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.
Categories: math.CO