The connection between the magical coloring of trees

Jing Su; Qiyue Zhang; Bing Yao; Jing Su; Qiyue Zhang; Bing Yao

doi:10.3934/math.20241354

AIMS Mathematics

2024, Volume 9, Issue 10: 27896-27907. doi: 10.3934/math.20241354

Previous Article Next Article

Research article

The connection between the magical coloring of trees

1.
College of Computing Science and Technology, Xi'an University of Science and Technology, Xi'an 710054, China
2.
College of Ulster, Shaanxi University of Science and Technology, Xi'an 710016, China
3.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received: 21 June 2024 Revised: 31 August 2024 Accepted: 12 September 2024 Published: 26 September 2024
MSC : 05C15

Let $ f $ be a set-ordered edge-magic labeling of a graph $ G $ from $ V(G) $ and $ E(G) $ to $ [0, p-1] $ and $ [1, p-1] $, respectively; it also satisfies the following conditions: $ |f(V(G))| = p $, $ \max f(X) < \min f(Y) $, and $ f(x)+f(y)+f(xy) = C $ for each edge $ xy\in E(G) $. In this paper, we removed the restriction that the labeling of vertices could not be repeated, and presented the concept of magical colorings including edge-magic coloring, edge-difference coloring, felicitous-difference coloring, and graceful-difference coloring. We studied the magical colorings on the tree and proved the existence of four kinds of magical colorings on the tree from a set-ordered edge-magic labeling. Further, we revealed the transformation relationship between these kinds of colorings.
- set-ordered edge-magic labeling,
- magical coloring,
- tree
Citation: Jing Su, Qiyue Zhang, Bing Yao. The connection between the magical coloring of trees[J]. AIMS Mathematics, 2024, 9(10): 27896-27907. doi: 10.3934/math.20241354

Related Papers:

Abstract

Let $ f $ be a set-ordered edge-magic labeling of a graph $ G $ from $ V(G) $ and $ E(G) $ to $ [0, p-1] $ and $ [1, p-1] $, respectively; it also satisfies the following conditions: $ |f(V(G))| = p $, $ \max f(X) < \min f(Y) $, and $ f(x)+f(y)+f(xy) = C $ for each edge $ xy\in E(G) $. In this paper, we removed the restriction that the labeling of vertices could not be repeated, and presented the concept of magical colorings including edge-magic coloring, edge-difference coloring, felicitous-difference coloring, and graceful-difference coloring. We studied the magical colorings on the tree and proved the existence of four kinds of magical colorings on the tree from a set-ordered edge-magic labeling. Further, we revealed the transformation relationship between these kinds of colorings.

References

[1]	A. Gheorghiu, T. Kapourniotis, E. Kashefi, Verification of quantum computation: An over view of existing approaches, Theory Math. Syst., 63 (2019), 715–808. https://doi.org/10.1007/s00224-018-9872-3 doi: 10.1007/s00224-018-9872-3
[2]	S. Suhail, R. Hussain, A. Khan, C. S. Hong, On the role of hash-based signatures in quantum-safe internet of things: Current solutions and future directions, IEEE Internet Things, 8 (2021), 1–17. https://doi.org/10.1109/JIOT.2020.3013019 doi: 10.1109/JIOT.2020.3013019
[3]	S. Katsumata, T. Matsuda, A. Takayasu, Lattice-based revocable (hierarchical) IBE with decryption key exposure resistance, Theor. Comput. Sci., 809 (2020), 103–136. https://doi.org/10.1016/j.tcs.2019.12.003 doi: 10.1016/j.tcs.2019.12.003
[4]	Z. R. Zheng, X. Y. Liu, L. Z. Yin, Z. C. Liu, A hybrid password authentication scheme based on shape and text, J. Comput., 5 (2010), 765–772. https://doi.org/10.4304/jcp.5.5.765-772 doi: 10.4304/jcp.5.5.765-772
[5]	Y. Z. Tian, L. X. Li, H. P. Peng, Y. X. Yang, Achieving flatness: Graph labelling can generate graphical honeywords, Comput. Secur., 104 (2021), 102212. https://doi.org/10.1016/j.cose.2021.102212 doi: 10.1016/j.cose.2021.102212
[6]	W. Chen, J. Li, Z. Huang, C. Z. Gao, S. M. Yiu, Lattice-based unidirectional infinite-use proxy resignatures with private re-signature key, J. Comput. Syst. Sci., 120 (2021), 137–148. https://doi.org/10.1016/j.jcss.2021.03.008 doi: 10.1016/j.jcss.2021.03.008
[7]	X. Zhang, C. Ye, S. Zhang, B. Yao, Graph colorings and labelings having multiple restrictive conditions in topological coding, Mathematics, 10 (2022), 1592. https://doi.org/10.3390/math10091592 doi: 10.3390/math10091592
[8]	J. Sedlacek, Problem 27, Theory of graphs and its applications, Proc Symp Smolenice, Praha: Academia, 1963, 163–169.
[9]	G. S. Bloom, S. W. Golomb, Numbered complete graphs, unusual rules, and assorted applications, In: Theory and Applications of Graphs, 2006, 53–65. https://doi.org/10.1007/BFb0070364
[10]	W. D. Wallis, Magic graphs, Berlin: Birkh-auser, 2001.
[11]	H. Y. Wang, J. Xu, B. Yao, Exploring new cryptographical construction of complex network data, In: 2016 IEEE First International Conference on Data Science in Cyberspace(DSC), 2016, 155–160. https://doi.org/10.1109/DSC.2016.76
[12]	H. Y. Wang, J. Xu, B. Yao, The key-models and their lock-models for designing new labellings of networks, In: 2016 IEEE Advanced Information Management, Communicates, Electronic and Automation Control Conference (IMCEC), 2016, 565–568. https://doi.org/10.1109/IMCEC.2016.7867273
[13]	B. Yao, M. M. Zhao, X. H. Zhang, Y. R. Mu, Y. R. Sun, Topological coding and topological matrices toward network overall security, arXiv preprint, 2019. https://doi.org/10.48550/arXiv.1909.01587
[14]	J. A. Bondy, U. S. R. Murty, Graph theory with applications, MacMillan Press Ltd., 1976.
[15]	J. A. Gallian, A dynamic survey of graph labelling, 23 Eds., Electron. J. Comb., 2020.

Reader Comments

Your name:*

Email:*
© 2024 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)