Chromaticity of Normal Graphs and Cyclic Graphs

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 these graphs, 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.