A note on k-uniform self-complementary hypergraphs of given order

Artur Szymański; A. Paweł Wojda

Displaying similar documents to “A note on k-uniform self-complementary hypergraphs of given order”

Self-complementary hypergraphs

A. Paweł Wojda (2006)

Discussiones Mathematicae Graph Theory

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A k-uniform hypergraph H = (V;E) is called self-complementary if there is a permutation σ:V → V, called self-complementing, such that for every k-subset e of V, e ∈ E if and only if σ(e) ∉ E. In other words, H is isomorphic with $H^{'} = (V; (\binom{V}{k}) - E)$ . In the present paper, for every k, (1 ≤ k ≤ n), we give a characterization of self-complementig permutations of k-uniform self-complementary hypergraphs of the order n. This characterization implies the well known results for self-complementing permutations...

Asymptotic Sharpness of Bounds on Hypertrees

Yi Lin, Liying Kang, Erfang Shan (2017)

Discussiones Mathematicae Graph Theory

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The hypertree can be defined in many different ways. Katona and Szabó introduced a new, natural definition of hypertrees in uniform hypergraphs and investigated bounds on the number of edges of the hypertrees. They showed that a k-uniform hypertree on n vertices has at most [...] (nk−1) $(n k - 1)$ edges and they conjectured that the upper bound is asymptotically sharp. Recently, Szabó verified that the conjecture holds by recursively constructing an infinite sequence of k-uniform hypertrees and...

The generalized Day norm. Part I. Properties

Monika Budzyńska, Aleksandra Grzesik, Mariola Kot (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we introduce a modification of the Day norm in $c_{0} (Γ)$ and investigate properties of this norm.

The sum number of d-partite complete hypergraphs

Hanns-Martin Teichert (1999)

Discussiones Mathematicae Graph Theory

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A d-uniform hypergraph is a sum hypergraph iff there is a finite S ⊆ IN⁺ such that is isomorphic to the hypergraph $⁺_{d} (S) = (V,)$ , where V = S and $= v ₁, . . ., v_{d} : (i \neq j \Rightarrow v_{i} \neq v_{j}) \land \sum_{i = 1}^{d} v_{i} \in S$ . For an arbitrary d-uniform hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices $w ₁, . . ., w_{σ} \notin V$ such that $\cup w ₁, . . ., w_{σ}$ is a sum hypergraph. In this paper, we prove $σ (_{n ₁, . . ., n_{d}}^{d}) = 1 + \sum_{i = 1}^{d} (n_{i} - 1) + m i n 0, ⌈ 1 / 2 (\sum_{i = 1}^{d - 1} (n_{i} - 1) - n_{d}) ⌉$ , where $_{n ₁, . . ., n_{d}}^{d}$ denotes the d-partite complete hypergraph; this generalizes the corresponding result of Hartsfield and Smyth [8] for complete bipartite graphs.

Sum labellings of cycle hypergraphs

Hanns-Martin Teichert (2000)

Discussiones Mathematicae Graph Theory

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A hypergraph is a sum hypergraph iff there are a finite S ⊆ IN⁺ and d̲, [d̅] ∈ IN⁺ with 1 < d̲ ≤ [d̅] such that is isomorphic to the hypergraph $_{d ̲, [d ̅]} (S) = (V,)$ where V = S and $= e \subseteq S : d ̲ \leq | e | \leq [d ̅] \land \sum_{v \in e} v \in S$ . For an arbitrary hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices $y ₁, . . ., y_{σ} \notin V$ such that $\cup y ₁, . . ., y_{σ}$ is a sum hypergraph. Generalizing the graph Cₙ we obtain d-uniform hypergraphs where any d consecutive vertices of Cₙ form an edge. We determine sum numbers and investigate properties of sum labellings...

A uniform dichotomy for generic $SL (2, ℝ)$ cocycles over a minimal base

Artur Avila, Jairo Bochi (2007)

Bulletin de la Société Mathématique de France

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We consider continuous $SL (2, ℝ)$ -cocycles over a minimal homeomorphism of a compact set $K$ of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform subexponential growth.

Some remarks on universality properties of $ℓ_{\infty} / c ₀$

Mikołaj Krupski, Witold Marciszewski (2012)

Colloquium Mathematicae

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We prove that if is not a Kunen cardinal, then there is a uniform Eberlein compact space K such that the Banach space C(K) does not embed isometrically into $ℓ_{\infty} / c ₀$ . We prove a similar result for isomorphic embeddings. Our arguments are minor modifications of the proofs of analogous results for Corson compacta obtained by S. Todorčević. We also construct a consistent example of a uniform Eberlein compactum whose space of continuous functions embeds isomorphically into $ℓ_{\infty} / c ₀$ , but fails to embed...

A condition equivalent to uniform ergodicity

Maria Elena Becker (2005)

Studia Mathematica

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Let T be a linear operator on a Banach space X with $s u p ₙ | | T ⁿ / n^{w} | | < \infty$ for some 0 ≤ w < 1. We show that the following conditions are equivalent: (i) $n^{- 1} \sum_{k = 0}^{n - 1} T^{k}$ converges uniformly; (ii) $c l (I - T) X = z \in X : l i m_{n} \sum_{k = 1}^{n} T^{k} z / k e x i s t s$ .

A note on perfect matchings in uniform hypergraphs with large minimum collective degree

Vojtěch Rödl, Andrzej Ruciński, Mathias Schacht, Endre Szemerédi (2008)

Commentationes Mathematicae Universitatis Carolinae

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For an integer $k \geq 2$ and a $k$ -uniform hypergraph $H$ , let $δ_{k - 1} (H)$ be the largest integer $d$ such that every $(k - 1)$ -element set of vertices of $H$ belongs to at least $d$ edges of $H$ . Further, let $t (k, n)$ be the smallest integer $t$ such that every $k$ -uniform hypergraph on $n$ vertices and with $δ_{k - 1} (H) \geq t$ contains a perfect matching. The parameter $t (k, n)$ has been completely determined for all $k$ and large $n$ divisible by $k$ by Rödl, Ruci’nski, and Szemerédi in [, submitted]. The values of $t (k, n)$ are very close to $n / 2 - k$ . In fact, the function $t (k, n) = n / 2 - k + c_{n, k}$ ,...

Color-bounded hypergraphs, V: host graphs and subdivisions

Csilla Bujtás, Zsolt Tuza, Vitaly Voloshin (2011)

Discussiones Mathematicae Graph Theory

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A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set = E₁,...,Eₘ, together with integers $s_{i}$ and $t_{i}$ satisfying $1 \leq s_{i} \leq t_{i} \leq | E_{i} |$ for each i = 1,...,m. A vertex coloring φ is proper if for every i, the number of colors occurring in edge $E_{i}$ satisfies $s_{i} \leq | φ (E_{i}) | \leq t_{i}$ . The hypergraph ℋ is colorable if it admits at least one proper coloring. We consider hypergraphs ℋ over a “host graph”, that means a graph G on the same vertex set X as ℋ, such that each $E_{i}$ induces a connected subgraph in G....

On the uniform convergence of double sine series

Péter Kórus, Ferenc Móricz (2009)

Studia Mathematica

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Let a single sine series (*) $\sum_{k = 1}^{\infty} a_{k} s i n k x$ be given with nonnegative coefficients $a_{k}$ . If $a_{k}$ is a “mean value bounded variation sequence” (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series (*) is that $k a_{k} \to 0$ as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series (**) $\sum_{k = 1}^{\infty} \sum_{l = 1}^{\infty} c_{k l} s i n k x s i n l y$ , even with complex coefficients $c_{k l}$ . We also...

Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅²₊

Péter Kórus, Ferenc Móricz (2010)

Studia Mathematica

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We investigate the convergence behavior of the family of double sine integrals of the form $\int_{0}^{\infty} \int_{0}^{\infty} f (x, y) s i n u x s i n v y d x d y$ , where (u,v) ∈ ℝ²₊:= ℝ₊ × ℝ₊, ℝ₊:= (0,∞), and f: ℝ²₊ → ℂ is a locally absolutely continuous function satisfying certain generalized monotonicity conditions. We give sufficient conditions for the uniform convergence of the remainder integrals $\int_{a ₁}^{b ₁} \int_{a ₂}^{b ₂}$ to zero in (u,v) ∈ ℝ²₊ as maxa₁,a₂ → ∞ and $b_{j} > a_{j} \geq 0$ , j = 1,2 (called uniform convergence in the regular sense). This implies the uniform convergence of the partial...

Classes of hypergraphs with sum number one

Hanns-Martin Teichert (2000)

Discussiones Mathematicae Graph Theory

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A hypergraph ℋ is a sum hypergraph iff there are a finite S ⊆ ℕ⁺ and d̲,d̅ ∈ ℕ⁺ with 1 < d̲ < d̅ such that ℋ is isomorphic to the hypergraph $ℋ_{d ̲, d ̅} (S) = (V,)$ where V = S and $= e \subseteq S : d ̲ < | e | < d ̅ \land \sum_{v \in e} v \in S$ . For an arbitrary hypergraph ℋ the sum number(ℋ ) is defined to be the minimum number of isolatedvertices $w ₁, . . ., w_{σ} \notin V$ such that $ℋ \cup w ₁, . . ., w_{σ}$ is a sum hypergraph. For graphs it is known that cycles Cₙ and wheels Wₙ have sum numbersgreater than one. Generalizing these graphs we prove for the hypergraphs ₙ and ₙ that under a certain condition...

Noncommutative uniform algebras

Mati Abel, Krzysztof Jarosz (2004)

Studia Mathematica

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We show that a real Banach algebra A such that ||a²|| = ||a||² for a ∈ A is a subalgebra of the algebra $C_{ℍ} (X)$ of continuous quaternion-valued functions on a compact set X.

On the double Lusin condition and convergence theorem for Kurzweil-Henstock type integrals

Abraham Racca, Emmanuel Cabral (2016)

Mathematica Bohemica

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Equiintegrability in a compact interval $E$ may be defined as a uniform integrability property that involves both the integrand $f_{n}$ and the corresponding primitive $F_{n}$ . The pointwise convergence of the integrands $f_{n}$ to some $f$ and the equiintegrability of the functions $f_{n}$ together imply that $f$ is also integrable with primitive $F$ and that the primitives $F_{n}$ converge uniformly to $F$ . In this paper, another uniform integrability property called uniform double Lusin condition introduced in the papers...

On some global and local geometric properties of Calderón-Lozanovskiĭ spaces

Agata Narloch (2005)

Banach Center Publications

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Criteria for full k-rotundity (k ∈ ℕ, k ≥ 2) and uniform rotundity in every direction of Calderón-Lozanovskiĭ spaces are formulated. A characterization of $H_{μ}$ -points in these spaces is also given.

Existence and upper semicontinuity of uniform attractors in $H ¹ (ℝ^{N})$ for nonautonomous nonclassical diffusion equations

Cung The Anh, Nguyen Duong Toan (2014)

Annales Polonici Mathematici

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We prove the existence of uniform attractors $_{ε}$ in the space $H ¹ (ℝ^{N})$ for the nonautonomous nonclassical diffusion equation $u_{t} - ε Δ u_{t} - Δ u + f (x, u) + λ u = g (x, t)$ , ε ∈ [0,1]. The upper semicontinuity of the uniform attractors ${_{ε}}_{ε \in [0, 1]}$ at ε = 0 is also studied.

Two variants of the size Ramsey number

Andrzej Kurek, Andrzej Ruciński (2005)

Discussiones Mathematicae Graph Theory

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Given a graph H and an integer r ≥ 2, let G → (H,r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let $m (G) = m a x_{F \subseteq G} | E (F) | / | V (F) |$ and define the Ramsey density $m_{i n f} (H, r)$ as the infimum of m(G) over all graphs G such that G → (H,r). In the first part of this paper we show that when H is a complete graph Kₖ on k vertices, then $m_{i n f} (H, r) = (R - 1) / 2$ , where R = R(k;r) is the classical Ramsey number. As a corollary we derive a new proof of the result credited...

Low-discrepancy point sets for non-uniform measures

Christoph Aistleitner, Josef Dick (2014)

Acta Arithmetica

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We prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure μ on the d-dimensional unit cube. We improve a theorem of Beck, by showing that for any d ≥ 1, N ≥ 1, and any non-negative, normalized Borel measure μ on ${[0, 1]}^{d}$ there exists a point set $x_{1}, . . ., x_{N} \in {[0, 1]}^{d}$ whose star-discrepancy with respect to μ is of order $D_{N} * (x_{1}, . . ., x_{N}; μ) ≪ ({(l o g N)}^{(3 d + 1) / 2}) / N$ . For the proof we use a theorem of Banaszczyk concerning the balancing of vectors, which implies an upper bound for the linear...