  Barnette conjectured that all cubic $3$-connected plane graphs with maximum
face size at most $6$ are hamiltonian. We provide a method of construction of a
hamiltonian cycle (in dual terms) in an arbitrary cubic, $3$-connected plane
graph possessing such a face $g$ that every face incident with $g$ has at most
$5$ edges and every other face has at most $6$ edges.
