Thoughts on Barnette's conjecture

Helmut Alt, Michael S. Payne, Jens M. Schmidt, David R Wood

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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.

Original languageEnglish
Pages (from-to)354-365
Number of pages12
JournalAustralasian Journal of Combinatorics
Issue number2
Publication statusPublished - 2016

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