On the connectivity of finite subset spaces

Jacob Mostovoy; Rustam Sadykov

Displaying similar documents to “On the connectivity of finite subset spaces”

On D-connected sets in the space $V^{q}$

S. Topa (1976)

Annales Polonici Mathematici

Similarity:

Does the endomorphism poset $P^{P}$ determine whether a finite poset $P$ is connected? An issue Duffus raised in 1978

Jonathan David Farley (2023)

Mathematica Bohemica

Similarity:

Duffus wrote in his 1978 Ph.D. thesis, “It is not obvious that $P$ is connected and $P^{P} ≅ Q^{Q}$ imply that $Q$ is connected”, where $P$ and $Q$ are finite nonempty posets. We show that, indeed, under these hypotheses $Q$ is connected and $P ≅ Q$ .

Domination numbers in graphs with removed edge or set of edges

Magdalena Lemańska (2005)

Discussiones Mathematicae Graph Theory

Similarity:

It is known that the removal of an edge from a graph G cannot decrease a domination number γ(G) and can increase it by at most one. Thus we can write that γ(G) ≤ γ(G-e) ≤ γ(G)+1 when an arbitrary edge e is removed. Here we present similar inequalities for the weakly connected domination number $γ_{w}$ and the connected domination number $γ_{c}$ , i.e., we show that $γ_{w} (G) \leq γ_{w} (G - e) \leq γ_{w} (G) + 1$ and $γ_{c} (G) \leq γ_{c} (G - e) \leq γ_{c} (G) + 2$ if G and G-e are connected. Additionally we show that $γ_{w} (G) \leq γ_{w} (G - E ₚ) \leq γ_{w} (G) + p - 1$ and $γ_{c} (G) \leq γ_{c} (G - E ₚ) \leq γ_{c} (G) + 2 p - 2$ if G and G - Eₚ are connected and Eₚ = E(Hₚ) where Hₚ of order...

On the domination of triangulated discs

Noor A&amp;#039;lawiah Abd Aziz, Nader Jafari Rad, Hailiza Kamarulhaili (2023)

Mathematica Bohemica

Similarity:

Let $G$ be a $3$ -connected triangulated disc of order $n$ with the boundary cycle $C$ of the outer face of $G$ . Tokunaga (2013) conjectured that $G$ has a dominating set of cardinality at most $\frac{1}{4} (n + 2)$ . This conjecture is proved in Tokunaga (2020) for $G - C$ being a tree. In this paper we prove the above conjecture for $G - C$ being a unicyclic graph. We also deduce some bounds for the double domination number, total domination number and double total domination number in triangulated discs.

On compactness and connectedness of the paratingent

Wojciech Zygmunt (2016)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

In this note we shall prove that for a continuous function $ϕ : Δ \to ℝ^{n}$ , where $Δ \subset ℝ$ , the paratingent of $ϕ$ at $a \in Δ$ is a non-empty and compact set in $ℝ^{n}$ if and only if $ϕ$ satisfies Lipschitz condition in a neighbourhood of $a$ . Moreover, in this case the paratingent is a connected set.

Making holes in the cone, suspension and hyperspaces of some continua

José G. Anaya, Enrique Castañeda-Alvarado, Alejandro Fuentes-Montes de Oca, Fernando Orozco-Zitli (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

A connected topological space $Z$ is unicoherent provided that if $Z = A \cup B$ where $A$ and $B$ are closed connected subsets of $Z$ , then $A \cap B$ is connected. Let $Z$ be a unicoherent space, we say that $z \in Z$ makes a hole in $Z$ if $Z - {z}$ is not unicoherent. In this work the elements that make a hole to the cone and the suspension of a metric space are characterized. We apply this to give the classification of the elements of hyperspaces of some continua that make them hole.

Proper connection number of bipartite graphs

Jun Yue, Meiqin Wei, Yan Zhao (2018)

Czechoslovak Mathematical Journal

Similarity:

An edge-colored graph $G$ is proper connected if every pair of vertices is connected by a proper path. The proper connection number of a connected graph $G$ , denoted by $pc (G)$ , is the smallest number of colors that are needed to color the edges of $G$ in order to make it proper connected. In this paper, we obtain the sharp upper bound for $pc (G)$ of a general bipartite graph $G$ and a series of extremal graphs. Additionally, we give a proper $2$ -coloring for a connected bipartite graph $G$ having $δ (G) \geq 2$ and a dominating...

On connectivity properties of hyperspaces of $R_{0}$ spaces

M. Mršević (1979)

Matematički Vesnik

Similarity:

On the bounds of Laplacian eigenvalues of $k$ -connected graphs

Xiaodan Chen, Yaoping Hou (2015)

Czechoslovak Mathematical Journal

Similarity:

Let $μ_{n - 1} (G)$ be the algebraic connectivity, and let $μ_{1} (G)$ be the Laplacian spectral radius of a $k$ -connected graph $G$ with $n$ vertices and $m$ edges. In this paper, we prove that $μ_{n - 1} (G) \geq \frac{2 n k^{2}}{(n (n - 1) - 2 m) (n + k - 2) + 2 k^{2}},$ with equality if and only if $G$ is the complete graph $K_{n}$ or $K_{n} - e$ . Moreover, if $G$ is non-regular, then $μ_{1} (G) < 2 Δ - \frac{2 (n Δ - 2 m) k^{2}}{2 (n Δ - 2 m) (n^{2} - 2 n + 2 k) + n k^{2}},$ where $Δ$ stands for the maximum degree of $G$ . Remark that in some cases, these two inequalities improve some previously known results.

Thompson’s conjecture for the alternating group of degree $2 p$ and $2 p + 1$

Azam Babai, Ali Mahmoudifar (2017)

Czechoslovak Mathematical Journal

Similarity:

For a finite group $G$ denote by $N (G)$ the set of conjugacy class sizes of $G$ . In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N (G) = N (L)$ , then $G ≅ L$ . We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z (G) = 1$ and $N (G) = N (A_{i})$ is necessarily isomorphic to $A_{i}$ , where $i \in {2 p, 2 p + 1}$ .

On maps preserving connectedness and/or compactness

István Juhász, Jan van Mill (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We call a function $f : X \to Y$ P-preserving if, for every subspace $A \subset X$ with property P, its image $f (A)$ also has property P. Of course, all continuous maps are both compactness- and connectedness-preserving and the natural question about when the converse of this holds, i.e. under what conditions such a map is continuous, has a long history. Our main result is that any nontrivial product function, i.e. one having at least two nonconstant factors, that has connected domain, $T_{1}$ range, and is connectedness-preserving...