Combinatorial analysis of line graphs: domination, chromaticity, and Hamiltoniancity

Line graphs are a fundamental class of graphs extensively studied for their structural properties and applications in diverse fields such as network design, optimization, and algorithm development. Pan and lollipop graphs, with their distinctive hybrid structures, offer a fertile ground for exploring combinatorial properties in their line graphs. Motivated by the need to better understand domination, chromaticity, and Hamiltonian properties in line graphs, this study examined the line graphs of pan and lollipop graphs. These investigations were inspired by their potential applications in connectivity analysis and optimization in networks. We derived analytical formulas for the domination and chromatic numbers of these line graphs, established relationships between these parameters and their corresponding original graphs, and proved that the line graph of a pan graph is Hamiltonian while that of a lollipop graph is traceable. The methodology combines established theoretical results and inequalities, including domination bounds and chromaticity relations, with rigorous combinatorial analysis. Our results not only contribute to the theoretical understanding of line graphs but also have implications for practical problems in network optimization and graph algorithm design, opening avenues for further research into hybrid graph structures.