  Barnette's conjecture asserts that every cubic $3$-connected plane bipartite
graph is hamiltonian. Although, in general, the problem is still open, some
partial results are known. In particular, let us call a face of a plane graph
big (small) if it has at least six edges (it has four edges, respectively).
Goodey proved for a $3$-connected bipartite cubic plane graph $P$, that if all
big faces in $P$ have exactly six edges, then $P$ is hamiltonian. In this paper
we prove that the same is true under the condition that no face in $P$ has more
than four big neighbours. We also prove, that if each vertex in $P$ is incident
both with a small and a big face, then~$P$ has at least $2^{k}$ different
Hamilton cycles, where $k = \left\lceil\frac{|B|-2}{4\Delta(B) -
7}\right\rceil$, $|B|$ is the number of big faces in $P$ and $\Delta(B)$ is the
maximum size of faces in $P$. 15 pages
