Complete pluripolar curves and graphs

Tomas Edlund

Displaying similar documents to “Complete pluripolar curves and graphs”

Edit distance measure for graphs

Tomasz Dzido, Krzysztof Krzywdziński (2015)

Czechoslovak Mathematical Journal

Similarity:

In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for $g (n, l)$ , the biggest number $k$ guaranteeing that there exist $l$ graphs on $n$ vertices, each two having edit distance at least $k$ . By edit distance of two graphs $G$ , $F$ we mean the number of edges needed to be added to or deleted from graph $G$ to obtain graph $F$ . This new extremal number $g (n, l)$ is closely linked to the edit distance of graphs. Using probabilistic methods we show...

Intrinsic linking and knotting are arbitrarily complex

Erica Flapan, Blake Mellor, Ramin Naimi (2008)

Fundamenta Mathematicae

Similarity:

We show that, given any n and α, any embedding of any sufficiently large complete graph in ℝ³ contains an oriented link with components Q₁, ..., Qₙ such that for every i ≠ j, $| l k (Q_{i}, Q_{j}) | \geq α$ and $| a ₂ (Q_{i}) | \geq α$ , where $a ₂ (Q_{i})$ denotes the second coefficient of the Conway polynomial of $Q_{i}$ .

Remarks on $D$ -integral complete multipartite graphs

Pavel Híc, Milan Pokorný (2016)

Czechoslovak Mathematical Journal

Similarity:

A graph is called distance integral (or $D$ -integral) if all eigenvalues of its distance matrix are integers. In their study of $D$ -integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on $D$ -integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs $K_{p_{1}, p_{2}, p_{3}}$ with $p_{1} < p_{2} < p_{3}$ , and $K_{p_{1}, p_{2}, p_{3}, p_{4}}$ with $p_{1} < p_{2} < p_{3} < p_{4}$ , as well as the infinite classes of distance integral...

Rational points on $X_{0}^{+} (p^{r})$

Yuri Bilu, Pierre Parent, Marusia Rebolledo (2013)

Annales de l’institut Fourier

Similarity:

Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of $X_{0}^{+} (p^{r}) (ℚ)$ , for $r > 1$ and $p$ a prime number exceeding $2 \cdot 10^{11}$ . This includes the case of the curves $X_{split} (p)$ . We then prove, with the help of computer calculations, that the same holds true for $p$ in the range $11 \leq p \leq 10^{14}$ , $p \neq 13$ . The combination of those results completes the qualitative study of rational points on $X_{0}^{+} (p^{r})$ undertook in our previous work, with the only exception of $p^{r} = 13^{2}$ .

Complete pluripolar graphs in $ℂ^{N}$

Nguyen Quang Dieu, Phung Van Manh (2014)

Annales Polonici Mathematici

Similarity:

Let F be the Cartesian product of N closed sets in ℂ. We prove that there exists a function g which is continuous on F and holomorphic on the interior of F such that $Γ_{g} (F) : = (z, g (z)) : z \in F$ is complete pluripolar in $ℂ^{N + 1}$ . Using this result, we show that if D is an analytic polyhedron then there exists a bounded holomorphic function g such that $Γ_{g} (D)$ is complete pluripolar in $ℂ^{N + 1}$ . These results are high-dimensional analogs of the previous ones due to Edlund [Complete pluripolar curves and graphs, Ann. Polon. Math....

Note on improper coloring of $1$ -planar graphs

Yanan Chu, Lei Sun, Jun Yue (2019)

Czechoslovak Mathematical Journal

Similarity:

A graph $G = (V, E)$ is called improperly $(d_{1}, \dots, d_{k})$ -colorable if the vertex set $V$ can be partitioned into subsets $V_{1}, \dots, V_{k}$ such that the graph $G [V_{i}]$ induced by the vertices of $V_{i}$ has maximum degree at most $d_{i}$ for all $1 \leq i \leq k$ . In this paper, we mainly study the improper coloring of $1$ -planar graphs and show that $1$ -planar graphs with girth at least $7$ are $(2, 0, 0, 0)$ -colorable.

Saturation numbers for linear forests $P_{6} + t P_{2}$

Jingru Yan (2023)

Czechoslovak Mathematical Journal

Similarity:

A graph $G$ is $H$ -saturated if it contains no $H$ as a subgraph, but does contain $H$ after the addition of any edge in the complement of $G$ . The saturation number, $sat (n, H)$ , is the minimum number of edges of a graph in the set of all $H$ -saturated graphs of order $n$ . We determine the saturation number $sat (n, P_{6} + t P_{2})$ for $n \geq \frac{10}{3} t + 10$ and characterize the extremal graphs for $n > \frac{10}{3} t + 20$ .

Embedding products of graphs into Euclidean spaces

Mikhail Skopenkov (2003)

Fundamenta Mathematicae

Similarity:

For any collection of graphs $G ₁, . . ., G_{N}$ we find the minimal dimension d such that the product $G ₁ \times . . . \times G_{N}$ is embeddable into $ℝ^{d}$ (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and $(K_{3, 3}) ⁿ$ are not embeddable into $ℝ^{2 n}$ , where K₅ and $K_{3, 3}$ are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is a reduction to a problem from so-called Ramsey link theory: we show that any embedding $L k O \to S^{2 n - 1}$ , where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.

Generalized connectivity of some total graphs

Yinkui Li, Yaping Mao, Zhao Wang, Zongtian Wei (2021)

Czechoslovak Mathematical Journal

Similarity:

We study the generalized $k$ -connectivity $κ_{k} (G)$ as introduced by Hager in 1985, as well as the more recently introduced generalized $k$ -edge-connectivity $λ_{k} (G)$ . We determine the exact value of $κ_{k} (G)$ and $λ_{k} (G)$ for the line graphs and total graphs of trees, unicyclic graphs, and also for complete graphs for the case $k = 3$ .

Note on a conjecture for the sum of signless Laplacian eigenvalues

Xiaodan Chen, Guoliang Hao, Dequan Jin, Jingjian Li (2018)

Czechoslovak Mathematical Journal

Similarity:

For a simple graph $G$ on $n$ vertices and an integer $k$ with $1 \leq k \leq n$ , denote by $𝒮_{k}^{+} (G)$ the sum of $k$ largest signless Laplacian eigenvalues of $G$ . It was conjectured that $𝒮_{k}^{+} (G) \leq e (G) + (\binom{k + 1}{2})$ , where $e (G)$ is the number of edges of $G$ . This conjecture has been proved to be true for all graphs when $k \in {1, 2, n - 1, n}$ , and for trees, unicyclic graphs, bicyclic graphs and regular graphs (for all $k$ ). In this note, this conjecture is proved to be true for all graphs when $k = n - 2$ , and for some new classes of graphs.

Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

Nazar Arakelian, Herivelto Borges (2015)

Acta Arithmetica

Similarity:

For each integer s ≥ 1, we present a family of curves that are $_{q}$ -Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case s=2, we give necessary and sufficient conditions for such curves to be $_{q}$ -Frobenius nonclassical with respect to the linear system of conics. In the $_{q}$ -Frobenius nonclassical cases, we determine the exact number of $_{q}$ -rational points. In the remaining cases, an upper bound for the number of $_{q}$ -rational points will follow from Stöhr-Voloch...

On ramified covers of the projective plane II: Generalizing Segre’s theory

Michael Friedman, Rebecca Lehman, Maxim Leyenson, Mina Teicher (2012)

Journal of the European Mathematical Society

Similarity:

The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in $ℙ^{3}$ . We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, $B$ and $E$ , we give a necessary and sufficient condition for $B$ to be the branch curve of a surface $X$ in $ℙ^{N}$ and $E$ to be the image of the double curve of a $ℙ^{3}$ -model of $X$ . In the classical Segre theory, a...

On distinguishing and distinguishing chromatic numbers of hypercubes

Werner Klöckl (2008)

Discussiones Mathematicae Graph Theory

Similarity:

The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d colors that is not preserved by any nontrivial automorphism. The restriction to proper labelings leads to the definition of the distinguishing chromatic number $χ_{D} (G)$ of G. Extending these concepts to infinite graphs we prove that $D (Q_{ℵ} ₀) = 2$ and $χ_{D} (Q_{ℵ} ₀) = 3$ , where $Q_{ℵ} ₀$ denotes the hypercube of countable dimension. We also show that $χ_{D} (Q ₄) = 4$ , thereby completing the investigation of finite hypercubes with respect to $χ_{D}$ . Our...

Degree sums of adjacent vertices for traceability of claw-free graphs

Tao Tian, Liming Xiong, Zhi-Hong Chen, Shipeng Wang (2022)

Czechoslovak Mathematical Journal

Similarity:

The line graph of a graph $G$ , denoted by $L (G)$ , has $E (G)$ as its vertex set, where two vertices in $L (G)$ are adjacent if and only if the corresponding edges in $G$ have a vertex in common. For a graph $H$ , define ${\bar{σ}}_{2} (H) = min {d (u) + d (v) : u v \in E (H)}$ . Let $H$ be a 2-connected claw-free simple graph of order $n$ with $δ (H) \geq 3$ . We show that, if ${\bar{σ}}_{2} (H) \geq \frac{1}{7} (2 n - 5)$ and $n$ is sufficiently large, then either $H$ is traceable or the Ryjáček’s closure $cl (H) = L (G)$ , where $G$ is an essentially $2$ -edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10...

On the multiplicity of Laplacian eigenvalues for unicyclic graphs

Fei Wen, Qiongxiang Huang (2022)

Czechoslovak Mathematical Journal

Similarity:

Let $G$ be a connected graph of order $n$ and $U$ a unicyclic graph with the same order. We firstly give a sharp bound for $m_{G} (μ)$ , the multiplicity of a Laplacian eigenvalue $μ$ of $G$ . As a straightforward result, $m_{U} (1) \leq n - 2$ . We then provide two graph operations (i.e., grafting and shifting) on graph $G$ for which the value of $m_{G} (1)$ is nondecreasing. As applications, we get the distribution of $m_{U} (1)$ for unicyclic graphs on $n$ vertices. Moreover, for the two largest possible values of $m_{U} (1) \in {n - 5, n - 3}$ , the corresponding graphs $U$ are...