Hamiltonian cycles in bipartite plane cubic maps

Abstract

The nature of bipartite cubic plane maps is investigated relative to questions concerning connectedness, local structure, and the line graph. Several contributions are made toward the solution of the conjecture of Barnette that each 3-connected bipartite cubic plane map is Hamiltonian, including a proof that if e is an edge in a bipartite cubic plane map M which has exactly six quadrilaterals, then there is a Hamiltonian cycle in M which passes through e. Hamiltonian cycles are also discussed relative to bipartite cubic plane maps of connectivity 2, and it is proved that every bipartite cubic plane map of connectivity 2 has at least eight quadrilaterals, and those with exactly eight quadrilaterals are Hamiltonian.