Sharkovskiĭ's theorem holds for some discontinuous functions

Piotr Szuca

Displaying similar documents to “Sharkovskiĭ's theorem holds for some discontinuous functions”

Proper connection number of bipartite graphs

Jun Yue, Meiqin Wei, Yan Zhao (2018)

Czechoslovak Mathematical Journal

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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...

Baire one functions and their sets of discontinuity

Jonald P. Fenecios, Emmanuel A. Cabral, Abraham P. Racca (2016)

Mathematica Bohemica

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A characterization of functions in the first Baire class in terms of their sets of discontinuity is given. More precisely, a function $f : ℝ \to ℝ$ is of the first Baire class if and only if for each $ϵ > 0$ there is a sequence of closed sets ${C_{n}}_{n = 1}^{\infty}$ such that $D_{f} = ⋃_{n = 1}^{\infty} C_{n}$ and $ω_{f} (C_{n}) < ϵ$ for each $n$ where $ω_{f} (C_{n}) = sup {| f (x) - f (y) | : x, y \in C_{n}}$ and $D_{f}$ denotes the set of points of discontinuity of $f$ . The proof of the main theorem is based on a recent $ϵ$ - $δ$ characterization of Baire class one functions as well as on a well-known theorem due to Lebesgue. Some direct applications...

On sets of discontinuities of functions continuous on all lines

Luděk Zajíček (2022)

Commentationes Mathematicae Universitatis Carolinae

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Answering a question asked by K. C. Ciesielski and T. Glatzer in 2013, we construct a $C^{1}$ -smooth function $f$ on $[0, 1]$ and a closed set $M \subset graph f$ nowhere dense in $graph f$ such that there does not exist any linearly continuous function on $ℝ^{2}$ (i.e., function continuous on all lines) which is discontinuous at each point of $M$ . We substantially use a recent full characterization of sets of discontinuity points of linearly continuous functions on $ℝ^{n}$ proved by T. Banakh and O. Maslyuchenko in 2020. As an easy consequence...

A remark on functions continuous on all lines

Luděk Zajíček (2019)

Commentationes Mathematicae Universitatis Carolinae

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We prove that each linearly continuous function $f$ on $ℝ^{n}$ (i.e., each function continuous on all lines) belongs to the first Baire class, which answers a problem formulated by K. C. Ciesielski and D. Miller (2016). The same result holds also for $f$ on an arbitrary Banach space $X$ , if $f$ has moreover the Baire property. We also prove (extending a known finite-dimensional result) that such $f$ on a separable $X$ is continuous at all points outside a first category set which is also null in any usual...

Nonempty intersection of longest paths in a graph with a small matching number

Fuyuan Chen (2015)

Czechoslovak Mathematical Journal

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A maximum matching of a graph $G$ is a matching of $G$ with the largest number of edges. The matching number of a graph $G$ , denoted by $α^{'} (G)$ , is the number of edges in a maximum matching of $G$ . In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Although this conjecture has been disproved, finding some nice classes of graphs that support this conjecture is still very meaningful and interesting. In this short note, we prove that Gallai’s conjecture...

The extremal irregularity of connected graphs with given number of pendant vertices

Xiaoqian Liu, Xiaodan Chen, Junli Hu, Qiuyun Zhu (2022)

Czechoslovak Mathematical Journal

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The irregularity of a graph $G = (V, E)$ is defined as the sum of imbalances $| d_{u} - d_{v} |$ over all edges $u v \in E$ , where $d_{u}$ denotes the degree of the vertex $u$ in $G$ . This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. In this paper, we completely determine the extremal values of the irregularity of connected graphs with $n$ vertices and $p$ pendant vertices ( $1 \leq p \leq n - 1$ ), and characterize the corresponding extremal graphs.

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

Xiaodan Chen, Yaoping Hou (2015)

Czechoslovak Mathematical Journal

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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.

Insertion of a Contra-Baire- $1$ (Baire- $. 5$ ) Function

Majid Mirmiran (2019)

Communications in Mathematics

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Necessary and sufficient conditions in terms of lower cut sets are given for the insertion of a Baire- $. 5$ function between two comparable real-valued functions on the topological spaces that $F_{σ}$ -kernel of sets are $F_{σ}$ -sets.

On the signless Laplacian and normalized signless Laplacian spreads of graphs

Emina Milovanović, Serife B. Bozkurt Altindağ, Marjan Matejić, Igor Milovanović (2023)

Czechoslovak Mathematical Journal

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Let $G = (V, E)$ , $V = {v_{1}, v_{2}, ..., v_{n}}$ , be a simple connected graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $d_{1} \geq d_{2} \geq \dots \geq d_{n}$ . Denote by $A$ and $D$ the adjacency matrix and diagonal vertex degree matrix of $G$ , respectively. The signless Laplacian of $G$ is defined as $L^{+} = D + A$ and the normalized signless Laplacian matrix as $ℒ^{+} = D^{- 1 / 2} L^{+} D^{- 1 / 2}$ . The normalized signless Laplacian spreads of a connected nonbipartite graph $G$ are defined as $r (G) = γ_{2}^{+} / γ_{n}^{+}$ and $l (G) = γ_{2}^{+} - γ_{n}^{+}$ , where $γ_{1}^{+} \geq γ_{2}^{+} \geq \dots \geq γ_{n}^{+} \geq 0$ are eigenvalues of $ℒ^{+}$ . We establish sharp lower and upper bounds for the normalized signless...