Exploring Edge-Neighbor Distinguishing Proper Coloring in Neutrosophic Graphs: Theory and Applications

Abstract

The article introduces the novel concept of the chromatic number of neutrosophic graphs, expanding upon the theoretical framework of graph coloring from crisp graphs to incorporate elements of neutrosophic logic. It explores the theoretical underpinnings of neutrosophic graphs and their coloring theory, aiming to broaden the understanding of graph coloring in uncertain environments. Definitions of neighboring vertex distinguishing suitable edge coloring for neutrosophic graphs are provided, leveraging  -cuts and distance functions. The article establishes lower bounds on the chromatic number based on edge coloring properties, showcasing distinctions from general chromatic numbers. Furthermore, it highlights differences between adjacent vertex distinguishing chromatic numbers and offers insights into creating (p,f)- edge-neighbor distinguishing proper coloring (ENDPC) for neutrosophic graphs. Recommendations for further research into differentiating coloring of neutrosophic graphs are provided, indicating the ongoing exploration and potential applications of this emerging area of study.