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