Zero-sum subsequences of distinct lengths

2015 ◽  
Vol 11 (07) ◽  
pp. 2141-2150 ◽  
Author(s):  
Weidong Gao ◽  
Pingping Zhao ◽  
Jujuan Zhuang
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Zero Sum ◽  
Positive Integers

Let G be an additive finite abelian group, and let disc (G) denote the smallest positive integer t such that every sequence S over G of length ∣S∣ ≥ t has two nonempty zero-sum subsequences of distinct lengths. We determine disc (G) for some groups including the groups [Formula: see text], the groups of rank at most two and the groups Cmpn ⊕ H, where m, n are positive integers, p is a prime and H is a p-group with pn ≥ D*(H).

EXTREMAL CAYLEY DIGRAPHS OF FINITE ABELIAN GROUPS

2011 ◽  
Vol 12 (01n02) ◽  
pp. 125-135 ◽  
Author(s):  
ABBY GAIL MASK ◽  
JONI SCHNEIDER ◽  
XINGDE JIA
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Communication Networks ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Abelian Groups ◽  
Element Subset ◽  
Cayley Digraph ◽  
Positive Integers ◽  
Element Set

Cayley digraphs of finite abelian groups are often used to model communication networks. Because of their applications, extremal Cayley digraphs have been studied extensively in recent years. Given any positive integers d and k. Let m*(d, k) denote the largest positive integer m such that there exists an m-element finite abelian group Γ and a k-element subset A of Γ such that diam ( Cay (Γ, A)) ≤ d, where diam ( Cay (Γ, A)) denotes the diameter of the Cayley digraph Cay (Γ, A) of Γ generated by A. Similarly, let m(d, k) denote the largest positive integer m such that there exists a k-element set A of integers with diam (ℤm, A)) ≤ d. In this paper, we prove, among other results, that [Formula: see text] for all d ≥ 1 and k ≥ 1. This means that the finite abelian group whose Cayley digraph is optimal with respect to its diameter and degree can be a cyclic group.

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.

On some weighted zero-sum constants

2017 ◽  
Vol 13 (02) ◽  
pp. 301-308 ◽  
Author(s):  
Mohan N. Chintamani ◽  
Prabal Paul
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Upper Bound ◽  
Structural Information ◽  
Finite Abelian Group ◽  
Text Length ◽  
Zero Sum

Let [Formula: see text] be a finite abelian group with exponent exp[Formula: see text]. Let [Formula: see text]. The constant [Formula: see text] is defined as the least positive integer [Formula: see text] such that for any given sequence [Formula: see text] of elements of [Formula: see text] with length [Formula: see text] it has a [Formula: see text] length [Formula: see text]-weighted zero-sum subsequence. In this article, we obtain the exact value of [Formula: see text] for [Formula: see text] and an upper bound for the case [Formula: see text], where [Formula: see text] is an odd prime, [Formula: see text] is an odd integer and [Formula: see text]. We also obtain the structural information on the extremal zero-sum free sequences.

scholarly journals On the Number of Subsequences with a Given Sum in a Finite Abelian Group

10.37236/620 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Gerard Jennhwa Chang ◽  
Sheng-Hua Chen ◽  
Yongke Qu ◽  
Guoqing Wang ◽  
Haiyan Zhang
Keyword(s):  
Abelian Group ◽  
Lower Bound ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Zero Sum

Suppose $G$ is a finite abelian group and $S$ is a sequence of elements in $G$. For any element $g$ of $G$, let $N_g(S)$ denote the number of subsequences of $S$ with sum $g$. The purpose of this paper is to investigate the lower bound for $N_g(S)$. In particular, we prove that either $N_g(S)=0$ or $N_g(S)\ge2^{|S|-D(G)+1}$, where $D(G)$ is the smallest positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a nonempty zero-sum subsequence. We also characterize the structures of the extremal sequences for which the equality holds for some groups.

scholarly journals The zero-sum constant, the Davenport constant and their analogues

2020 ◽  
pp. 1-14
Author(s):  
Maciej Zakarczemny
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Asymptotic Behaviour ◽  
Finite Abelian Group ◽  
Classical Case ◽  
Independent Interest ◽  
Davenport Constant ◽  
Smooth Numbers ◽  
Abelian Case ◽  
Zero Sum

Let D(G) be the Davenport constant of a finite Abelian group G. For a positive integer m (the case m=1, is the classical case) let Em(G) (or ηm(G)) be the least positive integer t such that every sequence of length t in G contains m disjoint zero-sum sequences, each of length |G| (or of length ≤exp(G), respectively). In this paper, we prove that if G is an Abelian group, then Em(G)=D(G)–1+m|G|, which generalizes Gao’s relation. Moreover, we examine the asymptotic behaviour of the sequences (Em(G))m≥1 and (ηm(G))m≥1. We prove a generalization of Kemnitz’s conjecture. The paper also contains a result of independent interest, which is a stronger version of a result by Ch. Delorme, O. Ordaz, D. Quiroz. At the end, we apply the Davenport constant to smooth numbers and make a natural conjecture in the non-Abelian case.

On the Erdős–Ginzburg–Ziv invariant and zero-sum Ramsey number for intersecting families

2014 ◽  
Vol 10 (07) ◽  
pp. 1637-1647 ◽  
Author(s):  
Haiyan Zhang ◽  
Guoqing Wang
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Ramsey Number ◽  
Uniform Hypergraph ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Zero Sum

Let G be a finite abelian group, and let m > 0 with exp (G) | m. Let sm(G) be the generalized Erdős–Ginzburg–Ziv invariant which denotes the smallest positive integer d such that any sequence of elements in G of length d contains a subsequence of length m with sum zero in G. For any integer r > 0, let [Formula: see text] be the collection of all r-uniform intersecting families of size m. Let [Formula: see text] be the smallest positive integer d such that any G-coloring of the edges of the complete r-uniform hypergraph [Formula: see text] yields a zero-sum copy of some intersecting family in [Formula: see text]. Among other results, we mainly prove that [Formula: see text], where Ω(sm(G)) denotes the least positive integer n such that [Formula: see text], and we show that if r | Ω(sm(G)) – 1 then [Formula: see text].

On some weighted zero-sum constants II

2018 ◽  
Vol 14 (02) ◽  
pp. 383-397 ◽  
Author(s):  
Mohan N. Chintamani ◽  
Prabal Paul
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Positive Integer ◽  
Upper Bound ◽  
Structure Theorem ◽  
Finite Abelian Group ◽  
Zero Sum

For a finite abelian group [Formula: see text] with exponent [Formula: see text], let [Formula: see text]. The constant [Formula: see text] (respectively [Formula: see text]) is defined to be the least positive integer [Formula: see text] such that given any sequence [Formula: see text] over [Formula: see text] with length [Formula: see text] has a [Formula: see text]-weighted zero-sum subsequence of length [Formula: see text] (respectively at most [Formula: see text]). In [M. N. Chintamani and P. Paul, On some weighted zero-sum constants, Int. J. Number Theory 13(2) (2017) 301–308], we proved the exact value of this constant for the group [Formula: see text] and proved the structure theorem for the extremal sequences related to this constant. In this paper, we prove the similar results for the group [Formula: see text] and we obtained an upper bound when [Formula: see text] is replaced by any integer [Formula: see text].

Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group

2021 ◽  
Author(s):  
Weidong Gao ◽  
Siao Hong ◽  
Wanzhen Hui ◽  
Xue Li ◽  
Qiuyu Yin ◽  
...  
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum

scholarly journals On Subsequence Sums of a Zero-sum Free Sequence

10.37236/970 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Fang Sun
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum ◽  
Open Question

Let $G$ be a finite abelian group with exponent $m$, and let $S$ be a sequence of elements in $G$. Let $f(S)$ denote the number of elements in $G$ which can be expressed as the sum over a nonempty subsequence of $S$. In this paper, we show that, if $|S|=m$ and $S$ contains no nonempty subsequence with zero sum, then $f(S)\geq 2m-1$. This answers an open question formulated by Gao and Leader. They proved the same result with the restriction $(m,6)=1$.

scholarly journals The Minimum Size of Signed Sumsets

10.37236/4881 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Béla Bajnok ◽  
Ryan Matzke
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Finite Abelian Group ◽  
Minimum Size ◽  
Positive Integers

For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$\rho(G, m, h) = \min \{ |hA| \; : \; A \subseteq G, |A|=m\}$$ and$$\rho_{\pm} (G, m, h) = \min \{ |h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$ where $hA$ and $h_{\pm} A$ denote the $h$-fold sumset and the $h$-fold signed sumset of $A$, respectively. The study of $\rho(G, m, h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. Here we prove that $\rho_{\pm}(G, m, h)$ equals $\rho (G, m, h)$ when $G$ is cyclic, and establish an upper bound for $\rho_{\pm} (G, m, h)$ that we believe gives the exact value for all $G$, $m$, and $h$.

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