Group distance magic labeling of the Cartesian product of two directed cycles

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Abstract

Let $\vec{G}$ be a finite simple directed graph with $n$ vertices, and let $\Gamma$ be a finite abelian group of order $n$. A $\Gamma$-distance magic labeling is a bijection $\phi : V(\vec{G}) \to \Gamma$ for which there exists $c \in \Gamma$ such that $\sum_{y \in N^+(x)} \phi(y) - \sum_{y \in N^-(x)} \phi(y) = c$ for any $x \in V(\vec{G})$, where $N^+(x)$ and $N^-(x)$ denote the set of the head and the tail of $x$, respectively. In this paper, we obtain a necessary and sufficient condition for that there exists a $\Gamma$-distance magic labeling for the Cartesian products of two directed cycles.

Keywords

group distance magic labeling / magic constant / cartesian product / directed cycles / exponent

CHANGE HISTORY

A Correction to this paper has been published:
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This article has been retracted.Please see the Retraction Notice for more detail:
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References

[1] D. Froncek, Group distance magic labeling of Cartesian product of cycles, Australas. J. Comb., 55 (2013), 167-174.

[2] S. Cichacz, P. Dyrlaga, D. Froncek, Group distance magic Cartesian product of two cycles, Discrete Math., 343 (2020), 111807. https://doi.org/10.1016/j.disc.2019.111807

[3] X. Zeng, G. Deng, C. Luo, Characterize group distance magic labeling of Cartesian product of two cycles, Discrete Math., 346 (2023), 113407. https://doi.org/10.1016/j.disc.2023.113407

[4] B. Freyberg, M. Keranen, Orientable Zn-distance magic labeling of the Cartesian product of two cycles, Australas. J. Comb., 69 (2017), 222-235.

[5] S. Cichacz, B. Freyberg, D. Froncek, Orientable ZN-distance magic graphs, Discuss. Math. Graph Theory, 39 (2019), 533-546. https://doi.org/10.7151/dmgt.2094

[6] P. Dyrlaga, K. Szopa, Orientable Zn-distance magic regular graphs, AKCE Int. J. Graphs Comb., 18 (2021), 60-63. https://doi.org/10.1016/j.akcej.2019.06.005

[7] B. Freyberg, M. Keranen, Orientable Zn-distance magic graphs via products, Australas. J. Comb., 70 (2018), 319-328.

[8] K. H. Rosen, Elementary Number Theory and Its Applications, 6th edition, Addison Wesley, 2010.