Bilangan Kromatik Lokasi Disjoint Union Graf Barbel Shadow Lintasan

Abstract

This study investigates the locating chromatic number of the disjoint union of barbell shadow path graphs. The locating chromatic number is a variation of the classical chromatic number, ensuring not only that adjacent vertices receive different colors but also that every vertex can be uniquely distinguished through its color code with respect to all color classes. The research focuses on determining the locating chromatic number for the disjoint union of barbell shadow path graphs with , odd , with . The proof method involves analyzing the graph structure, defining vertex and edge sets, and constructing consistent color codes for all vertices. The results demonstrate that the locating chromatic number of all such graphs is consistently equal to 6. This finding indicates that the complexity of barbell shadow path graph structures does not increase the required number of colors in the disjoint union case. Therefore, the research contributes to strengthening the theory of locating coloring in graphs and provides potential applications in network analysis, coding theory, and optimization systems that require unique identification based on graph structures.