Coloring picture fuzzy graphs through their cuts and its computation

In a fuzzy set (FS), there is a concept of alpha-cuts of the FS for alpha in [0,1]. Further, this concept was extended into (alpha,delta)-cuts in an intuitionistic fuzzy set (IFS) for delta in [0,1]. One of the expansions of FS and IFS is the picture fuzzy set (PFS). Hence, the concept of (alpha,delta)-cuts was developed into (alpha,delta,beta)-cuts in a PFS where beta is an element of [0,1]. Since a picture fuzzy graph (PFG) consists of picture fuzzy vertex or edge sets or both of them, we have an idea to construct the notion of the (alpha,delta,beta)-cuts in a PFG. The steps used in this paper are developing theories and algorithms. The objectives in this research are to construct the concept of (alpha,delta,beta)-cuts in picture fuzzy graphs (PFGs), to construct the (alpha,delta,beta)-cuts coloring of PFGs, and to design an algorithm for finding the cut chromatic numbers of PFGs. The first result is a definition of the (alpha,delta,beta)-cut in picture fuzzy graphs (PFGs) where (alpha,delta,beta) are elements of a level set of the PFGs. Further, some properties of the cuts are proved. The second result is a concept of PFG coloring and the chromatic number of PFG based on the cuts. The third result is an algorithm to find the cuts and the chromatic numbers of PFGs. Finally, an evaluation of the algorithm is done through Matlab programming. This research could be used to solve some problems related to theories and applications of PFGs.


Introduction
Zadeh first introduced the fuzzy set (FS) theory in 1965 that could be used to handle indeterminate phenomena in real-life problems [1]. Every member in the FS is assigned to a single value of membership degree. However, in real problems, sometimes people want to know a degree of non-membership of an element. To deal with this problem, Atanassov initiated an intuitionistic fuzzy set (IFS) where there are membership and non-membership degrees of each element [2]. Recently, Cuong constructed a picture fuzzy set (PFS), an extension of FS and IFS theories, by adding a neutral membership degree for each element [3]. In other words, there are three types of degrees assigned to each element in the PFS (i.e., positive, negative, and neutral membership degrees). Some aspects of the PFS have also been investigated [4] [5].
Based on FS's fuzzy relation, Rosenfeld proposed a fuzzy graph to deal with indeterminacy in a network [1]. After that, an intuitionistic fuzzy graph (IFG) has also been constructed through the concept of intuitionistic fuzzy relation [6], [7]. Nowadays, a concept of picture fuzzy graph (PFG) has been initiated by means of picture fuzzy relation [8] [9]. Further, Zuo et al. [10] gave some operations on PFG, i.e., union, join, and cartesian product. Meanwhile, Xiao et al. [11]  In a fuzzy set (FS), there is a concept of alpha-cuts of the FS for alpha in [0,1]. Further, this concept was extended into (alpha,delta)-cuts in an intuitionistic fuzzy set (IFS) for delta in [0,1]. One of the expansions of FS and IFS is the picture fuzzy set (PFS). Hence, the concept of (alpha,delta)cuts was developed into (alpha,delta,beta)-cuts in a PFS where beta is an element of [0,1]. Since a picture fuzzy graph (PFG) consists of picture fuzzy vertex or edge sets or both of them, we have an idea to construct the notion of the (alpha,delta,beta)-cuts in a PFG. The steps used in this paper are developing theories and algorithms. The objectives in this research are to construct the concept of (alpha,delta,beta)-cuts in picture fuzzy graphs (PFGs), to construct the (alpha,delta,beta)-cuts coloring of PFGs, and to design an algorithm for finding the cut chromatic numbers of PFGs. The first result is a definition of the (alpha,delta,beta)-cut in picture fuzzy graphs (PFGs) where (alpha,delta,beta) are elements of a level set of the PFGs. Further, some properties of the cuts are proved. The second result is a concept of PFG coloring and the chromatic number of PFG based on the cuts. The third result is an algorithm to find the cuts and the chromatic numbers of PFGs. Finally, an evaluation of the algorithm is done through Matlab programming. This research could be used to solve some problems related to theories and applications of PFGs. provided some properties of them. Furthermore, Ismayil et al. [12] investigated an edge domination problem in PFGs, Jayalakshmi and Vidhya [13] initiated a direct sum on PFGs, and Talebi et al. [14] determined energy in PFGs. In addition, some scholars verified applications of PFGs, such as in Koczy et al. [15], Mohanta et al. [16], Das and Ghorai [17][18], Das et al. [19][20], Sitara et al. [21], and Mani et al. [22].
The concept of α-cuts of an FS for α∈[0,1] was first proposed by Zadeh [1]. Further, the α-cuts were extended into α,δ-cuts in an IFS [2]. Furthermore, Cuong and Kreinovich expanded the concept into the α,δ,β-cuts in a PFS [3]. This background motivated us to develop the cuts in picture fuzzy graphs (PFGs). The concepts of fuzzy graph coloring and IFG coloring have been proposed by researchers [23]- [29]. However, no researchers discussed the concept of coloring in the PFGs until now. Therefore, the objectives of this research are as follows: to develop the concept of α,δ,β-cuts in picture fuzzy graphs (PFGs) because the PFGs consist of picture fuzzy vertex or edge sets or both of them; to investigate some properties of the α,δ,β-cuts in the PFGs; to construct a concept of PFG coloring and the chromatic number of PFG based on the α,δ,β-cuts; and to design an algorithm for determining the cuts and the chromatic number of the PFGs. The last objective is to evaluate the algorithm through Matlab programming. These are novelties of this research.
We organize this paper in the following: introduction and some basic concepts are summarized in section 1. Further, the methods used in this research are given in section 2. Moreover, the construction of the α,δ,β-cuts of PFGs and the cuts' properties, a coloring concept of PFGs based on the cuts, and an algorithm are described in section 3. The conclusion and upcoming research are given at the end section.
• A union of PFSs A  and B  , symbolized by ̃∪ � , is a set [3] " • An intersection of A  and B  is defined as [3] " • Given two PFSs • A height of positive membership is a set ℎ( + ) = sup { � ( )}. Whereas, the heights of neutral and negative memberships are the sets ℎ( Furthermore, the concept of picture fuzzy graph and related concepts are described below [8] [30].

Definition 4. Given a graph * ( , )
The picture fuzzy graph (PFG) in Definition 4 is a PFG with picture fuzzy vertex and edge sets. Meanwhile, a PFG with crisp vertex set is a graph � ( , � ) with picture fuzzy edge set which automatically satisfies (1).
Based on the concept of the subset of PFS, a subset of PFG is defined as follows. Given PFGs Example 2.1. The PFG in Fig. 1 strong. Meanwhile, the PFG in Fig. 2 is complete. However, both of the two PFGs are not regular.

Method
This research is theoretical research that includes the development of theories and the construction of an algorithm. The steps used in this research are as follows: 1) Developing a concept of α,δ,β-cut in the PFGs based on α,δ,β-cut of the PFS.

4)
Defining the chromatic number, i.e., the minimum number to color vertices of PFGs, through the cut chromatic numbers.

5)
Constructing an algorithm to determine the α,δ,β-cuts and the chromatic number of PFGs based on the previous steps' concepts.
6) Evaluating the algorithm through Matlab programming.

Results and Discussion
The main results in this paper are discussed in this section. Firstly, we develop the notion of level set and α,δ,β-cuts in picture fuzzy graphs (PFGs) as in Definition 3.2.1 and Definition 3.2.2.

α,δ,β-Cuts of picture fuzzy graphs (PFGs)
The concept of level set in Definition 3.1.1 is constructed based on the PFS level set concept as given in [5].

Properties of the , , -cuts of PFGs
We prove some properties of the , , -cuts of PFGs as stated in Theorem 3.2.1-3.2.2.

Coloring PFGs through , , -Cut Graphs
Given a graph * = ( , ) and � ( � , � ) is a picture fuzzy graph on * . Let Γ = { , , } be a family of , , -cuts of the PFG � , for , , ∈ ∪ {0}, 1 α δ β + + ≤ , and is the level set of � . We focus on vertex coloring of the PFG through crisp coloring on the family Γ. Further, we determine a crisp chromatic number , , of each cut , , . A chromatic number of the PFG � is defined below. An interpretation of the chromatic number of the PFG is as follows: • When the values and δ are low and the value is high, there are a lot of adjacent vertices (more incompatible vertices). Hence, more colors are used for vertex coloring of � . • Conversely, when the values and δ are high, and the value is low, there are not so many adjacent vertices (less incompatible vertices). Hence, fewer colors are used for vertex coloring of � .
An illustration of an application of coloring of a PFG is given in the example as follows. • The degree ( ) is the level of insecurity when we apply the number of phase in an intersection.
• The degree ( ) is the indeterminacy level whether the condition is secure or not when we apply the number of phase in an intersection.
The lower values and and the higher value in the network indicate that we need to arrange traffic flows at a maximum security level since there are much incompatible traffic flows in the intersection, and vice versa.
Further, some properties of the cut chromatic numbers of the PFGs are verified.
According to a property of crisp chromatic number, we get Proof. According to Theorem 4.2.2, we get , , is a complete crisp graph with vertices. Thus, its chromatic number is , , = . █ Example 3.3.2. Given a PFG in Fig. 6 (left). Some , , -cuts of the PFG and their chromatic numbers 71 Vol. 7, No. 1, March 2021, pp. 63-75 Rosyida and Suryono (Coloring picture fuzzy graphs through their cuts and its computation) , , are described in Fig. 8. Note that the vertex-labels 1 = , 2 = , 3 = , and 4 = . We can see in Fig Table 1. Moreover, we construct an algorithm to find the , , -cuts of a PFG and determine chromatic number , , of each cut , , G α δ β as in Fig. 9.  We evaluate the algorithm through a computer-based experiment using Matlab programming. Let us consider a PFG � ( � , � ) in Table 2 (No 1, column 3). Some of , , -Cuts of the PFG which are outputs of the Matlab program file, are shown in Table 2 (No 2-6). Column 1 contains the values y1,y2, and y3 in step 1. Meanwhile, column 2 and column 3 contain the output of step 2-20. Table 2. Some of , , -Cuts of the PFG in Fig. 4 No y1=α,y2=δ,y3=β , , G α δ β

Conclusion
We have developed the notion of α,δ,β-cuts of picture fuzzy graphs (PFGs) and have investigated some properties of the cuts. Further, we have used the concept to define PFGs coloring and the chromatic number of the PFGs based on the α,δ,β-cuts. Based on the concepts that have been constructed, we designed an algorithm for determining the α,δ,β-cuts and the PFGs chromatic number. Finally, evaluations of the algorithm have been done through a computer-based experiment. The results of this research will have an impact on problem-solving related to theories and applications of PFGs. In the upcoming research, we will propose PFGs coloring based on strong adjacency between vertices and define the strong chromatic number of the PFGs.