A note on hamiltonicity conditions of the coprime and non-coprime graphs of a finite group
DOI:
https://doi.org/10.25077/jmua.13.3.157-162.2024Keywords:
Coprime Graph, Non-coprime Graph, Hamiltonian GraphAbstract
Let $G$ be a group. The coprime and non-coprime graphs of $G$ are introduced by Ma et al. (2014) and Mansoori et al. (2016), respectively, when $G$ is finite. By their definitions, which refer to coprime and non-coprime terms of two positive integers, those graphs must be related. We prove that they are closely related through their graph complement and preserve the isomorphism groups. Furthermore, according to Cayley's theorem, which states that any group $G$ is isomorphic to a subgroup of the symmetric group on $G$, it implies that the studies of the coprime and non-coprime graphs of any group $G$ (especially, when $G$ is finite) can actually be represented by the coprime and non-coprime graphs of any subgroup of the symmetric group on $G$. This encourages us to specifically study the hamiltonicity of both kinds of graphs associated with $G$ when $G$ is isomorphic to the symmetric group on $G$.Downloads
References
Ma, X., Wei, H., Yang, L., 2014, The coprime graph of a group, textit{International Journal of Group Theory}, Vol. textbf{3} (3) : 13-23.
Mansoori, F., Erfanian, A., Tolue, B., 2016, Non-coprime graph of a finite group, in textit{AIP Conference Proceedings}, Vol. textbf{1750} (1) (AIP Publishing, 2016), https://doi.org/10.1063/1.4954605
Adkins, W.A., Weintraub S.H., 1992, textit{Algebra: An Approach Via Module Theory}, Springer-Verlag, New York.
Rhani, N.A., Ali, N.M.M., Sarmin, N.H., Erfanian, A., 2017, On the dominating number, independent number and the regularity of the relative co-prime graph of a group, textit{Malaysian Journal of Fundamental and Applied Sciences}, Vol. textbf{13} (2) : 72-74, https://doi.org/10.11113/mjfas.v13n2.602
Rajkumar, R., Devi, P., 2015, Coprime graph of subgroups of a group, textit{Arxiv: Group Theory}, https://doi.org/10.48550/arXiv.1510.00129
Erdős, P., Sarkozy, G.N., 1997, On cycles in the coprime graph of integers, textit{Electron. J. Combin.}, Vol. textbf{4} (2), https://doi.org/10.37236/1323
Mostafa, M.H.B., Ghorbani, E., 2021, Hamiltonicity of a coprime graph, textit{Graphs and Combinatorics}, Vol. textbf{37} : 2387-2395.
Mutharasu, S., Rilwan, N.M., Jebitha, M.K.A., Chelvam, T.T., 2014, On generalized coprime graphs, textit{Iranian Journal of Mathematical Sciences and Informatics}, Vol. textbf{9} (2) : 1-6, https://dx.doi.org/10.7508/ijmsi.2014.02.001
Kathirvel, S.A., Cameron, P.J., Chelvam, T.T., 2024, Generalized non-coprime graphs of groups, textit{J. Algebraic Combin.}, https://doi.org/10.1007/s10801-024-01310-5
Huda, M.N., Ali, S., Non-coprime graph energy of dihedral and symmetric group, to appear in textit{AIP Conference Proceedings of The 9th SEAMS-UGM} (AIP Publishing).
Downloads
Additional Files
Published
How to Cite
Issue
Section
License
All articles published in Jurnal Matematika UNAND (JMUA) are open access and licensed under the Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0). This license allows published content to be freely accessed, shared, redistributed, and adapted for any purpose, including commercial use, provided that the applicable license conditions are followed.
Copyright Holder
The copyright of articles published in Jurnal Matematika UNAND is held by the Departemen Matematika dan Sains Data, Fakultas Matematika dan Ilmu Pengetahuan Alam (FMIPA), Universitas Andalas (UNAND), in accordance with the journal's stated copyright policy.
Author Rights
Under the CC BY-SA 4.0 license, authors may share, distribute, and reuse their published articles, subject to the applicable license conditions. These rights include:
-
Self-Archiving: Authors may deposit their published articles in institutional repositories, personal websites, or academic platforms.
-
Reuse: Authors may reuse their published work in subsequent publications, research, teaching materials, or other lawful purposes, provided that appropriate attribution is maintained.
-
Distribution: Authors may share and distribute their published articles without requiring additional permission under the terms of the license.
Creative Commons License (CC BY-SA 4.0)
Under the CC BY-SA 4.0 license, users are free to:
-
Share: Copy and redistribute the material in any medium or format.
-
Adapt: Remix, transform, and build upon the material for any purpose, including commercial purposes.
These permissions are subject to the following conditions:
-
Attribution: Users must give appropriate credit to the original author(s), retain relevant copyright notices, provide a link to the license, and indicate if changes were made. Attribution must not imply endorsement by the author(s), copyright holder, or journal.
-
ShareAlike: If users remix, transform, or build upon the material, they must distribute their contributions under the same CC BY-SA 4.0 license or a compatible license.
-
No Additional Restrictions: Users may not apply legal terms or technological measures that legally restrict others from exercising the rights granted by the license.
For more information about the CC BY-SA 4.0 license, please visit:
https://creativecommons.org/licenses/by-sa/4.0/
Third-Party Content
Third-party materials, including figures, tables, photographs, and illustrations, are covered by the article's CC BY-SA 4.0 license only when their inclusion under that license is authorized.
Materials excluded from the license must be clearly identified with appropriate copyright notices and attribution. Authors are responsible for obtaining any necessary permissions, while users must comply with the applicable licensing terms or permissions when reusing excluded materials.
Copyright and Licensing Information
Copyright and licensing information should be clearly displayed on each published article's landing page and included in the full-text versions (HTML and PDF), where applicable, to ensure transparency regarding copyright ownership and permitted reuse.
