Abstract
Coloring of fuzzy graphs has many real-life applications in combinatorial optimization problems like traffic light system, exam scheduling, and register allocation. The coloring of total fuzzy graphs and its applications are well studied. This manuscript discusses the description of 2-quasitotal graph for fuzzy graphs. The proposed concept of 2-quasitotal fuzzy graph is explicated by several numerical examples. Moreover, some theorems related to the properties of 2-quasitotal fuzzy graphs are stated and proved. The results of these theorems are compared with the results obtained from total fuzzy graphs and 1-quasitotal fuzzy graphs. Furthermore, it defines 2-quasitotal coloring of fuzzy total graphs and which is justified.
Highlights
The graph theory rapidly moved into the mainstream of mathematics
In 1965, the total coloring of the graph was introduced by Behazad [3], which is followed by Harary, who contributed the concept of total graphs [4]
Many real-world problems cannot be properly modeled by a crisp graph theory as the problems contain uncertain information. e fuzzy set theory, anticipated by Zadeh [8], is used to handle the phenomena of uncertainty and real-life situation
Summary
The graph theory rapidly moved into the mainstream of mathematics. Coloring of fuzzy graphs plays a vital role in both theory and practical applications. Nevethana studied about fuzzy total coloring and its chromatic number of complete fuzzy graphs [16]. Sitara and Akram studied fuzzy graph structures and their applications [17]. Akram and Sitara introduced the concept of Residue Product of Fuzzy Graph Structures and studied their properties [20]. Akram covers both theories and applications of introduction to m-polar fuzzy graphs and m-polar fuzzy hypergraphs [21]. Is paper is being organized as follows: In Section 2, some basic definitions and elementary concepts of the fuzzy set, fuzzy graph, and coloring of fuzzy graphs have been reviewed.
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