49 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
On the minimum leaf number of cubic graphs
The \emph{minimum leaf number} of a connected graph is
defined as the minimum number of leaves of the spanning trees of . We
present new results concerning the minimum leaf number of cubic graphs: we show
that if is a connected cubic graph of order , then , improving on the best known result in [Inf. Process.
Lett. 105 (2008) 164-169] and proving the conjecture in [Electron. J. Graph
Theory and Applications 5 (2017) 207-211]. We further prove that if is also
2-connected, then , improving on the best
known bound in [Math. Program., Ser. A 144 (2014) 227-245]. We also present new
conjectures concerning the minimum leaf number of several types of cubic graphs
and examples showing that the bounds of the conjectures are best possible.Comment: 17 page
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