Chromaticity of Normal Graphs and Cyclic Graphs

Hussein, Abdullah

doi:10.25673/121022

Proceedings of International Conference on Applied Innovation in IT · 2025/07/26 · Vol. 13 · Issue 3 · pp. 279–285

Chromaticity of Normal Graphs and Cyclic Graphs

Abdullah AbdulRazzaq Obaid and Haneen Kareem Hussein

📄 Download PDF DOI: 10.25673/121022

Abstract

In our work, we will find new results about the chromaticity of Normal graphs and Cyclic Graphs of Dihedral group 𝐷2𝑛, (𝐺𝑁𝑜𝑟𝐷2𝑛 and 𝐺𝐶𝑦𝑐𝐷2𝑛, respectively). We will determine the new theorems about the chromaticuniqueness of these graphs, and we will study all cases of a positive integer number 𝑛 for 𝐺𝑁𝑜𝑟𝐷2𝑛 and𝐺𝐶𝑦𝑐𝐷2𝑛. Our primary objective is to establish new theorems regarding the chromatic uniqueness of thesegraphs, which will significantly enhance the understanding of their structural and combinational characteristics. We will thoroughly analyze all cases of positive integers 𝑛 to accurately determine the chromatic numbers and systematically explore the implications of these findings within graph theory. This research not only contributes to the theoretical framework of graph theory by enriching its foundational concepts but also opens promising avenues for future interdisciplinary studies examining the intricate interplay between group theory, combinatorial mathematics, and graph properties, thereby advancing both theoretical insights and practical computational applications.

Keywords

Chromaticity Normal Graphs Cyclic Graphs Dihedral Group Chromatic Uniqueness.

References

H. Al-Janabi and G. Bacso, "Integral-root Polynomials and Chromatic Uniqueness of Graphs," Journal of Discrete Mathematical Sciences and Cryptography, vol. 24, no. 4, pp. 1127–1147, 2021.
H. Al-Janabi and G. Bacso, "Chromatic Uniqueness of Zero-Divisor Graphs," The Art of Discrete and Applied Mathematics, vol. 6, no. 1, 2022.
I. Beck, "Coloring of Commutative Rings," Journal of Algebra, vol. 116, pp. 208–226, 1988.
G. D. Birkhoff, "A Determinate Formula for the Number of Ways of Coloring a Map," Annals of Mathematics, vol. 14, no. 2, pp. 42–46, 1912.
W. C. Calhoun, "Counting the Subgroups of Some Finite Groups," American Mathematical Monthly, vol. 94, no. 1, pp. 54–59, 1987.
S. R. Cavior, "The Subgroups of the Dihedral Groups," Mathematics Magazine, vol. 48, 1975.
C. Y. Chao and E. G. Whitehead Jr., "On Chromatic Equivalence of Graphs," in Theory and Applications of Graphs, Proc. Int. Conf., Western Michigan Univ., Kalamazoo, Mich., 1976, Lecture Notes in Mathematics, vol. 642, Springer, Berlin, pp. 121–131, 1978.
G. Chartrand and P. Zhang, Chromatic Graph Theory, Taylor and Francis Group, LLC, USA, 2009.
G. L. Chia, "A Bibliography on Chromatic Polynomials," Discrete Mathematics, vol. 172, pp. 175–191, 1997.
G. L. Chia, "On Graphs Unique l-Colorable under the Action of Their Automorphism."
F. M. Dong, K. M. Koh, and K. L. Teo, Chromatic Polynomials and Chromaticity of Graphs, World Scientific Publishing Co. Pte. Ltd., Singapore, 2005.
S. Sh. Gehet and A. M. Khalaf, "Chromatic Polynomials and Chromaticity of Zero-Divisor Graphs," in Proc. 3rd Int. Conf. on Applied Mathematics and Pharmaceutical Sciences, Singapore, Apr. 2013.
O. A. Hadi and H. B. Shelash, "Finite Groups with the Probability of Parameters," IOP Journal of Physics, 2nd Int. Conf. IICPS, Babelon University, Iraq, submitted 2021.
O. A. Hadi and H. B. Shelash, "Relation between Normality Degree and Normal Graph of Some Finite Groups," 1st Annual Iranian International Group Theory Conference (IGTC), submitted 2022.
A. N. Jasim and A. A. Najim, "Solving Edges Deletion Problem of Complete Graphs," Baghdad Science Journal, vol. 21, no. 12, 2024.
A. N. Jasim and A. A. Najim, "Edges Deletion Problem of Hypercube Graphs for Some n," World Scientific Connect, vol. 21, no. 12, 2024.
K. M. Koh and K. L. Teo, "The Search for Chromatically Unique Graphs-II," Discrete Mathematics, vol. 172, pp. 59–78, 1997.
K. M. Koh and K. L. Teo, "The Search for Chromatically Unique Graphs," Graphs and Combinatorics, vol. 6, no. 3, pp. 259–285, 1990.
L. Lovász, Combinatorial Problems and Exercises, 2nd ed., AMS Chelsea Publishing, American Mathematical Society, Island, 1993.
A. A. Obaid and H. Al-Janabi, "Chromaticity of Normal Graphs," Passer Journal of Basic and Applied Sciences, accepted for publication.
R. C. Read, "An Introduction to Chromatic Polynomials," Journal of Combinatorial Theory, vol. 4, 1968.
S. Roman, Fundamentals of Group Theory, pp. 52–71, 2011.
H. Whitney, "The Coloring of Graphs," Annals of Mathematics, vol. 33, pp. 688–718, 1932.
S. A. Al-Ameedee and A. J. Obaid, "New Results and Application of Differential Quasi Subordinations for Higher-Order Derivatives of Meromorphic Multivalent Functions," Journal of Interdisciplinary Mathematics, pp. 1–8, doi: 10.47974/JIM-1483.
S. A. Al-Ameedee, H. A. Ahmed, and A. J. Obaid, "Differential Subordinations of Meromorphic Multivalent Harmonic Functions," Journal of Interdisciplinary Mathematics, vol. 27, no. 4, pp. 947–952, 2024, doi: 10.47974/JIM-1915.