A computer-assisted proof of Barnette-Goodey conjecture: Not only   fullerene graphs are Hamiltonian
  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. 
Franti\v{s}ek Kardo\v{s}
math.CO