Neutrosophic Path-Coloring Numbers Based On Endpoints In Neutrosophic Graphs

New setting is introduced to study path-coloring number and neutrosophic path-coloring number arising from different types of paths based on shared endpoints amid them in neutrosophic graphs assigned to neutrosophic graphs. Consider two vertices. Minimum number of shared endpoints based on those vertices in the formations of all paths with those vertices as their starts and their ends to compare with other paths, is a number which is representative based on those vertices. Minimum neutrosophic number of latter endpoints corresponded to path-coloring set amid neutrosophic cardinality of all sets of latter endpoints corresponded to path-coloring set is called neutrosophic path-coloring number. Forming sets from special paths to figure out different types of number of paths having smallest number of colors from shared endpoints from two vertices are given in the terms of minimum number of paths to get minimum number to assign to neutrosophic graphs is key type of approach to have these notions namely path-coloring number and neutrosophic path-coloring number arising from different types of paths based on shared endpoints amid them in neutrosophic graphs assigned to neutrosophic graphs. Two numbers and one set are assigned to a neutrosophic graph, are obtained but now both settings lead to approach is on demand which is to compute and to find representatives of paths having smallest number of colors from shared endpoints from two vertices are given forming different types of sets of paths in the terms of minimum number and minimum neutrosophic number forming it to get minimum number to assign to a neutrosophic graph. Let NTG : (V,E,σ,μ) be a neutrosophic graph. Then for given two vertices, x and y, there are some paths from x to y. If two paths from x to y share an endpoint, then they’re assigned to different colors. The set of different colors, S, in this process is called path-coloring set from x to y. The minimum cardinality between all path-coloring sets from two given vertices is called path-coloring number and it’s denoted by V(NTG); for given two vertices, x and y, there are some paths from x to y. If two paths from x to y share an endpoint, then they’re assigned to different colors. The set S of different colors in this process is called path-coloring set from x to y. The minimum neutrosophic cardinality, 􏰀x∈Z 􏰀3i=1 σi(x), between all sets Zs including the latter endpoints corresponded to path-coloring set Ss, is called neutrosophic path-coloring number and it’s denoted by Vn(NTG). As concluding results, there are some statements, remarks, examples and clarifications about some classes of neutrosophic graphs namely path-neutrosophic graphs, cycle-neutrosophic graphs, complete-neutrosophic graphs, star-neutrosophic graphs, complete-bipartite-neutrosophic graphs, complete-t-partite-neutrosophic graphs, andbwheel-neutrosophic graphs. The clarifications are also presented in both sections “Setting of path-coloring number,” and “Setting of neutrosophic path-coloring number,” for introduced results and used classes. This approach facilitates identifying paths which form path-coloring number and neutrosophic path-coloring number arising from different types of paths based on shared endpoints amid them in neutrosophic graphs assigned to neutrosophic graphs. In both settings, some classes of well-known neutrosophic graphs are studied. Some clarifications for each result and each definition are provided. The cardinality of set of shared endpoints and neutrosophic cardinality of latter endpoints corresponded to path-coloring set have eligibility to define path-coloring number and neutrosophic path-coloring number but different types of shared endpoints have eligibility to define path-coloring sets. Some results get more frameworks and perspective about these definitions. The way in that, different types of shared endpoints having smallest number from all paths from two vertices are given forming different types of sets in the terms of minimum number of shared endpoints having smallest number of different paths from two vertices are given and smallest number of shared endpoints having smallest number of paths from two vertices are given forming it to get minimum number to assign to neutrosophic graphs or in other words, the way in that, consider two vertices, minimum number of shared endpoints based on those vertices in the formations of all paths with those vertices as their starts and their ends to compare with other paths, is a number which is representative based on those vertices; minimum neutrosophic number of latter endpoints corresponded to path-coloring set amid neutrosophic cardinality of all sets of latter endpoints corresponded to path-coloring set is called neutrosophic path-coloring number, opens the way to do some approaches. These notions are applied into neutrosophic graphs as individuals but not family of them as drawbacks for these notions. Finding special neutrosophic graphs which are well-known, is an open way to pursue this study. Neutrosophic path-coloring notion is applied to different settings and classes of neutrosophic graphs. Some problems are proposed to pursue this study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this article.