Realization of digraphs in Abelian groups and its consequences

Cichacz, Sylwia; Tuza, Źsolt

doi:10.1002/jgt.22782

Cited by 3 publications

(6 citation statements)

References 12 publications

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“…is injective, then we say that ϕ is a Γ-irregular labeling of − → G . We have the following result: Cichacz and Tuza also showed the following: 6]). Any directed graph − → G of order n with no weakly connected components of cardinality less than 3 has a Γ-irregular labeling for every Γ such that |Γ| ≥ 2n + 2 n − 1/2 − 1.…”

Section: Group Irregular Labeling For Directed Graphsmentioning

confidence: 55%

“…Therefore we improve the result form [6] and, towards proving Conjecture 1.9, we leave open only the question if, for (ρ mod 6) = 5, Γ has not only 4-ZSPP, but also 3-ZSPP. We also present a new infinite family of groups for which Conjecture 1.7 holds.…”

Section: Recently Cichacz and Tuzamentioning

confidence: 91%

“…Then Γ has 3-ZSPP. [6] proved that, if |Γ| is large enough and |I(Γ)| > 1, then Γ has 4-ZSPP. This result was improved for Γ such that |Γ| = 2 n for some integer n > 1.…”

Section: Main Problemmentioning

confidence: 99%

“…[6]). A directed graph − → G with no isolated vertices has a Γ-irregular labeling if and only if there exists an injective mapping ϕ from V ( − → G ) to Γ such that x∈C ϕ(x) = 0 for every weakly connected component C of − → G .…”

mentioning

confidence: 99%

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Zero-sum Partitions of Abelian Groups

Cichacz¹,

Suchan²

2022

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 } t i=1 , be positive integers such that t i=1 m i = m − 1. Determine when Γ * = Γ \ {0}, the set of non-zero elements of Γ, can be partitioned into disjoint subsets {S i } t i=1 such that |S i | = m i and s∈Si s = 0 for all i, 1 ≤ i ≤ t. Such subset partitions are called zero-sum partitions.|I(Γ)| = 1, where I(Γ) is the set of involutions of Γ, is a necessary condition for the existence of zero-sum partitions. In this paper we show that the condition: m i ≥ 4 for all i, 1 ≤ i ≤ t, is sufficient. Moreover, we present some applications of zero-sum partitions to irregular, magic-and anti-magic-type labelings of graphs.

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Section: Group Irregular Labeling For Directed Graphsmentioning

confidence: 55%

Section: Recently Cichacz and Tuzamentioning

confidence: 91%

“…Then Γ has 3-ZSPP. [6] proved that, if |Γ| is large enough and |I(Γ)| > 1, then Γ has 4-ZSPP. This result was improved for Γ such that |Γ| = 2 n for some integer n > 1.…”

Section: Main Problemmentioning

confidence: 99%

mentioning

confidence: 99%

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Zero-sum Partitions of Abelian Groups

Cichacz¹,

Suchan²

2022

Preprint

Self Cite

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show abstract

“…Cichacz and Tuza showed that if n is large enough with respect to an arbitrarily fixed ε > 0 then − → G has a Γ-irregular labeling for any Γ such that |Γ| > (1 + ε)n [6]. In the case of any graph they obtain the following:…”

Section: Theorem 12 ([9]mentioning

confidence: 98%

Irregular labeling on Abelian groups of digraphs

Cichacz¹

2023

Preprint

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

Cichacz¹,

Suchan²

2023

Discrete Mathematics &Amp; Theoretical Computer Science

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The following problem has been known since the 80's. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $m_i$, $1 \leq i \leq t$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of $\Gamma$, can be partitioned into disjoint subsets $S_i$, $1 \leq i \leq t$, such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $i$, $1 \leq i \leq t$. It is easy to check that $m_i\geq 2$ (for every $i$, $1 \leq i \leq t$) and $|I(\Gamma)|\neq 1$ are necessary conditions for the existence of such partitions, where $I(\Gamma)$ is the set of involutions of $\Gamma$. It was proved that the condition $m_i\geq 2$ is sufficient if and only if $|I(\Gamma)|\in\{0,3\}$. For other groups (i.e., for which $|I(\Gamma)|\neq 3$ and $|I(\Gamma)|>1$), only the case of any group $\Gamma$ with $\Gamma\cong(Z_2)^n$ for some positive integer $n$ has been analyzed completely so far, and it was shown independently by several authors that $m_i\geq 3$ is sufficient in this case. Moreover, recently Cichacz and Tuza proved that, if $|\Gamma|$ is large enough and $|I(\Gamma)|>1$, then $m_i\geq 4$ is sufficient. In this paper we generalize this result for every Abelian group of order $2^n$. Namely, we show that the condition $m_i\geq 3$ is sufficient for $\Gamma$ such that $|I(\Gamma)|>1$ and $|\Gamma|=2^n$, for every positive integer $n$. We also present some applications of this result to graph magic- and anti-magic-type labelings.

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Realization of digraphs in Abelian groups and its consequences

Abstract: Let → G be a directed graph of order n with no compo-

Cited by 3 publications

References 12 publications

Zero-sum Partitions of Abelian Groups

Zero-sum Partitions of Abelian Groups

Irregular labeling on Abelian groups of digraphs

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

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