Hamiltonian cycles in cubic 3-connected bipartite planar graphs (Q801087)

Barnette has conjectured that every cubic 3-connected bipartite planar graph is Hamiltonian. In this paper data generated by McKay on Hamiltonian cycles in graphs on 40 or fewer vertices is combined with arguments on cuts and reductions to prove the following result: Theorem. If G is a cubic 3-connected bipartite planar graph on n vertices then (a) \(n\leq 64\) implies G is Hamiltonian, (b) \(n\leq 52\) implies every edge of G lies on some Hamiltonian cycle, (c) \(n\leq 44\) implies that for any two edges e and f of G, there is a Hamiltonian cycle through e avoiding f, except possibly if e is a central edge of a unique subgraph \(P_ 2\times P_ 3\) and f has no vertex in that subgraph, (d) \(n\leq 40\) implies that for any two edges e and f of G, there is a Hamiltonian cycle through e avoiding f.