Research Article | Open Access
V. N. Srinivasa Rao Repalle, Fekadu Tesgera Agama, "2-Quasitotal Fuzzy Graphs and Their Total Coloring", Advances in Fuzzy Systems, vol. 2020, Article ID 8814220, 10 pages, 2020. https://doi.org/10.1155/2020/8814220
2-Quasitotal Fuzzy Graphs and Their Total Coloring
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.
As of its emerging, the graph theory rapidly moved into the mainstream of mathematics. It has diverse applications in the fields of science and technology [1, 2]. In 1965, the total coloring of the graph was introduced by Behazad , which is followed by Harary, who contributed the concept of total graphs . Jayaraman studied the total chromatic number of total graphs . Besides, Sastry and Raju defined quasitotal graphs , and Sirnivasarao and Rao introduced 1-quasitotal graphs and bounds for its total chromatic number . Nowadays, many real-world problems cannot be properly modeled by a crisp graph theory as the problems contain uncertain information. The fuzzy set theory, anticipated by Zadeh , is used to handle the phenomena of uncertainty and real-life situation. Coloring of fuzzy graphs plays a vital role in both theory and practical applications. It is mainly studied in combinatorial optimization problems such as traffic light control, exam scheduling, and register allocation.
After Zadeh’s paper on fuzzy sets , Rosenfeld introduced fuzzy graphs . Later, Bhattacharya  gave some remarks on fuzzy graphs. Some operations on fuzzy graphs were introduced by Mordeson and Peng . As an advancement, the fuzzy coloring of the fuzzy graph was defined by Eslahchi and Onagh in 2004 and later developed by themselves as fuzzy vertex coloring in 2006 . Lavanay and Sattanathan extended the concept of fuzzy vertex coloring into a family of fuzzy sets . Kavitha  defined the total fuzzy graph and studied the total chromatic number of total graphs of fuzzy graphs . Kavitha derived fuzzy chromatic numbers for various graphs of complete fuzzy graphs . Nevethana studied about fuzzy total coloring and its chromatic number of complete fuzzy graphs . Sitara and Akram studied fuzzy graph structures and their applications . The total coloring of 1-quasitotal graph for crisp graph was studied. Recently Fekadu and SrinivasaRao Repalle have established the definition of 1-quasitotal fuzzy graph and its total coloring . Koam and Akram described decision making analysis in the real-life applications like marine crimes and road crimes by using graph structures . Akram and Sitara introduced the concept of Residue Product of Fuzzy Graph Structures and studied their properties . Akram covers both theories and applications of introduction to m-polar fuzzy graphs and m-polar fuzzy hypergraphs .
This 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. In Section 3, 2-quasitotal fuzzy graph is defined and the concept is justified with numerous examples. Section 4 describes and proves some properties of 2-quasitotal fuzzy graphs and compares the result with the properties of total fuzzy graphs and 1-quasitotal fuzzy graphs. Furthermore, Section 5 introduces the concept of 2-quasitotal fuzzy coloring and deliberates some of its properties. Finally, the paper is concluded in Section 6.
In this section, some basic definitions that are necessary for this paper have been included. Unless otherwise mentioned, the concepts are from Mordeson and Nair (see ).
Definition 1. Fuzzy Graph
A fuzzy graph is defined as an ordered triple f, where is the set of vertices is a fuzzy subset of , such that and are a fuzzy relation on with and that such that.
Definition 2. Crisp Graph
The underlying crisp graph of the fuzzy graph is denoted by , where . The crisp graph is a special fuzzy graph with each vertex, and each edge of has the same degree of membership equal to 1.
Definition 3. Order and Size of Fuzzy Graph
Let be a fuzzy graph with the underlying set . Then, the order of denoted by is defined as follows: and the size of denoted by and defined as follows:
Definition 4. Degree of a Vertex.
Let be a fuzzy graph. The degree of a vertex is defined as follows:
Definition 5. Busy Value of a Vertex.
Let be a fuzzy graph. The busy value of the vertex in is where are neighbors of and the busy value of is where are the vertexes of .
Definition 6. Adjacent Vertices
If then and are said to be adjacent to each other and lie on the edge, A path in a fuzzy graph is a sequence of distinct nodes such that Here is called the length of the path.
Definition 7. (see ). Path in Fuzzy Graph
A path in a fuzzy graph is a sequence of distinct vertices (except possibly and ) such that . Here, n is called the length of the path.
Definition 8. Connected Vertices
If are vertices in and if they are connected by means of a path, then the strength of that path is defined as If are connected by means of paths of length , thenIf then, the strength of connectedness between and
Definition 9. Connected Fuzzy Graph
Let be a fuzzy graph. Then, is said to be connected if . An arc is said to be a strong arc if and a node is said to be an isolated node, if.
Definition 10. (see ) Cyclic Fuzzy Graph
is a fuzzy cycle if and only if is a cycle and there does not exist a unique such that .
Definition 11. (see ). Total Coloring
A family of fuzzy sets on is called a fuzzy total coloring of , if(a) for all and for all edges (b)(c)For every adjacent vertex of , The least value of for which there exists a fuzzy coloring is called the fuzzy total chromatic number of and is denoted by .
Definition 12. (see ). 1-Quasitotal Fuzzy Graph
Let be a fuzzy graph with its underlying set and crisp graph The pair of the fuzzy graph is defined as follows:
Let the node set of be , where is the vertex set and is the edge set of the underlying crisp graph. The fuzzy subset is defined on as follows:The fuzzy relation is defined on , called edges of as follows:By definition, for all Hence, is a fuzzy relation on the fuzzy subset . Thus, the pair is a fuzzy graph, and it is termed as 1-Quasitotal fuzzy graph of .
3. 2-Quasitotal Fuzzy Graph
This section introduces the definition of 2-quasitotal fuzzy graph and sketches the 2-quasitotal fuzzy graph of a given fuzzy graph.
Definition 13. Let be a fuzzy graph with its underlying set and crisp graph The pair of the fuzzy graph is defined as follows:
Let the node set of be the union of the vertex set and the edge set of the underlying crisp graph. That is .
Let the fuzzy subset be defined on as follows:Let the fuzzy relation be defined on , called edges of as follows:By the definition of the fuzzy graph, for all Hence, is a fuzzy relation on the fuzzy subset . Therefore, the pair is a fuzzy graph, and it is termed as 2-Quasitotal Fuzzy Graph of .
Example 1. Let be a fuzzy graph with its underlying crisp graph , where and edge set . Let the fuzzy vertex set defined on be as such thatLet the fuzzy relation defined on the fuzzy edge set be as such thatHowever,Then, we have for all , and hence the graph is a fuzzy graph and its graph is as shown in Figure 1.
Now, let us construct the 2-quasitotal fuzzy graph of the fuzzy graph in Example 1 as follows.
That is, of the fuzzy graph , where the node set of is , which is the set . Hence, we define the fuzzy subset as follows:Thus, we have the following fuzzy subsets :The fuzzy relations will be as follows:Hence,However,Thus, we conclude that for all ; thus the graph is a fuzzy graph and is called 2-quasitotal fuzzy graph of the fuzzy graph in Example 1.
Now, based on the node sets , fuzzy subsets , and fuzzy relations , the 2-quasitotal fuzzy graph of is as shown in Figure 2.
Example 2. Consider the following graph with the fuzzy vertex set:
, , , and fuzzy edge set:Clearly, for all , the graph is a fuzzy graph and its graph is as shown in Figure 3.
Now, the construction of 2-quasitotal fuzzy graph of the graph in Example 2 will be as follows.(i)The node set of will be as follows:(ii)The fuzzy subset will be as follows: Hence,(iii)The fuzzy relation will be as follows:Hence,Clearly, for all and hence the graph is a fuzzy graph and it is a 2-quasitotal fuzzy graph of a graph in the above Example 2, and its graph is as shown in Figure 4.
4. Properties of 2-Quasitotal Fuzzy Graph
Theorem 1. Let be a fuzzy graph. Then,
Proof. From the definition of 2-quasitotal fuzzy graph, we have the node set of as and the fuzzy subset and .
Now,(by the definition of the order of ).
Note 1. For any fuzzy graph, ,(1), where is the total fuzzy graph of .(2), where is 1-quasitotal fuzzy graph of :(3).
Theorem 2. Let be a fuzzy graph, then
Proof. By the definition of the size of a fuzzy graph, we have the following:(The third summation is zero since there is no fuzzy relation between in 2-quasitotal fuzzy graph)
Note 2. For any fuzzy graph ,(1), where is total fuzzy graph of (2), where is 1-quasitotal fuzzy graph of (3), where is 2-quasitotal fuzzy graph of
Theorem 3. Let be a fuzzy graph; then,
Proof. By the definition of the degree of a vertex of a fuzzy graph, we have the following two cases to prove the theorem.
Case 1. Let . Then,(where lies on the edge of in the second summation)
Case 2. Let ; then,(The second summation is zero since there is no fuzzy relation between in 2-quasitotal fuzzy graph)
Note 3. For any fuzzy graph ,(1), where is the total fuzzy graph of (2), if , if , and where is 1-quasitotal fuzzy graph of (3), ifand if and is 2-quasitotal fuzzy graph of
Theorem 4. 2-quasitotal fuzzy graph of any connected fuzzy graph is a connected graph.
Proof. Let be a fuzzy graph.
The fuzzy vertex set of consists of of . The fuzzy relation is defined only for and and lies on the edge of .
Since is connected and every edge of is also considered as a node for , there is at least one path that connects every vertex and in and .
Hence, is a connected fuzzy graph.
5. 2-Quasitotal Fuzzy Coloring
In this section, we introduce the concept of 2-quasitotal fuzzy total coloring and discuss some of its properties.
Definition 14. A family of a fuzzy set on is called a 2-quasi k-fuzzy total coloring of fuzzy graph , if the following three conditions are met.(i) and .(ii)For every adjacent vertex of , .The least number of colors possible is called 2-quasitotal fuzzy chromatic number of and it is denoted by .
Example 3. Consider a fuzzy graph as shown in Figure 5.
From the graph, we have the vertex set and edge set whose membership functions can be expressed as follows from the graph:The family of fuzzy sets on will be as follows:To justify that the family of fuzzy sets defined as above satisfies the definition of the total coloring of the fuzzy graph and determines its total chromatic number, , we use Table 1 to check for the three conditions of the total coloring of a fuzzy graph.
From Table 1, we observe that the family of the fuzzy set satisfies the definition of the total coloring of a fuzzy graph . Hence, .
When we come to our point of concern, we need to determine the chromatic number of 2-quasitotal fuzzy graph of the fuzzy graph in Example 3.
Now, to construct a 2-quasitotal fuzzy graph , where. . The fuzzy subset of will be as follows: The fuzzy relation will be as follows:Hence, the 2-quasitotal fuzzy graph of the fuzzy graph in Example 3 is as shown in Figure 6.
Let be a family of fuzzy subset defined on as follows:(i)For the vertex set: