Necessary Condition for Cubic Planar 3-Connected Graph to be Non-Hamiltonian with Proof of Barnette's Conjecture

A conjecture of Barnette states that, every three connected cubic bipartite pla nar graph is Hamiltonian. This problem has remained open since its formulation. This paper has a threefold purpose. The first is to provide survey of literature surrounding the conjecture. The second is to give the necessary condition for cubic planar three connected graph to be non-Hamiltonian and finally, we shall prove near about 50 year Barnett’s conjecture. For the proof of different results using to prove the results we illustrate most of the results by using counter examples