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We prove that every vertex v of a tournament T belongs to at least $maxmin\delta ⁺\left(T\right),2\delta ⁺\left(T\right)-d{⁺}_T\left(v\right)+1,min\delta ¯\left(T\right),2\delta ¯\left(T\right)-d{¯}_T\left(v\right)+1$ arc-disjoint cycles, where δ⁺(T) (or δ¯(T)) is the minimum out-degree (resp. minimum in-degree) of T, and $d{⁺}_T\left(v\right)$ (or $d{¯}_T\left(v\right)$) is the out-degree (resp. in-degree) of v.

Let be the family of all 2-connected plane triangulations with vertices of degree three or six. Grünbaum and Motzkin proved (in dual terms) that every graph P ∈ has a decomposition into factors P₀, P₁, P₂ (indexed by elements of the cyclic group Q = 0,1,2) such that every factor $P_q$ consists of two induced paths of the same length M(q), and K(q) - 1 induced cycles of the same length 2M(q). For q ∈ Q, we define an integer S⁺(q) such that the vector (K(q),M(q),S⁺(q)) determines the graph P (if P is...

On a non-trivial partially ordered real vector space (V,≤) the orthogonality relation is defined by incomparability and ζ(V,⊥) is a complete lattice of double orthoclosed sets. We say that A ⊆ V is an orthogonal set when for all a,b ∈ A with a ≠ b, we have a ⊥ b. In our earlier papers we defined an integrally open ordered vector space and two closure operations A → D(A) and $A\to A^{\perp \perp }$. It was proved that V is integrally open iff $D\left(A\right)=A^{\perp \perp }$ for every orthogonal set A ⊆ V. In this paper we generalize this result. We...