# Note on group distance magic complete bipartite graphs

Open Mathematics (2014)

• Volume: 12, Issue: 3, page 529-533
• ISSN: 2391-5455

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## Abstract

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A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ℓ from V to an Abelian group Γ of order n such that the weight $w\left(x\right)={\sum }_{y\in {N}_{G}\left(x\right)}\ell \left(y\right)$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|. In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ℤp-distance magic. Moreover we show that if p ≡ 2 (mod 4) and k is even, then there does not exist a group Γ of order p such that there exists a Γ-distance labeling for a k-partite complete graph of order p. We also prove that K m,n is a group distance magic graph if and only if n + m ≢ 2 (mod 4).

## How to cite

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Sylwia Cichacz. "Note on group distance magic complete bipartite graphs." Open Mathematics 12.3 (2014): 529-533. <http://eudml.org/doc/269065>.

@article{SylwiaCichacz2014,
abstract = {A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ℓ from V to an Abelian group Γ of order n such that the weight $w(x) = \sum \nolimits \_\{y \in N\_G (x)\} \{\ell (y)\}$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|. In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ℤp-distance magic. Moreover we show that if p ≡ 2 (mod 4) and k is even, then there does not exist a group Γ of order p such that there exists a Γ-distance labeling for a k-partite complete graph of order p. We also prove that K m,n is a group distance magic graph if and only if n + m ≢ 2 (mod 4).},
author = {Sylwia Cichacz},
journal = {Open Mathematics},
keywords = {Graph labeling; Abelian group; graph labeling; abelian group},
language = {eng},
number = {3},
pages = {529-533},
title = {Note on group distance magic complete bipartite graphs},
url = {http://eudml.org/doc/269065},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Sylwia Cichacz
TI - Note on group distance magic complete bipartite graphs
JO - Open Mathematics
PY - 2014
VL - 12
IS - 3
SP - 529
EP - 533
AB - A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ℓ from V to an Abelian group Γ of order n such that the weight $w(x) = \sum \nolimits _{y \in N_G (x)} {\ell (y)}$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|. In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ℤp-distance magic. Moreover we show that if p ≡ 2 (mod 4) and k is even, then there does not exist a group Γ of order p such that there exists a Γ-distance labeling for a k-partite complete graph of order p. We also prove that K m,n is a group distance magic graph if and only if n + m ≢ 2 (mod 4).
LA - eng
KW - Graph labeling; Abelian group; graph labeling; abelian group
UR - http://eudml.org/doc/269065
ER -

## References

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1. [1] Arumugam S., Froncek D., Kamatchi N., Distance magic graphs¶a survey, J. Indones. Math. Soc., 2011, Special edition, 11–26 Zbl1288.05216
2. [2] Beena S., On Σ and Σ′ labelled graphs, Discrete Math., 2009, 309(6), 1783–1787 http://dx.doi.org/10.1016/j.disc.2008.02.038
3. [3] Cichacz S., Note on group distance magic graphs G[C 4], Graphs Combin. (in press), DOI: 10.1007/s00373-013-1294-z Zbl1284.05122
4. [4] Combe D., Nelson A.M., Palmer W.D., Magic labellings of graphs over finite abelian groups, Australas. J. Combin., 2004, 29, 259–271 Zbl1050.05107
5. [5] Froncek D., Group distance magic labeling of Cartesian product of cycles, Australas. J. Combin., 2013, 55, 167–174 Zbl1278.05210
6. [6] Kaplan G., Lev A., Roditty Y., On zero-sum partitions and anti-magic trees, Discrete Math., 2009, 309(8), 2010–2014 http://dx.doi.org/10.1016/j.disc.2008.04.012 Zbl1229.05031

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