On zero sum-partition of Abelian groups into three sets and group distance magic labeling

Cichacz, Sylwia

doi:10.26493/1855-3974.1054.fcd

Cited by 4 publications

(5 citation statements)

References 5 publications

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“…We only know the following sufficient and necessary condition for Z m -distance magic labeling of a complete t-partite graph only for t or m odd. Theorem 5.1 ( [11,18]). Let G = K n 1 ,n 2 ,...,nt be a complete t-partite graph of order m such that 1 ≤ n 1 ≤ n 2 ≤ .…”

Section: Final Remarksmentioning

confidence: 99%

Zero-sum partitions of Abelian groups of order $2^n$

Cichacz¹,

Suchan²

2021

Preprint

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The following problem has been known since the 80's. Let Γ be an Abelian group of order m (denoted |Γ| = m), and let t and m i , 1 ≤ i ≤ t, be positive integers such thatIt was shown that the condition m i ≥ 2 (for any i, 24,31]), where I(Γ) is the set of involutions of Γ. In this paper we show that the condition m i ≥ 3 is sufficient for Γ such that |I(Γ)| > 1 and |Γ| = 2 n for any positive integer n. * This work was partially supported by the Faculty of Applied Mathematics AGH UST statutory tasks within subsidy of Ministry of Science and Higher Education.1 For standard notions, notations and results in finite algebra the reader is referred to the textbook by Gallian [21]. In this article we recall only the ones that are most directly related to the presentation of our work.

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Section: Final Remarksmentioning

confidence: 99%

Zero-sum partitions of Abelian groups of order $2^n$

Cichacz¹,

Suchan²

2021

Preprint

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“…Anholcer and Cichacz proved a lemma about a partition of the set of all elements of G of order at most 2 into two zero-sum sets (see [AC16], Lemma 2.4). Their result along with results proved by Cichacz (see [Cic17], Lemma 3.1) give the following lemma.…”

Section: Preliminariesmentioning

confidence: 67%

Group twin coloring of graphs

Cichacz,

Przybyło

2017

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For a given graph G, the least integer k ≥ 2 such that for every Abelian group G of order k there exists a proper edge labeling f :. This graph invariant is related to a few well-known problems in the field of neighbor distinguishing graph colorings. We conjecture that χ ′ g (G) ≤ ∆(G) + 3 for all graphs without isolated edges, where ∆(G) is the maximum degree of G, and provide an infinite family of connected graph (trees) for which the equality holds. We prove that this conjecture is valid for all trees, and then apply this result as the base case for proving a general upper bound for all graphs G without isolated edges: χ ′ g (G) ≤ 2(∆(G) + col(G)) − 5, where col(G) denotes the coloring number of G. This improves the best known upper bound known previously only for the case of cyclic groups Z k .

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“…It is an easy observation that the complete p-partite graph is not Z n -distance magic when p = 1. The next theorem completely classifies Z n -distance magic labelings of complete bipartite graphs [2], complete tripartite graphs [3], and complete p-partite graphs for n odd [2].…”

Section: Complete P-partite Graphsmentioning

confidence: 96%

On constant sum partitions and applications to distance magic-type graphs

Freyberg

2020

AKCE International Journal of Graphs and Combinatorics

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Let G be an additive abelian group of order n and let n = a 1 + a 2 + ... + a p be a partition of n where 1∑ a∈A i a = t, for some fixed t ∈ G and every 1 ≤ i ≤ p. In 2009, Kaplan, Lev, and Roditty proved that a 0-sum partition of the cyclic group Z n exists for n odd if and only if a 2 ≥ 2. In this paper, we address the case when n is even. In particular, we show that a n 2 -sum partition of Z n exists for n even and p odd if and only if a 2 ≥ 2. Moreover, we provide applications to distance magic-type graphs including the classification of Z n -distance magic complete p-partite graphs for p odd.

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On zero sum-partition of Abelian groups into three sets and group distance magic labeling

Cited by 4 publications

References 5 publications

Zero-sum partitions of Abelian groups of order $2^n$

Zero-sum partitions of Abelian groups of order $2^n$

Group twin coloring of graphs

On constant sum partitions and applications to distance magic-type graphs

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