### On Barnette's Conjecture and H^+- property [article]

Jan Florek
2012 arXiv   pre-print
A conjecture of Barnette states that every 3-connected cubic bipartite plane graph has a Hamilton cycle, which is equivalent to the statement that every simple even plane triangulation admits a partition of its vertex set into two subsets so that each induces a tree. Let G be a simple even plane triangulation and suppose that V_1, V_2, V_3 is a 3-coloring of the vertex set of G. Let B_i, i = 1, 2, 3, be the set of all vertices in V_i of the degree at least 6. We prove that if induced graphs
more » ... 1 ∪ B_2] and G[B_1 ∪ B_3] are acyclic, then the following properties are satisfied: [6pt] (1) For every path abc there is possible to partition the vertex set of G into two subsets so that each induces a tree, and one of them contains the edge ab and avoids the vertex c, [6pt] (2) For every path abc with vertices a, c of the same color there is possible to partition the vertex set of G into two subsets so that each induces a tree, and one of them contains the path abc.