Packing of nonuniform hypergraphs - product and sum of sizes conditions

Paweł Naroski

Displaying similar documents to “Packing of nonuniform hypergraphs - product and sum of sizes conditions”

The s-packing chromatic number of a graph

Wayne Goddard, Honghai Xu (2012)

Discussiones Mathematicae Graph Theory

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Let S = (a₁, a₂, ...) be an infinite nondecreasing sequence of positive integers. An S-packing k-coloring of a graph G is a mapping from V(G) to 1,2,...,k such that vertices with color i have pairwise distance greater than $a_{i}$ , and the S-packing chromatic number $χ_{S} (G)$ of G is the smallest integer k such that G has an S-packing k-coloring. This concept generalizes the concept of proper coloring (when S = (1,1,1,...)) and broadcast coloring (when S = (1,2,3,4,...)). In this paper, we consider...

Dimensions of non-differentiability points of Cantor functions

Yuanyuan Yao, Yunxiu Zhang, Wenxia Li (2009)

Studia Mathematica

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For a probability vector (p₀,p₁) there exists a corresponding self-similar Borel probability measure μ supported on the Cantor set C (with the strong separation property) in ℝ generated by a contractive similitude $h_{i} (x) = a_{i} x + b_{i}$ , i = 0,1. Let S denote the set of points of C at which the probability distribution function F(x) of μ has no derivative, finite or infinite. The Hausdorff and packing dimensions of S have been found by several authors for the case that $p_{i} > a_{i}$ , i = 0,1. However, when p₀ < a₀...

Some properties of packing measure with doubling gauge

Sheng-You Wen, Zhi-Ying Wen (2004)

Studia Mathematica

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Let g be a doubling gauge. We consider the packing measure $^{g}$ and the packing premeasure $₀^{g}$ in a metric space X. We first show that if $₀^{g} (X)$ is finite, then as a function of X, $₀^{g}$ has a kind of “outer regularity”. Then we prove that if X is complete separable, then $λ s u p ₀^{g} (F) \leq^{g} (B) \leq s u p ₀^{g} (F)$ for every Borel subset B of X, where the supremum is taken over all compact subsets of B having finite $₀^{g}$ -premeasure, and λ is a positive number depending only on the doubling gauge g. As an application, we show that for every doubling...

A note on the open packing number in graphs

Mehdi Mohammadi, Mohammad Maghasedi (2019)

Mathematica Bohemica

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A subset $S$ of vertices in a graph $G$ is an open packing set if no pair of vertices of $S$ has a common neighbor in $G$ . An open packing set which is not a proper subset of any open packing set is called a maximal open packing set. The maximum cardinality of an open packing set is called the open packing number and is denoted by $ρ^{o} (G)$ . A subset $S$ in a graph $G$ with no isolated vertex is called a total dominating set if any vertex of $G$ is adjacent to some vertex of $S$ . The total domination number...

Packing constant for Cesàro-Orlicz sequence spaces

Zhen-Hua Ma, Li-Ning Jiang, Qiao-Ling Xin (2016)

Czechoslovak Mathematical Journal

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The packing constant is an important and interesting geometric parameter of Banach spaces. Inspired by the packing constant for Orlicz sequence spaces, the main purpose of this paper is calculating the Kottman constant and the packing constant of the Cesàro-Orlicz sequence spaces ( ${ces}_{φ}$ ) defined by an Orlicz function $φ$ equipped with the Luxemburg norm. In order to compute the constants, the paper gives two formulas. On the base of these formulas one can easily obtain the packing constant...

Packing four copies of a tree into a complete bipartite graph

Liqun Pu, Yuan Tang, Xiaoli Gao (2022)

Czechoslovak Mathematical Journal

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In considering packing three copies of a tree into a complete bipartite graph, H. Wang (2009) gives a conjecture: For each tree $T$ of order $n$ and each integer $k \geq 2$ , there is a $k$ -packing of $T$ in a complete bipartite graph $B_{n + k - 1}$ whose order is $n + k - 1$ . We prove the conjecture is true for $k = 4$ .

Constructing universally small subsets of a given packing index in Polish groups

Taras Banakh, Nadya Lyaskovska (2011)

Colloquium Mathematicae

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A subset of a Polish space X is called universally small if it belongs to each ccc σ-ideal with Borel base on X. Under CH in each uncountable Abelian Polish group G we construct a universally small subset A₀ ⊂ G such that |A₀ ∩ gA₀| = for each g ∈ G. For each cardinal number κ ∈ [5,⁺] the set A₀ contains a universally small subset A of G with sharp packing index $p a c k ♯ (A_{κ}) = s u p | | ⁺ : \subset {g A}_{g \in G} i s d i s j o i n t$ equal to κ.

Perturbing the hexagonal circle packing: a percolation perspective

Itai Benjamini, Alexandre Stauffer (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the hexagonal circle packing with radius $1 / 2$ and perturb it by letting the circles move as independent Brownian motions for time $t$ . It is shown that, for large enough $t$ , if $𝛱_{t}$ is the point process given by the center of the circles at time $t$ , then, as $t \to \infty$ , the critical radius for circles centered at $𝛱_{t}$ to contain an infinite component converges to that of continuum percolation (which was shown – based on a Monte Carlo estimate – by Balister, Bollobás and Walters to be strictly...

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...

Continuous rearrangements of the Haar system in $H_{p}$ for 0 < p < ∞

Krzysztof Smela (2008)

Studia Mathematica

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We prove three theorems on linear operators $T_{τ, p} : H_{p} (ℬ) \to H_{p}$ induced by rearrangement of a subsequence of a Haar system. We find a sufficient and necessary condition for $T_{τ, p}$ to be continuous for 0 < p < ∞.

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....

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.

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...

Some results on packing in Orlicz sequence spaces

Y. Q. Yan (2001)

Studia Mathematica

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We present monotonicity theorems for index functions of N-fuctions, and obtain formulas for exact values of packing constants. In particular, we show that the Orlicz sequence space $l^{(N)}$ generated by the N-function N(v) = (1+|v|)ln(1+|v|) - |v| with Luxemburg norm has the Kottman constant $K (l^{(N)}) = N^{- 1} (1) / N^{- 1} (1 / 2)$ , which answers M. M. Rao and Z. D. Ren’s [8] problem.

On the asymptotics of counting functions for Ahlfors regular sets

Dušan Pokorný, Marc Rauch (2022)

Commentationes Mathematicae Universitatis Carolinae

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We deal with the so-called Ahlfors regular sets (also known as $s$ -regular sets) in metric spaces. First we show that those sets correspond to a certain class of tree-like structures. Building on this observation we then study the following question: Under which conditions does the limit ${lim}_{ε \to 0 +} ε^{s} N (ε, K)$ exist, where $K$ is an $s$ -regular set and $N (ε, K)$ is for instance the $ε$ -packing number of $K$ ?

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...

Characterization of local dimension functions of subsets of $ℝ^{d}$

L. Olsen (2005)

Colloquium Mathematicae

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For a subset $E \subseteq ℝ^{d}$ and $x \in ℝ^{d}$ , the local Hausdorff dimension function of E at x is defined by $d i m_{H, l o c} (x, E) = l i m_{r ↘ 0} d i m_{H} (E \cap B (x, r))$ where $d i m_{H}$ denotes the Hausdorff dimension. We give a complete characterization of the set of functions that are local Hausdorff dimension functions. In fact, we prove a significantly more general result, namely, we give a complete characterization of those functions that are local dimension functions of an arbitrary regular dimension index.

Mean value densities for temperatures

N. Suzuki, N. A. Watson (2003)

Colloquium Mathematicae

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A positive measurable function K on a domain D in $ℝ^{n + 1}$ is called a mean value density for temperatures if $u (0, 0) = \int \int_{D} K (x, t) u (x, t) d x d t$ for all temperatures u on D̅. We construct such a density for some domains. The existence of a bounded density and a density which is bounded away from zero on D is also discussed.