scholarly journals Forcing $k$-Repetitions in Degree Sequences

10.37236/3503 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Yair Caro ◽  
Asaf Shapira ◽  
Raphael Yuster
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Main Tool ◽  
Geometric Approach ◽  
Degree Sequences ◽  
Integer Sequences ◽  
Sum Theorem ◽  
Zero Sum

One of the most basic results in graph theory states that every graph with at least two vertices has two vertices with the same degree. Since there are graphs without $3$ vertices of the same degree, it is natural to ask if for any fixed $k$, every graph $G$ is "close" to a graph $G'$ with  $k$ vertices of the same degree. Our main result in this paper is that this is indeed the case. Specifically, we show that for any positive integer $k$, there is a constant $C=C(k)$, so that given any graph $G$, one can remove from $G$ at most $C$ vertices and thus obtain a new graph $G'$ that contains at least $\min\{k,|G|-C\}$ vertices of the same degree.Our main tool is a multidimensional zero-sum theorem for integer sequences, which we prove using an old geometric approach of Alon and Berman.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals On the sum of digits of some sequences of integers

2013 ◽  
Vol 11 (1) ◽  
Author(s):  
Javier Cilleruelo ◽  
Florian Luca ◽  
Juanjo Rué ◽  
Ana Zumalacárregui
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Integer Sequences ◽  
Sum Of Digits ◽  
Fixed Positive Integer ◽  
Almost All

AbstractLet b ≥ 2 be a fixed positive integer. We show for a wide variety of sequences {a n}n=1∞ that for almost all n the sum of digits of a n in base b is at least c b log n, where c b is a constant depending on b and on the sequence. Our approach covers several integer sequences arising from number theory and combinatorics.

scholarly journals A SUFFICIENT CONDITION FOR A PAIR OF SEQUENCES TO BE BIPARTITE GRAPHIC

2016 ◽  
Vol 94 (2) ◽  
pp. 195-200 ◽  
Author(s):  
GRANT CAIRNS ◽  
STACEY MENDAN ◽  
YURI NIKOLAYEVSKY
Keyword(s):  
Bipartite Graph ◽  
Sufficient Condition ◽  
Degree Sequences ◽  
Integer Sequences ◽  
Finite Integer

We present a sufficient condition for a pair of finite integer sequences to be degree sequences of a bipartite graph, based only on the lengths of the sequences and their largest and smallest elements.

On zero-sum subsequences of prescribed length

2017 ◽  
Vol 14 (01) ◽  
pp. 167-191 ◽  
Author(s):  
Dongchun Han ◽  
Hanbin Zhang
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Davenport Constant ◽  
Zero Sum

Let [Formula: see text] be an additive finite abelian group with exponent [Formula: see text]. For any positive integer [Formula: see text], let [Formula: see text] be the smallest positive integer [Formula: see text] such that every sequence [Formula: see text] in [Formula: see text] of length at least [Formula: see text] has a zero-sum subsequence of length [Formula: see text]. Let [Formula: see text] be the Davenport constant of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite abelian [Formula: see text]-group with [Formula: see text] then [Formula: see text] for every [Formula: see text], which confirms a conjecture by Gao et al. recently, where [Formula: see text] is a prime.

scholarly journals Region Coloring in Minahasa Regency Using Sequential Color Algorithm

d'CARTESIAN ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 86
Author(s):  
Yevie Ingamita ◽  
Nelson Nainggolan ◽  
Benny Pinontoan
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Chromatic Number ◽  
Human Life ◽  
Minimum Number ◽  
Mathematical Sciences ◽  
Map Coloring ◽  
Graph Application

Graph Theory is one of the mathematical sciences whose application is very wide in human life. One of theory graph application is Map Coloring. This research discusses how to color the map of Minahasa Regency by using the minimum color that possible. The algorithm used to determine the minimum color in coloring the region of Minahasa Regency that is Sequential Color Algorithm. The Sequential Color Algorithm is an algorithm used in coloring a graph with k-color, where k is a positive integer. Based on the results of this research was found that the Sequential Color Algorithm can be used to color the map of Minahasa Regency with the minimum number of colors or chromatic number χ(G) obtained in the coloring of 25 sub-districts on the map of Minahasa Regency are 3 colors (χ(G) = 3).

scholarly journals Monochromatic and Zero-Sum Sets of Nondecreasing Modified Diameter

10.37236/1054 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
David Grynkiewicz ◽  
Rasheed Sabar
Keyword(s):  
Positive Integer ◽  
Cyclic Group ◽  
Prime Divisor ◽  
Zero Sum ◽  
Equality Holding

Let $m$ be a positive integer whose smallest prime divisor is denoted by $p$, and let ${\Bbb Z}_m$ denote the cyclic group of residues modulo $m$. For a set $B=\{x_1,x_2,\ldots,x_m\}$ of $m$ integers satisfying $x_1 < x_2 < \cdots < x_m$, and an integer $j$ satisfying $2\leq j \leq m$, define $g_j(B)=x_j-x_1$. Furthermore, define $f_j(m,2)$ (define $f_j(m,{\Bbb Z}_m)$) to be the least integer $N$ such that for every coloring $\Delta : \{1,\ldots,N\}\rightarrow \{0,1\}$ (every coloring $\Delta : \{1,\ldots,N\}\rightarrow {\Bbb Z}_m$), there exist two $m$-sets $B_1,B_2\subset \{1,\ldots,N\}$ satisfying: (i) $\max(B_1) < \min(B_2)$, (ii) $g_j(B_1)\leq g_j(B_2)$, and (iii) $|\Delta (B_i)|=1$ for $i=1,2$ (and (iii) $\sum_{x\in B_i}\Delta (x)=0$ for $i=1,2$). We prove that $f_j(m,2)\leq 5m-3$ for all $j$, with equality holding for $j=m$, and that $f_j(m,{\Bbb Z}_m)\leq 8m+{m\over p}-6$. Moreover, we show that $f_j(m,2)\ge 4m-2+(j-1)k$, where $k=\left\lfloor\left(-1+\sqrt{{8m-9+j\over j-1}}\right){/2}\right\rfloor$, and, if $m$ is prime or $j\geq{m\over p}+p-1$, that $f_j(m,{\Bbb Z}_m)\leq 6m-4$. We conclude by showing $f_{m-1}(m,2)=f_{m-1}(m,{\Bbb Z}_m)$ for $m\geq 9$.

scholarly journals A variant of Niessen’s problem on degreesequences of graphs

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Jiyun Guo ◽  
Jianhua Yin
Keyword(s):  
Graph Theory ◽  
Discrete Math ◽  
Simple Graph ◽  
The Other ◽  
Degree Sequences ◽  
International Audience

Graph Theory International audience Let (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) be two sequences of nonnegative integers satisfying the condition that b1>=b2>=...>=bn, ai<= bi for i=1,2,\textellipsis,n and ai+bi>=ai+1+bi+1 for i=1,2,\textellipsis, n-1. In this paper, we give two different conditions, one of which is sufficient and the other one necessary, for the sequences (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) such that for every (c1,c2,\textellipsis,cn) with ai<=ci<=bi for i=1,2,\textellipsis,n and &#x2211;&limits;i=1n ci=0 (mod 2), there exists a simple graph G with vertices v1,v2,\textellipsis,vn such that dG(vi)=ci for i=1,2,\textellipsis,n. This is a variant of Niessen\textquoterights problem on degree sequences of graphs (Discrete Math., 191 (1998), 247&#x2013;253).

scholarly journals Shattering, Graph Orientations, and Connectivity

10.37236/3326 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
László Kozma ◽  
Shay Moran
Keyword(s):  
Graph Theory ◽  
Combinatorial Optimization ◽  
Main Tool ◽  
The Other ◽  
Vc Dimension ◽  
Forbidden Subgraphs ◽  
Set Systems ◽  
Flows In Networks ◽  
Extremal Systems

We present a connection between two seemingly disparate fields: VC-theory and graph theory. This connection yields natural correspondences between fundamental concepts in VC-theory, such as shattering and VC-dimension, and well-studied concepts of graph theory related to connectivity, combinatorial optimization, forbidden subgraphs, and others.In one direction, we use this connection to derive results in graph theory. Our main tool is a generalization of the Sauer-Shelah Lemma (Pajor, 1985; Bollobás and Radcliffe, 1995; Dress, 1997; Holzman and Aharoni). Using this tool we obtain a series of inequalities and equalities related to properties of orientations of a graph. Some of these results appear to be new, for others we give new and simple proofs.In the other direction, we present new illustrative examples of shattering-extremal systems - a class of set-systems in VC-theory whose understanding is considered by some authors to be incomplete Bollobás and Radcliffe, 1995; Greco, 1998; Rónyai and Mészáros, 2011). These examples are derived from properties of orientations related to distances and flows in networks.

scholarly journals Periodic solutions of Lienard differential equations via averaging theory of order two

2015 ◽  
Vol 87 (4) ◽  
pp. 1905-1913
Author(s):  
JAUME LLIBRE ◽  
DOUGLAS D. NOVAES ◽  
MARCO A. TEIXEIRA
Keyword(s):  
Positive Integer ◽  
Differential Equations ◽  
Periodic Function ◽  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Second Order ◽  
Averaging Theory ◽  
Existence Of Periodic Solutions

Abstract For ε ≠ 0sufficiently small we provide sufficient conditions for the existence of periodic solutions for the Lienard differential equations of the form x ′′ + f ( x ) x ′ + n 2 x + g ( x ) = ε 2 p 1 ( t ) + ε 3 p 2 ( t ) , where n is a positive integer, f : ℝ → ℝis a C 3function, g : ℝ → ℝis a C 4function, and p i : ℝ → ℝfor i = 1 , 2are continuous 2 π–periodic function. The main tool used in this paper is the averaging theory of second order. We also provide one application of the main result obtained.

scholarly journals The Adjacency Matrix of One Type of Directed Graph and the Jacobsthal Numbers and Their Determinantal Representation

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Fatih Yılmaz ◽  
Durmuş Bozkurt
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Directed Graph ◽  
Adjacency Matrix ◽  
Intensive Study ◽  
Integer Power ◽  
Determinantal Representation ◽  
Adjacency Matrices

Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we consider one type of directed graph. Then we obtain a general form of the adjacency matrices of the graph. By using the well-known property which states the(i,j)entry ofAm(Ais adjacency matrix) is equal to the number of walks of lengthmfrom vertexito vertexj, we show that elements ofmth positive integer power of the adjacency matrix correspond to well-known Jacobsthal numbers. As a consequence, we give a Cassini-like formula for Jacobsthal numbers. We also give a matrix whose permanents are Jacobsthal numbers.

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