Graphs with multi-$4$-cycles and the Barnette's conjecture

Let ${\cal H}$ denote the family of all graphs with multi-$4$-cycles and suppose that $G \in {\cal H}$. Then, $G$ is a bipartite graph with a vertex bipartition $\{V_{\alpha}, V_{\beta}\}$. We prove that for every vertex $v \in V_{\beta}$ and for every $2$-colouring $V_{\alpha} \rightarrow \{1, 2\}$ there exists a $2$-colouring $V_{\beta} \rightarrow \{1, 2\}$ such that every cycle in $G$ is not monochromatic and $b(v) = 1$ ($b(v) = 2$). Let now $G$ be a simple even plane triangulation with a vertex $3$-partition $\{V_{1}, V_{2}, V_{3}\}$. Denote by $B_{i}$, $i = 1, 2, 3$, the set of all vertices in $V_i$ of degree at least $6$ in $G$. Suppose that $G[B_{1}\cup B_{3}]$ ($G[B_{2}\cup B_{3}]$) is a subgraph of $G$ induced by the set $B_{1}\cup B_{3}$ ($B_{2}\cup B_{3}$, respectively). Let $G^{*}$ be the dual graph of $G$ with the following $3$-face-colouring: a face $f$ of $G^{*}$ is coloured with $i$ if and only if the vertex $v = f^{*} \in V_{i}$. We prove that if $H = G[B_{1}\cup B_{3}] \cup G[B_{2}\cup B_{3}] \in {\cal H}$, then, for any edge chosen on a face coloured $3$ and of size at least $6$ in $G^{*}$, there exists a Hamilton cycle of $G^{*}$ which avoids this edge. Moreover, if every component of $H$ is $2$-connected, then there exists a Hamilton cycle of $G^{*}$ such that for every face coloured $3$ it avoids every second edge of this face or it avoids at most two edges of this face.