showed that Barnette's conjecture is equivalent to a superficially stronger statement, that for every two edges e and f on the same face of a bipartite cubic polyhedron, there exists a Hamiltonian cycle that contains e but does not contain f.
Barnette's conjecture is also equivalent to the statement that the vertices of the dual of every cubic bipartite polyhedral graph can be partitioned into two subsets whose induced subgraphs are trees.