26 research outputs found

    Thoughts on Barnette's Conjecture

    Full text link
    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 GG be a planar triangulation. Then the dual G∗G^* is a cubic 3-connected planar graph, and G∗G^* is bipartite if and only if GG is Eulerian. We prove that if the vertices of GG are (improperly) coloured blue and red, such that the blue vertices cover the faces of GG, there is no blue cycle, and every red cycle contains a vertex of degree at most 4, then G∗G^* is Hamiltonian. This result implies the following special case of Barnette's Conjecture: if GG 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∗G^* is Hamiltonian. Our final result highlights the limitations of using a proper colouring of GG 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 figure

    A note on Barnette's conjecture

    Get PDF
    Barnette conjectured that each planar, bipartite, cubic, and 3-connected graph is hamiltonian. We prove that this conjecture is equivalent to the statement that there is a constant c > 0 such that each graph G of this class contains a path on at least c |V(G) | vertices

    Thoughts on Barnette's conjecture

    Get PDF
    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 Conjec- ture: 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 Bar- nette’s Conjecture that were obtained by Kelmans and for which detailed self-contained proofs have not been published

    Kalai's squeezed 3-spheres are polytopal

    Get PDF
    In 1988, Kalai extended a construction of Billera and Lee to produce many triangulated (d-1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack, he derived that for every dimension d>=5, most of these (d-1)-spheres are not polytopal. However, for d=4, this reasoning fails. We can now show that, as already conjectured by Kalai, all of his 3-spheres are in fact polytopal. Moreover, we can now give a shorter proof of Hebble & Lee's 2000 result that the dual graphs of these 4-polytopes are Hamiltonian. Therefore, the polars of these Kalai polytopes yield another family supporting Barnette's conjecture that all simple 4-polytopes admit a Hamiltonian circuit.Comment: 11 pages, 5 figures; accepted for publication in J. Discrete & Computational Geometr
    corecore