Remarks on Barnette's Conjecture

Abstract: 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 $P^{*}$ is the dual graph. We show that $P$ has at least $3^{\frac{2|P^{}|}{\Delta^{2}{(P^{})}}}$ different Hamilton cycles.