1-Quasi Total Fuzzy Graph and Its Total Coloring

Abstract

The fuzzy graph theory, its properties, total coloring and applications are currently climbing up. With this concept of fuzzy graph, total fuzzy graph is defined and its properties as well as fuzzy total colorings have been well discussed and studied. Similarly the theory of crisp graph, its properties, applications and colorings are well considered. Moreover, 1-quasi total graphs for crisp graphs, their properties and colorings were discussed by some researchers and the bounds for its total coloring have been established. In this manuscript, from the concept of fuzzy graph we introduced the definition of 1-quasi total graph for fuzzy graphs. To elaborate the definition we provide practical example of fuzzy graph and from this graph we construct the 1-quasi total fuzzy graph of the given fuzzy graph, so that the definition to be meaning full and their relationships can be easily observed from the sketched graphs. In addition some theorems related to the properties of 1-quasi total fuzzy graphs are stated and proved. The results of these theorems are compared with the results obtained from total fuzzy graphs, so that the differences and similarities that 1-quasi total fuzzy graph can have with that of total fuzzy graphs are revealed. Moreover, we define 1-quasi total coloring of fuzzy total graphs and give an example of total coloring of 1-quasi total graphs.