Alt, HelmutHelmutAltPayne, Michael StuartMichael StuartPayneSchmidt, Jens M.Jens M.SchmidtWood, David R.David R.Wood2020-10-202020-10-202016The Australasian journal of combinatorics (2016)http://hdl.handle.net/11420/7620We 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.en1034-49422016354365Centre for Discrete Mathematics and Computing, Univ. of QueenslandMathematikJournal Articlehttp://ajc.maths.uq.edu.au/pdf/64/ajc_v64_p354.pdfOther