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

Guixin Deng; Li Wang; Caimei Luo; Guixin Deng; Li Wang; Caimei Luo

doi:10.3934/era.2025178

Electronic Research Archive

2025, Volume 33, Issue 6: 4014-4026. doi: 10.3934/era.2025178

Previous Article Next Article

Research article

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

Center for Applied Mathematics of Guangxi, Nanning Normal University, Nanning 530100, China

Received: 25 March 2025 Revised: 26 May 2025 Accepted: 06 June 2025 Published: 24 June 2025

Let $ \overrightarrow{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 $ \varphi:V(\overrightarrow{G})\longrightarrow \Gamma $ for which there exists $ c\in\Gamma $ such that $ \sum_{y\in N^+(x)}\varphi(y)-\sum_{y\in N^-(x)}\varphi(y) = c $ for any $ x\in V(\overrightarrow{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.
- group distance magic labeling,
- magic constant,
- cartesian product,
- directed cycles,
- exponent
Citation: Guixin Deng, Li Wang, Caimei Luo. Group distance magic labeling of the Cartesian product of two directed cycles[J]. Electronic Research Archive, 2025, 33(6): 4014-4026. doi: 10.3934/era.2025178

Related Papers:

Abstract

Let $ \overrightarrow{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 $ \varphi:V(\overrightarrow{G})\longrightarrow \Gamma $ for which there exists $ c\in\Gamma $ such that $ \sum_{y\in N^+(x)}\varphi(y)-\sum_{y\in N^-(x)}\varphi(y) = c $ for any $ x\in V(\overrightarrow{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.

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 doi: 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 doi: 10.1016/j.disc.2023.113407
[4]	B. Freyberg, M. Keranen, Orientable $\mathbb{Z}_n$-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 $\mathbb{Z}_N$-distance magic graphs, Discuss. Math. Graph Theory, 39 (2019), 533–546. https://doi.org/10.7151/dmgt.2094 doi: 10.7151/dmgt.2094
[6]	P. Dyrlaga, K. Szopa, Orientable $\mathbb{Z}_n$-distance magic regular graphs, AKCE Int. J. Graphs Comb., 18 (2021), 60–63. https://doi.org/10.1016/j.akcej.2019.06.005 doi: 10.1016/j.akcej.2019.06.005
[7]	B. Freyberg, M. Keranen, Orientable $\mathbb{Z}_n$-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.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)