Remarks on Barnette's Conjecture

Let $P$ be a cubic $3$-connected bipartite plane graph which has a $2$-factor which consists only of facial $4$-cycles, and suppose that $P^{\ast }$ is the dual graph. We show that $P$ has at least $3^{\frac{2\mathrm{\mid }{P}^{\ast }\mathrm{\mid }}{{\mathrm{\Delta }}^2(P^{\ast })}}$ different Hamilton cycles.