Algorithmic proof of Barnette's Conjecture

TL;DR

This paper presents an algorithmic proof confirming that all 3-connected bipartite cubic planar graphs are Hamiltonian, using a novel spiral-chain method that transforms the problem into finding specific Hamiltonian paths.

Contribution

It introduces a new spiral-chain based algorithmic approach to prove Barnette's conjecture, differing from traditional methods.

Findings

Confirmed Barnette's conjecture for all 3-connected bipartite cubic planar graphs.

Developed a spiral-chain algorithm to identify Hamiltonian cycles.

Showed that non-Hamiltonian graphs have an (n-1)-vertex cycle structure.

Abstract

In this paper we have given an algorithmic proof of an long standing Barnette's conjecture (1969) that every 3-connected bipartite cubic planar graph is hamiltonian. Our method is quite different than the known approaches and it rely on the operation of opening disjoint chambers, bu using spiral-chain like movement of the outer-cycle elastic-sticky edges of the cubic planar graph. In fact we have shown that in hamiltonicity of Barnette graph a single-chamber or double-chamber with a bridge face is enough to transform the problem into finding specific hamiltonian path in the cubic bipartite graph reduced. In the last part of the paper we have demonstrated that, if the given cubic planar graph is non-hamiltonian then the algorithm which constructs spiral-chain (or double-spiral chain) like chamber shows that except one vertex there exists (n-1)-vertex cycle.