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 G[B 1 ∪ B 2 ] and G[B 1 ∪ B 3 ] are acyclic, then the following properties are satisfied:

No abstract

In a partly ordered space the orthogonality relation is defined by incomparability. We define integrally open and integrally semi-open ordered real vector spaces. We prove: if an ordered real vector space is integrally semi-open, then a complete lattice of double orthoclosed sets is orthomodular. An integrally open concept is closely related to an open set in the Euclidean topology in a finite dimensional ordered vector space. We prove: if V is an ordered Euclidean space, then V is integrally open and directed (and is also Archimedean) if and only if its positive cone, without vertex 0, is an open set in the Euclidean topology (and also the family of all order segments {z ∈ V : a < z < b}, a < b, is a base for the Euclidean topology).

No abstract