  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.
