Thoughts on Barnette's Conjecture

TL;DR

This paper introduces a new sufficient condition involving vertex coloring and face coverage in planar triangulations that guarantees the dual graph's Hamiltonicity, providing insights into Barnette's Conjecture.

Contribution

It establishes a novel sufficient condition for Hamiltonicity of the dual graph based on vertex coloring and face coverage in planar triangulations, advancing understanding of Barnette's Conjecture.

Findings

Proves a new sufficient condition for Hamiltonian dual graphs based on vertex coloring.

Derives a special case of Barnette's Conjecture involving red-green cycles and degree constraints.

Highlights limitations of proper vertex coloring approaches for Barnette's Conjecture.

Abstract

We prove a new sufficient condition for a cubic 3-connected planar graph to be Hamiltonian. This condition is most easily described as a property of the dual graph. Let $G$ be a planar triangulation. Then the dual $G^*$ is a cubic 3-connected planar graph, and $G^*$ is bipartite if and only if $G$ is Eulerian. We prove that if the vertices of $G$ are (improperly) coloured blue and red, such that the blue vertices cover the faces of $G$, there is no blue cycle, and every red cycle contains a vertex of degree at most 4, then $G^*$ is Hamiltonian. This result implies the following special case of Barnette's Conjecture: if $G$ is an Eulerian planar triangulation, whose vertices are properly coloured blue, red and green, such that every red-green cycle contains a vertex of degree 4, then $G^*$ is Hamiltonian. Our final result highlights the limitations of using a proper colouring of $G$ as a starting point for proving Barnette's Conjecture. We also explain related results on Barnette's Conjecture that were obtained by Kelmans and for which detailed self-contained proofs have not been published.