EUDML | The instability of nonseparable complete Erdős spaces and representations in ℝ-trees

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Displaying similar documents to “The instability of nonseparable complete Erdős spaces and representations in ℝ-trees”

The tree property at both $ℵ_{ω + 1}$ and $ℵ_{ω + 2}$

Laura Fontanella, Sy David Friedman (2015)

Fundamenta Mathematicae

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We force from large cardinals a model of ZFC in which $ℵ_{ω + 1}$ and $ℵ_{ω + 2}$ both have the tree property. We also prove that if we strengthen the large cardinal assumptions, then in the final model $ℵ_{ω + 2}$ even satisfies the super tree property.

A lower bound for the 3-pendant tree-connectivity of lexicographic product graphs

Yaping Mao, Christopher Melekian, Eddie Cheng (2023)

Czechoslovak Mathematical Journal

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For a connected graph $G = (V, E)$ and a set $S \subseteq V (G)$ with at least two vertices, an $S$ -Steiner tree is a subgraph $T = (V^{'}, E^{'})$ of $G$ that is a tree with $S \subseteq V^{'}$ . If the degree of each vertex of $S$ in $T$ is equal to 1, then $T$ is called a pendant $S$ -Steiner tree. Two $S$ -Steiner trees are if they share no vertices other than $S$ and have no edges in common. For $S \subseteq V (G)$ and $| S | \geq 2$ , the pendant tree-connectivity $τ_{G} (S)$ is the maximum number of internally disjoint pendant $S$ -Steiner trees in $G$ , and for $k \geq 2$ , the $k$ -pendant tree-connectivity $τ_{k} (G)$ is the...

The relation between the number of leaves of a tree and its diameter

Pu Qiao, Xingzhi Zhan (2022)

Czechoslovak Mathematical Journal

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Let $L (n, d)$ denote the minimum possible number of leaves in a tree of order $n$ and diameter $d .$ Lesniak (1975) gave the lower bound $B (n, d) = ⌈ 2 (n - 1) / d ⌉$ for $L (n, d) .$ When $d$ is even, $B (n, d) = L (n, d) .$ But when $d$ is odd, $B (n, d)$ is smaller than $L (n, d)$ in general. For example, $B (21, 3) = 14$ while $L (21, 3) = 19 .$ In this note, we determine $L (n, d)$ using new ideas. We also consider the converse problem and determine the minimum possible diameter of a tree with given order and number of leaves.

On a characterization of $k$ -trees

De-Yan Zeng, Jian Hua Yin (2015)

Czechoslovak Mathematical Journal

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A graph $G$ is a $k$ -tree if either $G$ is the complete graph on $k + 1$ vertices, or $G$ has a vertex $v$ whose neighborhood is a clique of order $k$ and the graph obtained by removing $v$ from $G$ is also a $k$ -tree. Clearly, a $k$ -tree has at least $k + 1$ vertices, and $G$ is a 1-tree (usual tree) if and only if it is a $1$ -connected graph and has no $K_{3}$ -minor. In this paper, motivated by some properties of 2-trees, we obtain a characterization of $k$ -trees as follows: if $G$ is a graph with at least $k + 1$ vertices, then $G$ is...

Quasi-tree graphs with the minimal Sombor indices

Yibo Li, Huiqing Liu, Ruiting Zhang (2022)

Czechoslovak Mathematical Journal

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The Sombor index $S O (G)$ of a graph $G$ is the sum of the edge weights $\sqrt{d_{G}^{2} (u) + d_{G}^{2} (v)}$ of all edges $u v$ of $G$ , where $d_{G} (u)$ denotes the degree of the vertex $u$ in $G$ . A connected graph $G = (V, E)$ is called a quasi-tree if there exists $u \in V (G)$ such that $G - u$ is a tree. Denote $𝒬 (n, k) = {G : G$ is a quasi-tree graph of order $n$ with $G - u$ being a tree and $d_{G} (u) = k}$ . We determined the minimum and the second minimum Sombor indices of all quasi-trees in $𝒬 (n, k)$ . Furthermore, we characterized the corresponding extremal graphs, respectively.

Spanning trees whose reducible stems have a few branch vertices

Pham Hoang Ha, Dang Dinh Hanh, Nguyen Thanh Loan, Ngoc Diep Pham (2021)

Czechoslovak Mathematical Journal

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Let $T$ be a tree. Then a vertex of $T$ with degree one is a leaf of $T$ and a vertex of degree at least three is a branch vertex of $T$ . The set of leaves of $T$ is denoted by $L (T)$ and the set of branch vertices of $T$ is denoted by $B (T)$ . For two distinct vertices $u$ , $v$ of $T$ , let $P_{T} [u, v]$ denote the unique path in $T$ connecting $u$ and $v .$ Let $T$ be a tree with $B (T) \neq \emptyset$ . For each leaf $x$ of $T$ , let $y_{x}$ denote the nearest branch vertex to $x$ . We delete $V (P_{T} [x, y_{x}]) ∖ {y_{x}}$ from $T$ for all $x \in L (T)$ . The resulting subtree of $T$ is called the reducible stem...

On graceful colorings of trees

Sean English, Ping Zhang (2017)

Mathematica Bohemica

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A proper coloring $c : V (G) \to {1, 2, ..., k}$ , $k \geq 2$ of a graph $G$ is called a graceful $k$ -coloring if the induced edge coloring $c^{'} : E (G) \to {1, 2, ..., k - 1}$ defined by $c^{'} (u v) = | c (u) - c (v) |$ for each edge $u v$ of $G$ is also proper. The minimum integer $k$ for which $G$ has a graceful $k$ -coloring is the graceful chromatic number $χ_{g} (G)$ . It is known that if $T$ is a tree with maximum degree $Δ$ , then $χ_{g} (T) \leq ⌈ \frac{5}{3} Δ ⌉$ and this bound is best possible. It is shown for each integer $Δ \geq 2$ that there is an infinite class of trees $T$ with maximum degree $Δ$ such that $χ_{g} (T) = ⌈ \frac{5}{3} Δ ⌉$ . In particular, we investigate for each...

Shadow trees of Mandelbrot sets

Virpi Kauko (2003)

Fundamenta Mathematicae

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The topology and combinatorial structure of the Mandelbrot set $ℳ^{d}$ (of degree d ≥ 2) can be studied using symbolic dynamics. Each parameter is mapped to a kneading sequence, or equivalently, an internal address; but not every such sequence is realized by a parameter in $ℳ^{d}$ . Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, $Λ^{d}$ . In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence...

Iterating along a Prikry sequence

Spencer Unger (2016)

Fundamenta Mathematicae

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We introduce a new method which combines Prikry forcing with an iteration between the Prikry points. Using our method we prove from large cardinals that it is consistent that the tree property holds at ℵₙ for n ≥ 2, $ℵ_{ω}$ is strong limit and $2^{ℵ_{ω}} = ℵ_{ω + 2}$ .

On non-normality points, Tychonoff products and Suslin number

Sergei Logunov (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let a space $X$ be Tychonoff product $\prod_{α < τ} X_{α}$ of $τ$ -many Tychonoff nonsingle point spaces $X_{α}$ . Let Suslin number of $X$ be strictly less than the cofinality of $τ$ . Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $β X$ . In particular, this is true if $X$ is either $R^{τ}$ or $ω^{τ}$ and a cardinal $τ$ is infinite and not countably cofinal.

On operators which factor through $l_{p}$ or c₀

Bentuo Zheng (2006)

Studia Mathematica

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Let 1 < p < ∞. Let X be a subspace of a space Z with a shrinking F.D.D. (Eₙ) which satisfies a block lower-p estimate. Then any bounded linear operator T from X which satisfies an upper-(C,p)-tree estimate factors through a subspace of ${(\sum F ₙ)}_{l_{p}}$ , where (Fₙ) is a blocking of (Eₙ). In particular, we prove that an operator from $L_{p}$ (2 < p < ∞) satisfies an upper-(C,p)-tree estimate if and only if it factors through $l_{p}$ . This gives an answer to a question of W. B. Johnson. We also prove...

Spaces with property $(D C (ω_{1}))$

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove that if $X$ is a first countable space with property $(D C (ω_{1}))$ and with a $G_{δ}$ -diagonal then the cardinality of $X$ is at most $𝔠$ . We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$ .

$ℓ_{\infty}$ -sums and the Banach space $ℓ_{\infty} / c ₀$

Christina Brech, Piotr Koszmider (2014)

Fundamenta Mathematicae

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This paper is concerned with the isomorphic structure of the Banach space $ℓ_{\infty} / c ₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that $ℓ_{\infty} / c ₀$ does not have an orthogonal $ℓ_{\infty}$ -decomposition, that is, it is not of the form $ℓ_{\infty} (X)$ for any Banach space X. The main local result is that it is consistent that $ℓ_{\infty} (c ₀ ())$ does not embed isomorphically into $ℓ_{\infty} / c ₀$ , where is the cardinality of the continuum,...

On γ-labelings of trees

Gary Chartrand, David Erwin, Donald W. VanderJagt, Ping Zhang (2005)

Discussiones Mathematicae Graph Theory

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Let G be a graph of order n and size m. A γ-labeling of G is a one-to-one function f:V(G) → 0,1,2,...,m that induces a labeling f’: E(G) → 1,2,...,m of the edges of G defined by f’(e) = |f(u)-f(v)| for each edge e = uv of G. The value of a γ-labeling f is $v a l (f) = Σ_{e \in E (G)} f^{'} K (e)$ . The maximum value of a γ-labeling of G is defined as $v a l_{m a x} (G) = m a x v a l (f) : f i s a γ - l a b e l i n g o f G$ ; while the minimum value of a γ-labeling of G is $v a l_{m i n} (G) = m i n v a l (f) : f i s a γ - l a b e l i n g o f G$ ; The values $v a l_{m a x} (S_{p, q})$ and $v a l_{m i n} (S_{p, q})$ are determined for double stars $S_{p, q}$ . We present characterizations of connected graphs G of order n for which...

Horocyclic products of trees

Laurent Bartholdi, Markus Neuhauser, Wolfgang Woess (2008)

Journal of the European Mathematical Society

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Let $T_{1}, \dots, T_{d}$ be homogeneous trees with degrees $q_{1} + 1, \dots, q_{d} + 1 \geq 3$ , respectively. For each tree, let $𝔥 : T_{j} \to ℤ$ be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of $T_{1}, \dots, T_{d}$ is the graph $𝖣𝖫 (q_{1}, \dots, q_{d})$ consisting of all $d$ -tuples $x_{1} \dots x_{d} \in T_{1} \times \dots \times T_{d}$ with $𝔥 (x_{1}) + \dots + 𝔥 (x_{d}) = 0$ , equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If $d = 2$ and $q_{1} = q_{2} = q$ then we obtain a Cayley graph...