  We study the existence of hamiltonian cycles in plane cubic graphs G having a
facial 2-factor Q. Thus hamiltonicity in G is transformed into the existence of
a (quasi) spanning tree of faces in the contraction G/Q. In particular, we
study the case where G is the leapfrog extension (called vertex envelope in
(Discrete Math., 309(14):4793-4809, 2009)) of a plane cubic graph G_0. As a
consequence we prove hamiltonicity in the leapfrog extension of planar cubic
cyclically 4-edge-connected bipartite graphs. This and other results of this
paper establish partial solutions of Barnette's Conjecture according to which
every 3-connected cubic planar bipartite graph is hamiltonian. These results go
considerably beyond Goodey's result on this topic (Israel J. Math., 22:52-56,
1975).
