hamiltonian, using a computer-assisted proof.

Contribution

It provides the first computer-assisted proof confirming that all such graphs are Hamiltonian, resolving a long-standing conjecture.

Findings

Barnette-Goodey conjecture is proven true

All 3-connected planar graphs with faces of size ≤6 are Hamiltonian

Fullerene graphs are Hamiltonian

Abstract

Fullerene graphs, i.e., 3-connected planar cubic graphs with pentagonal and hexagonal faces, are conjectured to be Hamiltonian. This is a special case of a conjecture of Barnette and Goodey, stating that 3-connected planar graphs with faces of size at most 6 are Hamiltonian. We prove the conjecture.