On the \(b\)-Chromatic Number of Power Graphs of Finite Groups

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-320
Published: Oct 5, 2026
Cite as:
Khaled Mustafa Al-Jamal, Ayser Nasir Tahat, Mowafaq Omar Al-Qadri, On the \(b\)-Chromatic Number of Power Graphs of Finite Groups, Int. J. Anal. Appl., 24 (2026), 320.

Main Article Content

Khaled Mustafa Al-Jamal, Ayser Nasir Tahat, Mowafaq Omar Al-Qadri

Abstract

In this paper, we study the \(b\)-chromatic number \(\varphi(\Gamma(G))\) of power graphs of finite groups. We present new bounds for non-cyclic \(p\)-groups in terms of the number of maximal cyclic subgroups, and we establish exact values for cyclic groups, elementary abelian groups, and certain direct products. Our results are supported by illustrative examples and computational evidence obtained using GAP, which also suggest directions for further research. This work emphasizes the strong interplay between subgroup structure and graph coloring invariants.

Article Details

References

  1. F. Ali, S. Fatima, W. Wang, On the Power Graphs of Certain Finite Groups, Linear Multilinear Algebra 70 (2022), 3803–3817. https://doi.org/10.1080/03081087.2020.1856028.
  2. A.M. Alotaibi, K.M. Aljamal, Exploring the Associated Groups of Quasi-Free Groups, Eur. J. Pure Appl. Math. 17 (2024), 2329–2335. https://doi.org/10.29020/nybg.ejpam.v17i3.5258.
  3. A. Alotaibi, K. Al-Tahat, K.M. Aljamal, On Finite Groups with Transfer Maps and Weak Closure, Eur. J. Pure Appl. Math. 18 (2025), 6859. https://doi.org/10.29020/nybg.ejpam.v18i4.6859.
  4. K.M. Aljamal, A.T. Ab Ghani, K.A. Al-Sharo, Finite Groups in Which Maximal Subgroups of Sylow p-Subgroups Are Nearly S-Permutable, Int. J. Math. Comput. Sci. 15 (2020), 877–884.
  5. K.M. Al-Jamal, A.T. Ab Ghani, On the Relation between CT-Groups and NSP-Groups on Finite Groups, J. Phys.: Conf. Ser. 1366 (2019), 012069. https://doi.org/10.1088/1742-6596/1366/1/012069.
  6. D. Božović, D.M. Štesl, I. Peterin, On the Star b-Chromatic Number of a Graph, arXiv:2504.00553, 2025. https://doi.org/10.48550/arXiv.2504.00553.
  7. P.J. Cameron, S. Ghosh, The Power Graph of a Finite Group, Discret. Math. 311 (2011), 1220–1222. https://doi.org/10.1016/j.disc.2010.02.011.
  8. B. Curtin, G.R. Pourgholi, H. Yousefi-Azari, On the Punctured Power Graph of a Finite Group, Australas. J. Comb. 62 (2015), 1–7. https://ajc.maths.uq.edu.au/pdf/62/ajc_v62_p001.pdf.
  9. B. Effantin, H. Kheddouci, The b-Chromatic Number of Some Power Graphs, Discrete Math. Theor. Comput. Sci. 6 (2003), 45–54. https://doi.org/10.46298/dmtcs.333.
  10. R.W. Irving, D.F. Manlove, The b-Chromatic Number of a Graph, Discrete Appl. Math. 91 (1999), 127–141. https://doi.org/10.1016/s0166-218x(98)00146-2.
  11. M. Kouider, M. Mahéo, Some Bounds for the b-Chromatic Number of a Graph, Discret. Math. 256 (2002), 267–277. https://doi.org/10.1016/s0012-365x(01)00469-1.
  12. C.V.G. Lima, N.A. Martins, L. Sampaio, M.C. Santos, A. Silva, b-Chromatic Index of Graphs, Electron. Notes Discrete Math. 44 (2013), 9–14. https://doi.org/10.1016/j.endm.2013.10.003.
  13. S. Shaebani, The b-Chromatic Number of Regular Graphs via the Edge Connectivity, arXiv:1105.2909, 2011. https://doi.org/10.48550/arXiv.1105.2909.
  14. A.N. Tahat, J.H. Ahmed, A. Hazaymeh, Shape Preserving Monotonic and Convex Data Interpolation Using Rational Cubic Ball Functions, Int. J. Neutrosophic Sci. 25 (2025), 489–500. https://doi.org/10.54216/ijns.250340.
  15. A.N. Tahat, A.R.M. Piah, Z.R. Yahya, Rational Cubic Ball Curves for Monotone Data, AIP Conf. Proc. 1750 (2016), 030021. https://doi.org/10.1063/1.4954557.