  A conjecture of Barnette states that every 3-connected cubic bipartite plane
graph has a Hamilton cycle, which is equivalent to the statement that every
simple even plane triangulation admits a partition of its vertex set into two
subsets so that each induces a tree.
  Let $G$ be a simple even plane triangulation and suppose that ${V_1, V_2,
V_3}$ is a 3-coloring of the vertex set of $G$. Let $B_{i}$, $i = 1, 2, 3$, be
the set of all vertices in $V_i$ of the degree at least 6. We prove that if
induced graphs $G[B_1 \cup B_2]$ and $G[B_1 \cup B_3]$ are acyclic, then the
following properties are satisfied: [6pt] (1) For every path $abc$ there is
possible to partition the vertex set of $G$ into two subsets so that each
induces a tree, and one of them contains the edge $ab$ and avoids the vertex
$c$, [6pt] (2) For every path $abc$ with vertices $a$, $c$ of the same color
there is possible to partition the vertex set of $G$ into two subsets so that
each induces a tree, and one of them contains the path $abc$.
