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.

Binomial subsampling

2006 ◽  
Vol 462 (2068) ◽  
pp. 1181-1195 ◽  
Author(s):  
Carsten Wiuf ◽  
Michael P.H Stumpf
Keyword(s):  
Mathematical Biology ◽  
Power Series ◽  
Generating Functions ◽  
Binomial Distribution ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Family ◽  
Necessary And Sufficient ◽  
Finite Moments

In this paper, we discuss statistical families with the property that if the distribution of a random variable X is in , then so is the distribution of Z ∼Bi( X ,  p ) for 0≤ p ≤1. (Here we take Z ∼Bi( X ,  p ) to mean that given X = x ,  Z is a draw from the binomial distribution Bi( x ,  p ).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probability generating functions and for families with finite moments of all orders we give a necessary and sufficient condition for the family to be closed under binomial subsampling. The results are illustrated with power series and other examples, and related to examples from mathematical biology. Finally, some issues concerning inference are discussed.

scholarly journals Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

A study on distance in graph complement

2020 ◽  
Vol 12 (03) ◽  
pp. 2050045
Author(s):  
A. Chellaram Malaravan ◽  
A. Wilson Baskar
Keyword(s):  
Positive Integer ◽  
Complete Graph ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

The aim of this paper is to determine radius and diameter of graph complements. We provide a necessary and sufficient condition for the complement of a graph to be connected, and determine the components of graph complement. Finally, we completely characterize the class of graphs [Formula: see text] for which the subgraph induced by central (respectively peripheral) vertices of its complement in [Formula: see text] is isomorphic to a complete graph [Formula: see text], for some positive integer [Formula: see text].

On the Structure of G(H, k)

2014 ◽  
Vol 21 (02) ◽  
pp. 317-330 ◽  
Author(s):  
Guixin Deng ◽  
Pingzhi Yuan
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Explicit Formula ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Directed Edge ◽  
Necessary And Sufficient

Let H be an abelian group written additively and k be a positive integer. Let G(H, k) denote the digraph whose set of vertices is just H, and there exists a directed edge from a vertex a to a vertex b if b = ka. In this paper we give a necessary and sufficient condition for G(H, k1) ≃ G(H, k2). We also discuss the problem when G(H1, k) is isomorphic to G(H2, k) for a given k. Moreover, we give an explicit formula of G(H, k) when H is a p-group and gcd (p, k)=1.

scholarly journals Dunkl harmonic analysis and fundamental sets of continuous functions on the unit sphere

2015 ◽  
Vol 6 (3) ◽  
Author(s):  
Roman A. Veprintsev
Keyword(s):  
Continuous Function ◽  
Harmonic Analysis ◽  
Unit Sphere ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Family ◽  
Necessary And Sufficient

AbstractWe establish a necessary and sufficient condition on a continuous function on [-1,1] under which the family of functions on the unit sphere 𝕊

scholarly journals A generalization of the practical numbers

2018 ◽  
Vol 14 (05) ◽  
pp. 1487-1503
Author(s):  
Nicholas Schwab ◽  
Lola Thompson
Keyword(s):  
Positive Integer ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Asymptotic Density ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
First Time

A positive integer [Formula: see text] is practical if every [Formula: see text] can be written as a sum of distinct divisors of [Formula: see text]. One can generalize the concept of practical numbers by applying an arithmetic function [Formula: see text] to each of the divisors of [Formula: see text] and asking whether all integers in a certain interval can be expressed as sums of [Formula: see text]’s, where the [Formula: see text]’s are distinct divisors of [Formula: see text]. We will refer to such [Formula: see text] as “[Formula: see text]-practical”. In this paper, we introduce the [Formula: see text]-practical numbers for the first time. We give criteria for when all [Formula: see text]-practical numbers can be constructed via a simple necessary-and-sufficient condition, demonstrate that it is possible to construct [Formula: see text]-practical sets with any asymptotic density, and prove a series of results related to the distribution of [Formula: see text]-practical numbers for many well-known arithmetic functions [Formula: see text].

Comparison of density topologies on the real line

2018 ◽  
Vol 68 (1) ◽  
pp. 173-180
Author(s):  
Renata Wiertelak
Keyword(s):  
Real Line ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
The Real ◽  
Closed Intervals ◽  
The Family ◽  
Necessary And Sufficient

Abstract In this paper will be considered density-like points and density-like topology in the family of Lebesgue measurable subsets of real numbers connected with a sequence 𝓙= {Jn}n∈ℕ of closed intervals tending to zero. The main result concerns necessary and sufficient condition for inclusion between that defined topologies.

Pairs of Consecutive Power Residues or Non-Residues

1964 ◽  
Vol 16 ◽  
pp. 310-314 ◽  
Author(s):  
J. H. Jordan
Keyword(s):  
Positive Integer ◽  
Cyclic Group ◽  
Proper Subgroup ◽  
Multiplicative Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Root Of Unity ◽  
Necessary And Sufficient

For a positive integer k and a prime p ≡ 1 (mod k), there is a proper subgroup, R, of the multiplicative group (mod p) consisting of the kth power residues (mod p). A necessary and sufficient condition that an integer t be an element of R is that the congruence xk ≡ t (mod p) be solvable. The cosets, not R, formed with respect to R are called classes of kth power nonresidues, and form with R a cyclic group of order k. Let ρ be a primitive kth root of unity and let S be a class of non-residues that is a generator of this cyclic group. There is a kth power character X (mod p) such that

REGULARITY AND STRONG REGULARITY IN THE CONTEXT OF CERTAIN CLASSES OF RINGS

2013 ◽  
Vol 12 (05) ◽  
pp. 1250205 ◽  
Author(s):  
MICHAŁ ZIEMBOWSKI
Keyword(s):  
Positive Integer ◽  
Sufficient Condition ◽  
Strong Regularity ◽  
Necessary And Sufficient Condition ◽  
Weak Dimension ◽  
Ring Of Polynomials ◽  
Necessary And Sufficient ◽  
Semihereditary Rings ◽  
The Ideal

We consider the ring R[x]/(xn+1), where R is a ring, R[x] is the ring of polynomials in an indeterminant x, (xn+1) is the ideal of R[x] generated by xn+1 and n is a positive integer. The aim of this paper is to show that regularity or strong regularity of a ring R is necessary and sufficient condition under which the ring R[x]/(xn+1) is an example of a ring which belongs to some important classes of rings. In this context, we discuss distributive rings, Bézout rings, Gaussian rings, quasi-morphic rings, semihereditary rings, and rings which have weak dimension less than or equal to one.

SYSTOLIC TREE WITH BASE AUTOMATA

1991 ◽  
Vol 02 (03) ◽  
pp. 221-236 ◽  
Author(s):  
A. MONTI ◽  
D. PARENTE
Keyword(s):  
Sufficient Condition ◽  
Tree Automata ◽  
Necessary And Sufficient Condition ◽  
The Family ◽  
Emptiness Problem ◽  
Necessary And Sufficient ◽  
Natural Way

Different systolic tree automata (STA) with base (T(b)−STA) are compared. This is a subclass of STA with interesting properties of modularity. We give a necessary and sufficient condition for the inclusion between classes of languages accepted by T(b)− STA, (L(T(b)−STA)), as b varies. We focus on T(b)−STA obtained by varying the base b in a natural way. We prove that for every base b within this framework there exists an a such that L(T(a)−STA) is not contained in L(T(b)−STA). We characterize the family of languages accepted by T(b)−STA when the input conditions are relaxed. Moreover we show that the emptiness problem is decidable for T(b)−STA.

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