On Barnette’s conjecture

Barnette’s conjecture is the statement that every cubic 3-connected bipartite planar graph is Hamiltonian. We show that if such a graph has a 2-factor FF which consists only of facial 4-cycles, then the following properties are satisfied: (1)If an edge is chosen on a face and this edge is in FF, there is a Hamilton cycle containing all other edges of this face.(2)If any face is chosen, there is a Hamilton cycle which avoids all edges of this face which are not in FF.(3)If any two edges are chosen on the same face, there is a Hamilton cycle through one and avoiding the other.(4)If any two edges are chosen which are an even distance apart on the same face, there is a Hamilton cycle which avoids both.