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1
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A note on arc-disjoint cycles in tournaments

100%
Florek J.
Colloquium Mathematicum
|
2014
|
tom 136
|
nr 2
259-262
EN
We prove that every vertex v of a tournament T belongs to at least $max{min{δ⁺(T), 2δ⁺(T) - d⁺_{T}(v) + 1}, min{δ¯(T), 2δ¯(T) - d¯_{T}(v) + 1}}$ arc-disjoint cycles, where δ⁺(T) (or δ¯(T)) is the minimum out-degree (resp. minimum in-degree) of T, and $d⁺_{T}(v)$ (or $d¯_{T}(v)$) is the out-degree (resp. in-degree) of v.
2
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Equations relating factors in decompositions into factors of some family of plane triangulations, and applications (with an appendix by Andrzej Schinzel)

100%
Florek J.
Colloquium Mathematicum
|
2015
|
tom 138
|
nr 1
23-42
EN
Let 𝓟 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 ∈ 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 simple) uniquely up to orientation-preserving isomorphism. We establish arithmetic equations that will allow calculating (K(q+1),M(q+1),S⁺(q+1)) from (K(q),M(q),S⁺(q)), q ∈ Q. We present some applications of these equations. The set {(K(q),M(q),S⁺(q)): q ∈ Q} is called the orbit of P. If P has a one-point orbit, then there is an orientation-preserving automorphism σ such that $σ(P_{i}) = P_{i+1}$ for every i ∈ Q (where P₃ = P₀). We characterize one-point orbits of graphs in 𝓟. It is known that every graph in 𝓟 has an even order. We prove that if P is of order 4n + 2, n ∈ ℕ, then it has two disjoint induced trees of the same order, which are equitable 2-colorable and together cover all vertices of P.
3
Content available remote

Orthomodular lattices and closure operations in ordered vector spaces

100%
Florek J.
Banach Center Publications
|
2010
|
tom 89
|
nr 1
129-133
EN
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 $A → A^{⊥⊥}$. It was proved that V is integrally open iff $D(A) = A^{⊥⊥}$ for every orthogonal set A ⊆ V. In this paper we generalize this result. We prove that V is integrally open iff D(A) = W for every W ∈ ζ(V,⊥) and every maximal orthogonal set A ⊆ W. Hence it follows that the lattice ζ(V,⊥) is orthomodular.
4
Content available remote

Billiards and the five distance theorem

88%
Florek J.
Acta Arithmetica
|
2009
|
tom 139
|
nr 3
229-239
5
Content available remote

Billiard and diophantine approximation

88%
Florek J.
Acta Arithmetica
|
2008
|
tom 134
|
nr 4
317-327
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