3.1. The AVDPEC Function of a Fuzzy Graph

A fuzzy set A defined on X can be characterized from its -cuts family , . This family of sets is monotone, i.e., it verifies for any satisfying .

Let be the family of -cuts set of , where the -cut of a fuzzy graph is the crisp graph with . In particular, with , is the support set of the fuzzy edge set , i.e., . can be called the support graph of the fuzzy graph .

Based on the family of -cuts set of , the definition of the adjacent vertex distinguishing proper edge coloring of fuzzy graph is given as follows.

Definition 17.

Given a fuzzy graph . Let be the family of -cuts set of , and for all , is a simple graph with no isolated edge. An adjacent vertex distinguishing proper edge coloring of is a family of functions satisfied is the AVDPEC function of the crisp graph .

In other words, the AVDPEC of fuzzy graph

can be expressed as

, where

,

. From the results in [

33

,

34

,

35

,

39

,

40

,

41

,

42

,

43

],

k

is a positive integer related to

. For example, when

has neither isolated edges nor triangular subgraphs,

; when

has no isolated edges and

,

.

A k-AVDPEC is an adjacent vertex distinguishing proper edge coloring using at most k colors.

Definition 18.

Given a fuzzy graph , be the family of -cuts set of , an adjacent vertex distinguishing proper edge chromatic number of is .

The adjacent vertex distinguishing proper edge chromatic number of is . This means that when k is less than , there is at least one such that has no k-AVDPEC. Similar to the notation of symbols in a crisp graph, for a k-AVDPEC of , the set of colors incident to v denoted by of the vertex v. Hence, k-AVDPEC function can be defined on . The k-AVDPEC function of is defined through this sequence.

Remark 3.

We note that the fuzzy graph degenerates to a crisp graph Z called a zero graph when the membership function μ of is a zero matrix. Any two vertices in the graph Z are not adjacent, so its chromatic number can be defined as 0. In addition, if there are isolated edges in the fuzzy graph , or a in the -cuts set of contains isolated edges, the fuzzy graph does not have the adjacent vertex distinguishing proper edge coloring mentioned in Definition 17. For fuzzy graphs containing rings and parallel edges, the discussion of adjacent vertex distinguishing coloring is consistent with the case of distinct graphs. Therefore, the adjacent vertex distinguishing coloring is only discussed for simple fuzzy graphs without isolated edges.

Example 5.

A fuzzy graph is depicted in Figure 5, where and the membership function of is μ as follows.

In some sense, the traffic light problem can be modeled as the above vertex coloring of ; refer to [16]. and t denote the incompatibility null, low, medium, high and total, respectively, , . For an adjacent vertex distinguishing proper edge coloring of fuzzy graph, the vertex in graph can be interpreted as the object that the artificial intelligence equipment needs to recognize, such as chicken, duck, dog, goose, and wild goose. If two objects have some common properties, connect an edge between them; we can refer to graph in Figure 4. The image of edge represents the attribute difference between and .

The fuzzy coloring problem consists of determining the chromatic number of a fuzzy graph and an associated coloring function. In our approach, for any level α, the minimum number of colors needed to color the crisp graph will be computed. In this way, the adjacent vertex distinguishing proper edge fuzzy chromatic number will be defined through its -cuts.

In fuzzy graph , five crisp graphs are obtained by considering the values . For each , Table 1 shows the adjacent vertex distinguishing proper edge chromatic number of , and the color set of vertex v under the coloring function .

Crisp graph

in column 2 of

Table 1

and its specific coloring method (or coloring function

) are shown in

Figure 6

.

The crisp graph is a 5 order complete graph with by Lemma 3. The crisp graph is a 3 order odd circle with by Lemma 4. The adjacent vertex distinguishing proper edge chromatic number of and are both the lower bounds in Lemma 1. is the zero graph. Therefore, the adjacent vertex distinguishing proper edge coloring of is a family of functions ,,,. It can be shown that the adjacent vertex distinguishing proper edge chromatic number of is .

Consider an artificial intelligence recognition problem for animals represented by vertices

of the fuzzy graph

in

Figure 5

. Let the edges represent the conflicting nature of the same attribute between pairs of vertices, and the degree of membership of fuzzy edge indicates the degree of conflict. Depending on the objective attributes of different animals, the degree of conflict of edges is assigned as High (

h

), Low (

l

), and Medium (

m

). Let the following fuzzy graph

in

Figure 5

represents the situation. The color number

of the fuzzy graph

can be interpreted as: lower values of

are associated with lower attribute difference between two animals; artificial intelligence equipment needs more information (i.e., color) to distinguish them and the chromatic number is high. On the other hand, for higher values of

, the higher the difference between the two animals when the chromatic number is lower, the less information is needed to distinguish them.

In order to solve the fuzzy adjacent vertex distinguishing proper edge coloring problem, any algorithm which computes the chromatic number of every crisp graph

can be used. For the graph with fixed order, an exact algorithm can be used, see [

44

].

In Example 5, we observe that the adjacent vertex distinguishing proper edge chromatic number of

seems to decrease with the increase in

, which is true for the point staining of fuzzy graphs in [

16

], but not necessarily for the adjacent vertex distinguishing proper edge coloring of the fuzzy graph. We illustrate this with Example 6; it is constructed based on the interesting properties of the adjacent vertex distinguishing proper edge coloring of crisp graph that we mentioned in the introduction.

Example 6.

Let be a fuzzy graph. and the membership function of is defined as

and is shown in Figure 7.

In our approach, for any level

, the minimum number of colors needed to color the crisp graph

will be computed. In this way, the adjacent vertex distinguishing proper edge chromatic number of a fuzzy graph will be defined through its

-cuts set. For each

,

Table 2

shows the adjacent vertex distinguishing proper edge chromatic number

of

, and the color set

of vertex

v

in

under the coloring function

.

Crisp graph

in column 2 of

Table 2

and its specific coloring method (or coloring function

) are shown in

Figure 8

.

The crisp graph is a 5 order complete graph with by Lemma 3. The crisp graph is a 5 order odd circle with by Lemma 4. The adjacent vertex distinguishing proper edge chromatic number of reaches the lower bounds in Lemma 1. is a zero graph. From the above process, we can get that the adjacent vertex distinguishing proper edge coloring of is a family of functions ,,. It can be shown that the adjacent vertex distinguishing proper edge chromatic number of is . Obviously, the adjacent vertex distinguishing proper edge chromatic number of the crisp graph does not decrease with the increase in . Therefore, for two fuzzy graphs and , it is not necessarily true that when .

Based on the family of -cuts set of , the definition of the maximum degree of can be defined naturally.

Definition 19.

Let be a fuzzy graph, and be the support graph of . The maximum degree of is the number .

Based on Definition 17, some of the conclusions in the crisp graph can be naturally generalized to the fuzzy graphs with crisp vertices and fuzzy edges. As shown in Theorems 2 and 3, an adjacent vertex distinguishing proper edge chromatic number of two isomorphic fuzzy graphs is equal, and an adjacent vertex distinguishing proper edge chromatic number of either fuzzy graph is not less than its maximum degree.

Theorem 2.

Let and be two fuzzy graphs. If , then .

Proof.

Obviously, this theorem can be obtained by Theorem 1 and the definition of the adjacent vertex distinguishing proper edge chromatic number of fuzzy graphs. □

Theorem 3.

Given a fuzzy graph . Let be the family of -cuts set of . If crisp graph without isolated edges for all , then .

Proof.

According to Definition 19, . By the Lemma 1 and the definition of the adjacent vertex distinguishing proper edge chromatic number of fuzzy graph, it is obvious that . □

3.2. The ()-Extended AVDPEC Function of Fuzzy Graph

Consider an examination scheduling problem with exams represented as vertices of a fuzzy graph. Fuzzy edges indicate that at least one student takes the two exams corresponding to the vertices. The exam scheduling problem is usually solved by transforming it into a vertex coloring problem for graphs, while satisfying the constraint that no student is required to write two examinations at the same time, and some other constraints, such as that certain groups of exams may be required to take place at the same time, the priority of different exam limits, exam ordering, etc. There are dozens of such constraints summarized in reference [

45

], but these constraints do not include the case of avoiding students taking multiple consecutive exams. In order to deal effectively with this new constraint, Susana Muñoz et al. [

16

] gave a new definition of (

)-extended coloring for fuzzy graphs by introducing a dissimilarity measure

d

defined on the color set and a scale function

f

. In the following, we restrict this new coloring problem to the edge coloring problem, and we obtain the second method for graph coloring in fuzzy graph by the distance description. Moreover, the adjacent vertex distinguishing proper edge coloring of the fuzzy graph defined in this way can avoid some of the cases mentioned in Remark 3. Since the fuzzy graphs containing isolated edges do not have an adjacent vertex distinguishing proper edge coloring, Definition 17 cannot be performed for the fuzzy graphs containing isolated edges in the cuts set, while the

-extended adjacent vertex distinguishing proper edge coloring of fuzzy graphs can avoid this situation.

Definition 20

([

16

]).

Let N be the available color set. d is a distance measure defined by with the following properties.

(i) , ,

(ii) ,

(iii) .

The basic idea of the graph coloring problem is to group edges or vertices in a graph as little as possible. Edges or vertices subject to different constraints are called incompatible, and these incompatible objects will not appear in the same group eventually. The distance measure function d can reflect the degree of incompatibility of adjacent fuzzy edges, that is, the more incompatible two edges are, the farther away their related colors are. In this way, an extended coloring function is introduced.

Given , the image of the membership function of a fuzzy graph . We assume that there is an order < defined on the elements of I to give the definition of the scale function.

Definition 21.

Let . The function is called a scale function if

The distance measure and scale function introduced above lead to the following definition.

Definition 22.

Given a fuzzy graph , where the membership function of is μ. A -extended AVDPEC function of , denoted by is a mapping satisfying the following conditions.

(i) for all edges and ,

(ii) if .

The minimum value k for which a -extended k-AVDPEC of exists is the -extended adjacent vertex distinguishing proper edge chromatic number of and it is denoted by .

Remark 4.

It should also be noted that the two ways of writing the same fuzzy edge in the fuzzy graph are not distinguished, i.e., for all , and indicate the same fuzzy edge. Both and indicate that two fuzzy edges and are adjacent to each other. Therefore, condition (i) in Definition 21 can also be written in the following form.

(ii) for all edges and .

A -extended AVDPEC function is a -extended AVDPEC function which takes maximum t different colors. In other words, where which satisfies the following conditions.

(i)

(ii)

The

-extended AVDPEC of a fuzzy graph

can be considered as a generalization of the AVDPEC of a crisp graph

. Take

,

,

, and

where

is defined as

The coloring given in Definition 22 differs from the adjacent vertex distinguishing proper edge coloring of a crisp graph in the following sense. In the classical graph coloring theory, any color that does not exceed its corresponding chromatic number is used. However, this does not necessarily hold in a -extended adjacent vertex distinguishing proper edge coloring, as we illustrate by the following Example 7.

Example 7.

Let , where . Let be a fuzzy graph where and the membership function of is

Let be color set and d be a distance measure defined as . Let the scale function be defined as follows (Table 3).
The fuzzy graph is shown in Figure 9. Crisp graph T is a connected acyclic graph, which we call a tree. From Lemma 5, the adjacent vertex distinguishing proper edge chromatic number of tree T is 3.

Consider the following cases for .

Case 1. and .

This assignment is possible because

whereas . It is not possible to color with 2 because of

Therefore, we notice that the constraint (i) in Definition 22 is not satisfied.

Case 2. When and , it is exactly the same as Case 1.

Case 3. There are four assignment methods.

(i) and ;

(ii) and ;

(iii) and ;

(iv) and .

Above assignments of colors are not possible because in each case whereas . Thus, it is not possible to color the above graph.

Similarly, it can be proved that there is no 4–extended adjacent vertex distinguishing proper edge coloring for when color set . If color set , we can obtain a 5–extended AVDPEC function of .

A 5–extended AVDPEC function is

and , whereas . Therefore, .

From the example above, we note that k–extended adjacent vertex distinguishes proper edge coloring; unlike the classical adjacent vertices distinguishing proper edge coloring, there can be some colors that are not assigned to any vertices, such as , . In other words, color 2 and color 4 are not assigned to any edges of .

3.3. An Simple Algorithm

The algorithm given in reference [

44

] can rapidly and efficiently calculate the adjacent vertex distinguishing proper edge chromatic number of crisp graphs with fixed order. Calling the adjacent vertex distinguishing proper edge coloring algorithm in [

43

], adjacency matrix

of the adjacent vertex distinguishing proper edge coloring of a crisp graph

G

can be output. The different distance functions

d

and the scale functions

f

define a new

-extended AVDTC function. From the properties of the adjacency matrix, it is clear that the matrix

is a symmetric matrix. On the basis of this matrix, for the given distance function

and scale function

satisfying Definitions 20 and 21, respectively, the

k

-extended adjacent vertex distinguishing proper edge coloring of the fuzzy graph and its chromatic number can be obtained according to the following algorithm.

According to Definition 22, a

k

-extended adjacent vertex distinguishing proper edge coloring of

should satisfy the following two constraint conditions: (i) The distance function value of adjacent fuzzy edges is not less than its scale function value; (ii) the color sets of two adjacent vertices are not equal. The basic idea of this algorithm is to transform the adjacent vertex distinguishing proper edge coloring of the support graph

of the fuzzy graph

so that they satisfy the constraint (i). In this paper, we design a

-extended adjacent vertex distinguishing proper edge coloring algorithm for any fuzzy graph with no isolated edge and also give the algorithm steps, and a simple example is used to demonstrate the results. Implementing Algorithm 1 has the following ideas.

Algorithm 1: The algorithm for k–extended AVDPEC of fuzzy graphs.

Input: Adjacency matrix of AVDPEC;

Output: Adjacency matrix of k–extended AVDPEC.

Step 1: We input the adjacency matrix of the adjacent vertex distinguishing

proper edge coloring of the support graph of the fuzzy graph , then we store

the maximum value of the elements in the matrix in k;

Step 2: Check the matrix ,

For to begin

For to begin

For to begin

When , compute ,

If

then while

else ,

End

End

End

Step 3: Step 2 is re-implemented until the elements of the new matrix all

satisfy condition (i);

Step 4: ;

Step 5: Output , k.

Remark 5.

The above algorithm uses only the latest color for each coloring change, so it can be guaranteed that the final output coloring is still distinguishable by adjacent vertices, i.e., the color sets of any pair of adjacent fuzzy edges satisfy constraint (ii).

For the above algorithm, we choose the fuzzy tree in Example 7 as the experimental object to demonstrate. The adjacency matrix of the adjacent vertex distinguishing proper edge coloring of the fuzzy graph

is shown in

Table 4

below, where

.

Execute Step 2 in the algorithm, calculate

and

. Since

and

, we rewrite the color of the fuzzy edge

as

. After the first adjustment, the coloring result of

is shown in

Table 5

.

Check the adjacency matrix in

Table 5

again, where

. Calculate

and

. Since

and

, the color of the fuzzy edge

needs to be rewritten to

(shown in

Table 6

).

It can be computed to verify that the adjacency matrix in

Table 6

satisfies constraint (ii) and

. Thus, we get 5-

-extended adjacent vertex distinguishing proper edge coloring of fuzzy graph

, i.e.,

The color set of each vertex is

, respectively.

The factors that mainly affect the time complexity of the algorithm can be classified as two aspects: one is to generate the adjacency matrix for a fuzzy graph of order

n

; the worst time complexity is

. The other is to adjust the color, the worst complexity about finishing adjacent vertex distinguishing proper edge coloring is

, Therefore, the time complexity of the algorithm in

Section 3.3

is

.

Remark 6.

The above algorithm has the disadvantage that it can only ensure that , not equal to . This defect comes from two aspects: on the one hand, for a crisp graph with no isolated edge, there is no algorithm that can calculate accurately its adjacent vertex distinguishing proper edge chromatic number until now. Unfortunately, as the number of vertices of the fuzzy graph increases, there is a large gap between the theoretical results and the number of colors simulated with the algorithm. On the other hand, in Step 2 of the above algorithm, each time the color is rewritten, only the new color is used and the old color is not fully utilized, which also leads to the algorithm outputting a larger number of colors than the theoretical result.

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