The Locating Chromatic Number for the New Operation on Generalized Petersen Graphs N_P(m,1)

Abstract

The locating chromatic number is a graph invariant that quantifies the minimum number of colors required for proper vertex coloring, ensuring that any two vertices with the same color have distinct sets of neighbors. This study introduces a new operation on generalized Petersen graphs denoted by N_(P(m,1)), exploring its impact on locating chromatic numbers. Through systematic analysis, we aim to determine the specific conditions under which this operation influences the locating chromatic number and provide insights into the underlying graph-theoretical properties. The method for computing the locating chromatic number for the new operation on generalized Petersen graphs, denoted by N_(P(m,1)), entails determining the lower and upper limits. The results indicate that the locating chromatic number for the new operation on the generalized Petersen graph is 4 for m=4 and 5 for m≥5. The findings contribute to a broader understanding of graph coloring.