scholarly journals Tiling Tripartite Graphs with $3$-Colorable Graphs

10.37236/198 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Ryan Martin ◽  
Yi Zhao
Keyword(s):  
Positive Integer ◽  
Positive Real Number ◽  
Minimum Degree ◽  
Discrete Math ◽  
The Other ◽  
Sufficient Condition ◽  
Positive Real ◽  
Tripartite Graph ◽  
Colorable Graph ◽  
Tripartite Graphs

For any positive real number $\gamma$ and any positive integer $h$, there is $N_0$ such that the following holds. Let $N\ge N_0$ be such that $N$ is divisible by $h$. If $G$ is a tripartite graph with $N$ vertices in each vertex class such that every vertex is adjacent to at least $(2/3+ \gamma) N$ vertices in each of the other classes, then $G$ can be tiled perfectly by copies of $K_{h,h,h}$. This extends the work in [Discrete Math. 254 (2002), 289–308] and also gives a sufficient condition for tiling by any fixed 3-colorable graph. Furthermore, we show that the minimum-degree $(2/3+ \gamma) N$ in our result cannot be replaced by $2N/3+ h-2$.

scholarly journals Detection number of bipartite graphs and cubic graphs

2014 ◽  
Vol Vol. 16 no. 3 ◽  
Author(s):  
Frederic Havet ◽  
Nagarajan Paramaguru ◽  
Rathinaswamy Sampathkumar
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Connected Graph ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
International Audience ◽  
Np Complete

International audience For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.

scholarly journals GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

2018 ◽  
Vol 107 (02) ◽  
pp. 272-288
Author(s):  
TOPI TÖRMÄ
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Rational Numbers ◽  
Positive Real ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Continued Fraction Expansions ◽  
Positive Integers

We study generalized continued fraction expansions of the form $$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$ where $N$ is a fixed positive integer and the partial numerators $a_{i}$ are positive integers for all $i$ . We call these expansions $\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many $\operatorname{dn}_{N}$ expansions for each $N$ . In particular, we study the $\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each $N$ , a $\operatorname{dn}_{N}$ expansion with bounded partial numerators.

scholarly journals The product ofr−kand∇δonℝm

2000 ◽  
Vol 24 (6) ◽  
pp. 361-369 ◽  
Author(s):  
C. K. Li
Keyword(s):  
Positive Integer ◽  
Positive Real Number ◽  
Dimensional Space ◽  
Particle Interactions ◽  
Transition Rates ◽  
Normalization Procedure ◽  
Positive Real ◽  
Need To Evaluate ◽  
Delta Sequence ◽  
Neutrix Product

In the theory of distributions, there is a general lack of definitions for products and powers of distributions. In physics (Gasiorowicz (1967), page 141), one finds the need to evaluateδ2when calculating the transition rates of certain particle interactions and using some products such as(1/x)⋅δ. In 1990, Li and Fisher introduced a “computable” delta sequence in anm-dimensional space to obtain a noncommutative neutrix product ofr−kandΔδ(Δdenotes the Laplacian) for any positive integerkbetween 1 andm−1inclusive. Cheng and Li (1991) utilized a netδϵ(x)(similar to theδn(x)) and the normalization procedure ofμ(x)x+λto deduce a commutative neutrix product ofr−kandδfor any positive real numberk. The object of this paper is to apply Pizetti's formula and the normalization procedure to derive the product ofr−kand∇δ(∇is the gradient operator) onℝm. The nice properties of theδ-sequence are fully shown and used in the proof of our theorem.

scholarly journals On the average distance property in finite dimensional real Banach spaces

1995 ◽  
Vol 51 (1) ◽  
pp. 87-101 ◽  
Author(s):  
Reinhard Wolf
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Positive Real Number ◽  
Real Banach Space ◽  
Average Distance ◽  
Positive Real ◽  
Finite Dimensional ◽  
Distance Property ◽  
Average Distance Property ◽  
Image Position

The average distance Theorem of Gross implies that for each N-dimensional real Banach space E (N ≥ 2) there is a unique positive real number r(E) with the following property: for each positive integer n and for all (not necessarily distinct) x1, x2, …, xn, in E with ‖x1‖ = ‖x2‖ = … = ‖xn‖ = 1, there exists an x in E with ‖x‖ = 1 such that.In this paper we prove that if E has a 1-unconditional basis then r(E)≤2−(l/N) and equality holds if and only if E is isometrically isomorphic to Rn equipped with the usual 1-norm.

scholarly journals A SPARSITY RESULT FOR THE DYNAMICAL MORDELL–LANG CONJECTURE IN POSITIVE CHARACTERISTIC

2021 ◽  
pp. 1-10
Author(s):  
DRAGOS GHIOCA ◽  
ALINA OSTAFE ◽  
SINA SALEH ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Positive Real Number ◽  
Positive Characteristic ◽  
Arithmetic Progressions ◽  
Partial Result ◽  
Positive Real

Abstract We prove a quantitative partial result in support of the dynamical Mordell–Lang conjecture (also known as the DML conjecture) in positive characteristic. More precisely, we show the following: given a field K of characteristic p, a semiabelian variety X defined over a finite subfield of K and endowed with a regular self-map $\Phi :X{\longrightarrow } X$ defined over K, a point $\alpha \in X(K)$ and a subvariety $V\subseteq X$ , then the set of all nonnegative integers n such that $\Phi ^n(\alpha )\in V(K)$ is a union of finitely many arithmetic progressions along with a subset S with the property that there exists a positive real number A (depending only on X, $\Phi $ , $\alpha $ and V) such that for each positive integer M, $$\begin{align*}\scriptsize\#\{n\in S\colon n\le M\}\le A\cdot (1+\log M)^{\dim V}.\end{align*}$$

scholarly journals On 1-rotational decompositions of complete graphs into tripartite graphs

2019 ◽  
Vol 39 (5) ◽  
pp. 623-643
Author(s):  
Ryan C. Bunge
Keyword(s):  
Positive Integer ◽  
Simple Graph ◽  
Vertex Coloring ◽  
Complete Graphs ◽  
Tripartite Graph ◽  
Tripartite Graphs ◽  
Proper Vertex

Consider a tripartite graph to be any simple graph that admits a proper vertex coloring in at most 3 colors. Let \(G\) be a tripartite graph with \(n\) edges, one of which is a pendent edge. This paper introduces a labeling on such a graph \(G\) used to achieve 1-rotational \(G\)-decompositions of \(K_{2nt}\) for any positive integer \(t\). It is also shown that if \(G\) with a pendent edge is the result of adding an edge to a path on \(n\) vertices, then \(G\) admits such a labeling.

scholarly journals On $(\delta, \chi)$-Bounded Families of Graphs

10.37236/595 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
András Gyárfás ◽  
Manouchehr Zaker
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Minimum Degree ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Induced Subgraph ◽  
The Family ◽  
Infinite Collection ◽  
Delta G ◽  
Necessary And Sufficient

A family ${\mathcal{F}}$ of graphs is said to be $(\delta,\chi)$-bounded if there exists a function $f(x)$ satisfying $f(x)\rightarrow \infty$ as $x\rightarrow \infty$, such that for any graph $G$ from the family, one has $f(\delta(G))\leq \chi(G)$, where $\delta(G)$ and $\chi(G)$ denotes the minimum degree and chromatic number of $G$, respectively. Also for any set $\{H_1, H_2, \ldots, H_k\}$ of graphs by $Forb(H_1, H_2, \ldots, H_k)$ we mean the class of graphs that contain no $H_i$ as an induced subgraph for any $i=1, \ldots, k$. In this paper we first answer affirmatively the question raised by the second author by showing that for any tree $T$ and positive integer $\ell$, $Forb(T, K_{\ell, \ell})$ is a $(\delta, \chi)$-bounded family. Then we obtain a necessary and sufficient condition for $Forb(H_1, H_2, \ldots, H_k)$ to be a $(\delta, \chi)$-bounded family, where $\{H_1, H_2, \ldots, H_k\}$ is any given set of graphs. Next we study $(\delta, \chi)$-boundedness of $Forb({\mathcal{C}})$ where ${\mathcal{C}}$ is an infinite collection of graphs. We show that for any positive integer $\ell$, $Forb(K_{\ell,\ell}, C_6, C_8, \ldots)$ is $(\delta, \chi)$-bounded. Finally we show a similar result when ${\mathcal{C}}$ is a collection consisting of unicyclic graphs.

The Order of Monochromatic Subgraphs with a Given Minimum Degree

10.37236/1725 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

Some remarks on the solvability of non-local elliptic problems with the Hardy potential

2014 ◽  
Vol 16 (04) ◽  
pp. 1350046 ◽  
Author(s):  
B. Barrios ◽  
M. Medina ◽  
I. Peral
Keyword(s):  
Real Number ◽  
Sobolev Inequality ◽  
Positive Real Number ◽  
Fractional Power ◽  
Existence Of Solution ◽  
Sharp Constant ◽  
Hardy Potential ◽  
Positive Real ◽  
Existence And Multiplicity ◽  
Non Local

The aim of this paper is to study the solvability of the following problem, [Formula: see text] where (-Δ)s, with s ∈ (0, 1), is a fractional power of the positive operator -Δ, Ω ⊂ ℝN, N > 2s, is a Lipschitz bounded domain such that 0 ∈ Ω, μ is a positive real number, λ < ΛN,s, the sharp constant of the Hardy–Sobolev inequality, 0 < q < 1 and [Formula: see text], with αλ a parameter depending on λ and satisfying [Formula: see text]. We will discuss the existence and multiplicity of solutions depending on the value of p, proving in particular that p(λ, s) is the threshold for the existence of solution to problem (Pμ).

Sign in / Sign up

Export Citation Format

Share Document