  In this paper, the concept of cyclic subsets in graph theory is introduced.
An interesting theorem which relates to the collective Hamiltonicity of these
cyclic subsets in graphs is also presented. This paper uses this theorem to
construct an inductive proof of Barnette's long-standing conjecture, which asks
whether every cubic, polyhedral, bipartite graph is Hamiltonian. Finding a
class of graphs that are certain to be Hamiltonian is one of the biggest
unsolved problems in Hamiltonian graph theory today.
