  Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite
graphs are Hamiltonian. We give a translation of this conjecture into the
matching-theoretic setting. This allows us to relax the requirement of
planarity to give the equivalent conjecture that all cubic, 3-connected,
Pfaffian, bipartite graphs are Hamiltonian.
  A graph, other than the path of length three, is a brace if it is bipartite
and any two disjoint edges are part of a perfect matching. Our perspective
allows us to observe that Barnette's Conjecture can be reduced to cubic, planar
braces. We show a similar reduction to braces for cubic, 3-connected, bipartite
graphs regarding four stronger versions of Hamiltonicity. Note that in these
cases we do not need planarity. As a practical application of these results, we
provide some supplements to a generation procedure for cubic, 3-connected,
planar, bipartite graphs discovered by Holton et al. [Hamiltonian Cycles in
Cubic 3-Connected Bipartite Planar Graphs, JCTB, 1985]. These allow us to check
whether a graph we generated is a brace.
