Cyclic Subsets and Barnette's Conjecture

TL;DR

This paper introduces cyclic subsets in graph theory and uses a new theorem to provide an inductive proof for Barnette's conjecture, a major open problem about Hamiltonian cycles in certain graphs.

Contribution

It presents a novel theorem relating cyclic subsets to Hamiltonicity and applies it to prove Barnette's conjecture.

Findings

Established a relation between cyclic subsets and Hamiltonian cycles.

Provided an inductive proof of Barnette's conjecture.

Identified conditions under which certain cubic, bipartite, polyhedral graphs are Hamiltonian.

Abstract

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.