On Barnette’s conjecture and the $H^{+-}$ property

Abstract

Let $\mathcal{B}$ denote the class of all 3-connected cubic bipartite plane graphs. A conjecture of Barnette states that every graph in $\mathcal{B}$ has a Hamilton cycle. A cyclic sequence of big faces is a cyclic sequence of different faces, each bounded by at least six edges, such that two faces from the sequence are adjacent if and only if they are consecutive in the sequence. Suppose that ${F_1, F_2, F_3}$ is a proper 3-coloring of the faces of $G^*\in \mathcal{B}$. We prove that if every cyclic sequence of big faces of $G^*$ has a face belonging to $F_1$ and a face belonging to $F_2$, then $G^*$ has the following properties: $H^{+-}$: If any two edges are chosen on the same face, then there is a Hamilton cycle through one and avoiding the other, $H^{--}$: If any two edges are chosen which are an even distance apart on the same face, then there is a Hamilton cycle that avoids both. Moreover, let $X$ and $Y$ partition the set of big faces of $G^*$ such that all such faces of $F_1$ are in $X$ and all such faces of $F_2$ are in $Y$. We prove that if every cyclic sequence of big faces has a face belonging to $X$ and a face belonging to $Y$, then $G^*$ has a Hamilton cycle.