  We prove a new sufficient condition for a cubic 3-connected planar graph to
be Hamiltonian. This condition is most easily described as a property of the
dual graph. Let $G$ be a planar triangulation. Then the dual $G^*$ is a cubic
3-connected planar graph, and $G^*$ is bipartite if and only if $G$ is
Eulerian. We prove that if the vertices of $G$ are (improperly) coloured blue
and red, such that the blue vertices cover the faces of $G$, there is no blue
cycle, and every red cycle contains a vertex of degree at most 4, then $G^*$ is
Hamiltonian.
  This result implies the following special case of Barnette's Conjecture: if
$G$ is an Eulerian planar triangulation, whose vertices are properly coloured
blue, red and green, such that every red-green cycle contains a vertex of
degree 4, then $G^*$ is Hamiltonian. Our final result highlights the
limitations of using a proper colouring of $G$ as a starting point for proving
Barnette's Conjecture. We also explain related results on Barnette's Conjecture
that were obtained by Kelmans and for which detailed self-contained proofs have
not been published.
