On a Combinatorial Theorem of Erdös, Ginzburg and Ziv

Hamidoune, Yahya Ould; Ordaz, Óscar; Ortuño, Asdrubal

doi:10.1017/s0963548398003721

Cited by 27 publications

(16 citation statements)

References 11 publications

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“…The study of zero-sum sequences in ÿnite abelian groups has been a topic of interest in the recent mathematical literature (see [1,2,11]). A natural generalization of zero-sum sequences are the barycentric sequences which are deÿned in the following way: Deÿnition 1.…”

Section: Introductionmentioning

confidence: 99%

Existence conditions for barycentric sequences

Delorme

Márquez

Ordaz

et al. 2004

Discrete Mathematics

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Section: Introductionmentioning

confidence: 99%

Existence conditions for barycentric sequences

Delorme

Márquez

Ordaz

et al. 2004

Discrete Mathematics

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“…Let g(m, k) denote the least integer n such that any sequence of elements from Z m with length n and k distinct residues must contain an m-term zero-sum subsequence. The function g(m, k) was introduced by Bialostocki and Lotspeich [10], and has been studied by several authors, [9], [27], [30], [31], [37]. The value of g(m, k) was known completely for k 4, [10].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On a partition analog of the Cauchy-Davenport theorem

Grynkiewicz

2005

Acta Math Hung

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Let G be a finite abelian group, and let n be a positive integer. From the Cauchy-Davenport Theorem it follows that if G is a cyclic group of prime order, then any collection of n subsets A1, A2, . . . , An of G satisfiesM. Kneser generalized the Cauchy-Davenport Theorem for any abelian group. In this paper, we prove a sequence-partition analog of the Cauchy-Davenport Theorem along the lines of Kneser's Theorem. A particular case of our theorem was proved by J. E. Olson in the context of the Erdős-Ginzburg-Ziv Theorem.

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“…. , a r , for general r. Hamidoune, Ordaz and Ortuño [11] gave a sufficient condition for 0 to be a k-sum from a sequence a 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Sums and k-sums in abelian groups of order k

Gao

Leader²

2006

Journal of Number Theory

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Let G be an abelian group of order k. How is the problem of minimizing the number of sums from a sequence of given length in G related to the problem of minimizing the number of k-sums? In this paper we show that the minimum number of k-sums for a sequence a 1 , . . . , a r that does not have 0 as a k-sum is attained at the sequence b 1 , . . . , b r−k+1 , 0, . . . , 0, where b 1 , . . . , b r−k+1 is chosen to minimise the number of sums without 0 being a sum. Equivalently, to minimise the number of k-sums one should repeat some value k − 1 times. This proves a conjecture of Bollobás and Leader, and extends results of Gao and of Bollobás and Leader. IntroductionGiven a sequence a 1 , . . . , a r in Z k , the integers modulo k, a k-sum is a sum of the form a i 1 + · · · + a i k , where i 1 < · · · < i k . How large can r be without 0 being a k-sum? It is clear that we may have r = 2k − 2, by taking a 1 = · · · = a k−1 = 0 and a k = · · · = a 2k−2 = 1. Erdös, Ginzburg and Ziv [5] showed that this is best possible. In other words, they showed that if we have a 1 , . . . , a 2k−1 in Z k then some k-sum is 0. Since then, numerous other proofs of this result have been found-see Alon and Dubiner [1] for a general survey.

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On a Combinatorial Theorem of Erdös, Ginzburg and Ziv

Cited by 27 publications

References 11 publications

Existence conditions for barycentric sequences

Existence conditions for barycentric sequences

On a partition analog of the Cauchy-Davenport theorem

Sums and k-sums in abelian groups of order k

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