Thoughts on Barnette's Conjecture
| Autor: | Alt, Helmut, Payne, Michael S., Schmidt, Jens M., Wood, David R. |
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| Rok vydání: | 2013 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 12 pages, 7 figures |
| Databáze: | arXiv |
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