# 2-Quasitotal Fuzzy Graphs and Their Total Coloring

Authors: V. N. Srinivasa Rao Repalle; Fekadu Tesgera Agama
Journal: Advances in Fuzzy Systems
 (2020)

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## 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.


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## Body

## 1. Introduction

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 [3], which is followed by Harary, who contributed the concept of total graphs [4]. Jayaraman studied the total chromatic number of total graphs [5]. Besides, Sastry and Raju defined quasitotal graphs [6], and Sirnivasarao and Rao introduced 1-quasitotal graphs and bounds for its total chromatic number [7]. 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 [8], 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 [9], Rosenfeld introduced fuzzy graphs [10]. Later, Bhattacharya [11] gave some remarks on fuzzy graphs. Some operations on fuzzy graphs were introduced by Mordeson and Peng [12]. 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 [13]. Lavanay and Sattanathan extended the concept of fuzzy vertex coloring into a family of fuzzy sets [14]. Kavitha [15] defined the total fuzzy graph and studied the total chromatic number of total graphs of fuzzy graphs [1]. Kavitha derived fuzzy chromatic numbers for various graphs of complete fuzzy graphs [15]. 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]. 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 [18]. Koam and Akram described decision making analysis in the real-life applications like marine crimes and road crimes by using graph structures [19]. 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].This paper is being organized as follows: In Section2, 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.


## 2. Preliminaries

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 [22]).Definition 1.
Fuzzy Graph
A fuzzy graph is defined as an ordered triple f, whereV is the set of vertices v1,v2,…,vn,σ is a fuzzy subset of V, such that σ:V⟶0,1 and μ are a fuzzy relation on σ with μ:V⟶0,1 and that μ:V×V⟶0,1 such that. μu,v≤σuΛσv∀u,v∈V.Definition 2.
Crisp Graph
The underlying crisp graph of the fuzzy graphG=V,σ,μ is denoted by G∗=V,E, where E⊆V×V. The crisp graph V,E is a special fuzzy graph G with each vertex, and each edge of G has the same degree of membership equal to 1.Definition 3.
Order and Size of Fuzzy Graph
LetG=V,σ,μ be a fuzzy graph with the underlying set V. Then, the order of G denoted by OrderG is defined as follows: (1)OrderG=∑u∈Vσu,and the size of G denoted by SizeG and defined as follows: (2)Sizeu=∑u,v∈Vμu,v.Definition 4.
Degree of a Vertex.
LetG=V,σ,μ be a fuzzy graph. The degree of a vertex u∈V is defined as follows:(3)dGu=∑v≠u,v∈Vμu,v.Definition 5.
Busy Value of a Vertex.
LetG=V,σ,μ be a fuzzy graph. The busy value of the vertex v in G is Dv=∑iσvΛσvi where vi are neighbors of v and the busy value of G is DG=∑iDvi where vi are the vertexes of G.Definition 6.
Adjacent Vertices
Ifμu,v>0, then u and v are said to be adjacent to each other and lie on the edge, e=u,v. A path ρ in a fuzzy graph G=V,σ,μ is a sequence of distinct nodes v0,v1,v2,…,vn such that μvi−1,vi>0,1≤i≤n. Here n is called the length of the path.Definition 7.
(see [23]). Path in Fuzzy Graph
A pathP in a fuzzy graph G=σ,μ is a sequence of distinct vertices u0,u1,…,un (except possibly u0 and un ) such that μui−1,ui>0,i=1,2,…,n. Here, n is called the length of the path.Definition 8.
Connected Vertices
Ifu,v are vertices in G and if they are connected by means of a path, then the strength of that path is defined as ∧ni=1μvi−1,vi. If u,v are connected by means of paths of length k, then(4)μku,v=supμu,v1Λμv1,v2Λμv2,v3Λ,…,Λμvk−1,v:u,v1,v2,…,vk−1,v∈V,
Ifu,v∈V, then, the strength of connectedness between u and vμ∞u,v=supμku,v:k=1,2,…,Definition 9.
Connected Fuzzy Graph
LetG=V,σ,μ be a fuzzy graph. Then, G is said to be connected if μ∞u,v>0for allu,v∈σ∗. An arc u,v is said to be a strong arc if μu,v≥μ∞u,v and a node u is said to be an isolated node, if. μu,v=0,for allu≠v.Definition 10.
(see [24]) Cyclic Fuzzy Graph
G=V,σ,μ is a fuzzy cycle if and only if σ∗,μ∗ is a cycle and there does not exist a unique x,y∈μ∗ such that μx,y=Λμu,v:u,v∈μ∗.Definition 11.
(see [25]). Total Coloring
A familyΓ=γ1,γ2,γ3,…,γk of fuzzy sets on V∪E is called a k− fuzzy total coloring of G=V,σ,μ, if(a)
Maxγiv=σv for all v∈V and Maxγiu,v=μu,v for all edges u,v∈E(b)
γiΛγj=0(c)
For every adjacent vertexu,v of G, Minγiu,γiv=0
The least value ofk for which there exists a k− fuzzy coloring is called the fuzzy total chromatic number of G and is denoted by χTfG.Definition 12.
(see [18]). 1-Quasitotal Fuzzy Graph
LetG=V,σ,μ be a fuzzy graph with its underlying set V and crisp graph G∗=σ∗,μ∗. The pair Q1TfG=σQ1Tf,μQ1Tf of the fuzzy graph G is defined as follows:
Let the node set ofQ1TfG be V∪E, where V is the vertex set and E is the edge set of the underlying crisp graph. The fuzzy subset σQ1Tf is defined on V∪E as follows:(5)σQ1Tfu=σu,ifu∈V,σQ1Tfe=μe,ife∈E.
The fuzzy relationμQ1Tf is defined on V∪E×V∪E, called edges of Q1TfG as follows:(6)μQ1Tfu,v=μu,v,ifu,v∈VμQ1Tfei,ej=μeiΛμej,ifeiandejhave a node in common between them=0,Otherwise.
By definition,μQ1Tfu,v≤σQ1TfuΛσQ1Tfv for all u,v∈V∪E. Hence, μQ1Tf is a fuzzy relation on the fuzzy subset σQ1Tf. Thus, the pair Q1TfσQ1Tf,μQ1Tf is a fuzzy graph, and it is termed as 1-Quasitotal fuzzy graph of G.


## 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.
LetG=V,σ,μ be a fuzzy graph with its underlying set V and crisp graph G∗=σ∗,μ∗. The pair Q2TfG=σQ2Tf,μQ2Tf of the fuzzy graph G is defined as follows:
Let the node set ofQ2TfG be the union of the vertex set and the edge set of the underlying crisp graph. That is V∪E.
Let the fuzzy subsetσQ2Tf be defined on V∪E as follows:(7)σQ2Tfu=σu,ifu∈V,σQ2Tfe=μe,ife∈E.
Let the fuzzy relationμQ2Tf be defined on V∪E×V∪E, called edges of Q2TfG as follows:(8)μQ2Tfu,v=μu,v,ifu,v∈VμQ2Tfu,e=σuΛμe,ifu∈Ve∈Eand the nodeulies on the edgee=0,otherwise.
By the definition of the fuzzy graph,μQ2Tfu,v≤σQ2TfuΛσQ2Tfv for all u,v∈V∪E. Hence, μQ2Tf is a fuzzy relation on the fuzzy subset σQ2Tf. Therefore, the pair Q2TfσQ2Tf,μQ2Tf is a fuzzy graph, and it is termed as 2-Quasitotal Fuzzy Graph of G.Example 1.
LetG=V,σ,μ be a fuzzy graph with its underlying crisp graph G∗=V,E, where V=v1,v2,v3 and edge set E=v1v2,v2v3,v3v1. Let the fuzzy vertex set defined on V be as σ:S⟶0,1 such that(9)σv1=13,σv2=12,σv3=14.
Let the fuzzy relation defined on the fuzzy edge set be asμ:SXS⟶0,1 such that(10)μv1,v2=13,μv2,v3=15,μv3,v1=14.
However,(11)13=μv1,v2≤σv1Λσv2=13Λ12=13,15=μv2,v3≤σv2Λσv3=12Λ14=14,14=μv3,v1≤σv3Λσv1=14Λ13=14.
Then, we haveμvi,vj≤σviΛσvj for all vi,vj∈V, and hence the graph G=V,σ,μ 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 Example1 as follows.
That is,Q2TfσQ2Tf,μQ2Tf of the fuzzy graph G, where the node set of Q2Tf is V∪E, which is the set v1,v2,v3,v1v2,v2v3,v3v1. Hence, we define the fuzzy subset δQ2Tf as follows:(12)σQ2Tfu=σu,ifu∈V,σQ2Tfe=μe,ife∈E.
Thus, we have the following fuzzy subsetsσQ2T:(13)σQ2Tfv1=σv1=13,σQ2Tfv2=σv2=12,σQ2Tfv3=σv3=14,σQ2Tfv1v2=μv1,v2=13,σQ2Tfv2v3=μv2,v3=15,σQ2Tfv3v1=μv3,v1=14.
The fuzzy relationsμQ2Tf will be as follows:(14)μQ2Tfu,v=μu,v,ifu,v∈VμQ2Tfu,e=σuΛμe,ifu∈V,e∈Eandulies on the edgee=0,otherwise.
Hence,(15)μQ2Tfv1,v2=μv1,v2=13,μQ2Tfv2,v3=μv2,v3=15,μQ2Tfv3,v1=μv3,v1=14,μQ2Tfv1,v1v2=σv1Λμv1,v2=13,μQ2Tfv1,v3v1=σv1Λμv3,v1=14,μQ2Tfv1,v2v3=0,μQ2Tfv2,v2v3=σv2Λμv2,v3=15,μQ2Tfv2,v2v1=σv2Λμv2,v1=13,μQ2Tfv2,v3v1=0,μQ2Tfv3,v3v1=σv3Λμv3,v1=14,μQ2Tfv3,v3v2=σv3Λμv3,v2=15,μQ2Tfv3,v1v2=0.
However,(16)13=μQ2Tfv1,v2≤σv1Λσv2=13,15=μQ2Tfv2,v3≤σv2Λσv3=14,14=μQ2Tfv3,v1≤σv3Λσv1=14,13=μQ2Tfv1,v1v2≤σv1Λμv1,v2=13,14=μQ2Tfv1,v3v1,≤σv1Λμv3,v1=14,15=μQ2Tfv2,v2v3≤σv2Λμv2,v3=15,13=μQ2Tfv2,v2v1≤σv2Λμv2,v1=13,14=μQ2Tfv3,v3v1≤σv3Λμv3,v1=14,15=μQ2Tfv3,v3v2≤σv3Λμv3,v2=15.
Thus, we conclude thatμQ2Tfvi,vj≤σQ2TfviΛσQ2Tfvj for all vi,vj∈V∪E; thus the graph Q2TfσQ2Tf,μQ2Tf is a fuzzy graph and is called 2-quasitotal fuzzy graph of the fuzzy graph G in Example 1.
Now, based on the node setsV∪E, fuzzy subsets σQ2Tf, and fuzzy relations μQ2Tf, the 2-quasitotal fuzzy graph of G is as shown in Figure 2.Figure 1
Fuzzy graphG.Figure 2
2-Quasitotal fuzzy graph ofG.Example 2.
Consider the following graphG=V,σ,μ with the fuzzy vertex set:
σv1=1, σv2=0.75, σv3=1, σv4=0.25 and fuzzy edge set:(17)μv1,v2=0.5,μv2,v3=0.5,μv3,v4=0.25,μv4,v1=0.25.
Clearly,μvi,vj≤σviΛσvj for all vi,vj∈V, the graph G=V,σ,μ is a fuzzy graph and its graph is as shown in Figure 3.
Now, the construction of 2-quasitotal fuzzy graphQ2TfσQ2Tf,μQ2Tf of the graph G in Example 2 will be as follows.(i)
The node set ofσQ2Tf will be as follows:(18)V∪E=v1,v2,v3,v4,v1v2,v2v3,v3v4,v4v1.(ii)
The fuzzy subsetσQ2TfG will be as follows:(19)σQ2Tfu=σu,ifu∈V,σQ2Tfe=μe,ife∈E.Hence,(20)σQ2Tfv1=σv1=1,σQ2Tfv2=σv2=0.75,σQ2Tfv3=σv3=1,σQ2Tfv4=σv4=0.25,σQ2Tfv1v2=μv1,v2=0.5,σQ2Tfv2v3=μv2,v3=0.5,σQ2Tfv3v4=μv3,v4=0.25,σQ2Tfv4v1=μv4,v1=0.25.(iii)
The fuzzy relationμQ2Tf will be as follows:(21)μQ2Tfu,v=μu,v,ifu,v∈V,μQ2Tfu,e=σuΛμe,ifulies on the edge ofe,=0,Otherwise.
Hence,(22)μQ2Tfv1,v2=μv1,v2=0.5,μQ2Tfv2,v3=μv2,v3=0.5,μQ2Tfv3,v4=μv3,v4=0.25,μQ2Tfv4,v1=μv4,v1=0.25,μQ2Tfv1,v1v2=σv1Λμv1,v2=1Λ0.5=0.5,μQ2Tfv1,v2v3=0,μQ2Tfv1,v3v4=0,μQ2Tfv1,v4v1=σv1Λμv4,v1=1Λ0.25=0.25,μQ2Tfv2,v2v3=σv2Λμv2,v3=0.75Λ0.5=0.5,μQ2Tfv2,v3v4=0,μQ2Tfv2,v4v1=0,μQ2Tfv2,v1v2=σv2Λμv1,v2=0.75Λ0.5=0.5,μQ2Tfv3,v3v4=σv3Λμv3,v4=1Λ0.25=0.25,μQ2Tfv3,v4v1=0,μQ2Tfv2,v1v2=0,μQ2Tfv3,v2v3=σv3Λμv2,v3=1Λ0.5=0.5,μQ2Tfv4,v4v1=σv4Λμv4,v1=0.25Λ0.25=0.25,μQ2Tfv4,v1v2=0,μQ2Tfv4,v2v3=0,μQ2Tfv4,v3v4=σv4Λμv3,v4=0.25Λ0.25=0.25.
Clearly,μQ2Tfvi,vj≤σQ2TfviΛσQ2Tfvj for all vi,vj∈V∪E and hence the graph Q2TfδQ2Tf,μQ2Tf 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.Figure 3
Fuzzy graph.GFigure 4
2-Quasitotal fuzzy graph ofG.


## 4. Properties of 2-Quasitotal Fuzzy Graph

Theorem 1.
LetG=V,σ,μ be a fuzzy graph. Then,(23)OrderQ2TfG=OrderG+SizeG.Proof.
From the definition of 2-quasitotal fuzzy graph, we have the node set ofQ2TfG as VUE and the fuzzy subset σQ2Tfu=σu,ifu∈V and σQ2Tfe=μe,ife∈E.
Now,(24)OrderQ2TfG=∑u∈V∪EσQ2Tfu,(by the definition of the order of G).(25)=∑u∈VσQ2Tfu+∑u∈EσQ2Tfu=∑u∈Vσu+∑u∈Eσu,by definition ofσQ2Tfu,=OrderG+SizeG,OrderQ2TfG=OrderG+SizeG.Note 1.
For any fuzzy graph,G=V,σ,μ,(1)
OrderTG=OrderG+SizeG, where TG is the total fuzzy graph of G.(2)
OrderQ1TfG=OrderG+SizeG, where Q1TfG is 1-quasitotal fuzzy graph of G:(26)OrderQ2TfG=OrderG+SizeG.(3)
OrderTG=OrderQ1TfG=OrderQ2TfG=OrderG+SizeG.Theorem 2.
LetG=V,σ,μ be a fuzzy graph, then(27)SizeQ2TfG=SizeG+∑u∈V,e∈EσuΛμe.Proof.
By the definition of the size of a fuzzy graph, we have the following:(28)SizeQ2TfG=∑u,v∈V∪EμQ2Tfu,v=∑u,v∈VμQ2Tfu,v+∑u∈V,e∈EμQ2Tfu,e+∑ei,ej∈EμQ2Tfei,ej=∑u,v∈VμQ2Tfu,v+∑u∈V,e∈EμQ2Tfu,e+0.
(The third summation is zero since there is no fuzzy relation betweenei,ej∈E in 2-quasitotal fuzzy graph)(29)=∑u,v∈Vμu,v+∑u∈V,e∈EμQ2Tfu,e,=∑u,v+∑μu,e,=SizeG+∑σuΛμe,SizeQ2TfG=SizeG+∑σuΛμe.Note 2.
For any fuzzy graphG=V,σ,μ,(1)
SizeTG=3SizeG+∑ei,ej∈EμeiΛμej, where TG is total fuzzy graph of G(2)
SizeQ1TfG=SizeG+∑ei,ej∈EμeiΛμej, where Q1Tf is 1-quasitotal fuzzy graph of G(3)
SizeQ2TfG=SizeG+∑σuΛμe, where Q2Tf is 2-quasitotal fuzzy graph of GTheorem 3.
LetG=V,σ,μ be a fuzzy graph; then,(30)dQ2Tfu=dGu+∑u∈V,e∈EσuΛμe,ifu∈V,=∑ei∈E,u∈VμeiΛσu,ifei∈E.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.
Letu∈V. Then,(31)dQ2TfGu=∑u,v∈VμQ2TfGu,v+∑u∈V,e∈EμQ2TfGu,e,(where u lies on the edge of e in the second summation)(32)=∑u,v∈Vμu,v+∑u∈V,e∈Eμu,e,=dGu+∑u∈V,e∈EσuΛμe,dQ2TfGu=dGu+∑u∈V,e∈EσuΛμe.Case 2.
Letei∈E; then,(33)dQ2TfGei=∑u∈VμQ2TfGei,u+∑ej∈EμQ2TfGei,ej=∑u∈Vμei,u+0.
(The second summation is zero since there is no fuzzy relation betweenei,ej∈E in 2-quasitotal fuzzy graph)(34)=∑u∈Vμei,u=∑u∈VμeiΛσu,dQ2TfGei=∑u∈VμeiΛσu.Note 3.
For any fuzzy graphG=V,σ,μ,(1)
dTGu=2dGu,ifu∈V, where TGu is the total fuzzy graph of G=busy value ofeiinTG,ifu∈E(2)
dQ1TfGu=dGu, if u∈V=bussy value ofeiinQ1TfG, if u∈E , and where Q1TfG is 1-quasitotal fuzzy graph of G(3)
dQ2Tfu=dGu+∑u∈V,e∈EσuΛμe, ifu∈V=∑u∈VμeiΛσuand if ei∈E and Q2Tf is 2-quasitotal fuzzy graph of GTheorem 4.
2-quasitotal fuzzy graph of any connected fuzzy graph is a connected graph.Proof.
LetG=V,σ,μ be a fuzzy graph.
The fuzzy vertex set ofQ2TfG consists of V∪E of G. The fuzzy relation μQ2TfG is defined only for μQ2Tfu,v,whereu,v∈V and μQ2Tfu,e,whereu∈V,e∈E and u lies on the edge of E.
SinceG is connected and every edge of G is also considered as a node for Q2TfG, there is at least one path that connects every vertex u and v in Q2TfG and μQ2Tf∞u,v≠0.
Hence,Q2TfG 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 familyΓ=γ1,γ2,…,γk, of a fuzzy set on V∪E is called a 2-quasi k-fuzzy total coloring of fuzzy graph G=V,σ,μ, if the following three conditions are met.(i)
Maxγiv=σvfor allv∈V and Maxγiu,v=μu,v,for all edgesu,v∈E.γiΛγj=0(ii)
For every adjacent vertexu,v of Q2TfG, Minγiu,γiv=0.
The least number of colors possible is called 2-quasitotal fuzzy chromatic number ofQ2TfG and it is denoted by χQ2fG.Example 3.
Consider a fuzzy graphG=V,σ,μ as shown in Figure 5.
From the graph, we have the vertex setV=v1,v2,v3,v4,v5,v6 and edge set E=v1v2,v2v3,v3v4,v4v5,v5v6,v6v1, whose membership functions can be expressed as follows from the graph:(35)σvi=0.2,fori=1,0.7,fori=2,0.5,fori=3,0.4,fori=4,0.6,fori=5,0.3,fori=6,μvi,vj=0.2,fori,j=1,2,0.5,fori,j=2,3,0.1,fori,j=3,4,0.4,fori,j=4,5,0.3,fori,j=5,6,0.1,fori,j=6,1.
The family of fuzzy setsΓ=γ1,γ2 on V∪E will be as follows:(36)γ1vi=0.2,fori=1,0.5,fori=3,0.6,fori=5,0,Otherwise,γ2vi=0.7,fori=20.4,fori=40.3,fori=60,Otherwise,γ1vi,vj=0.2,fori,j=1,2,0.1,fori,j=3,4,0.3,fori,j=5,6,0,Otherwise,γ2vi,vj=0.5,fori,j=2,3,0.4,fori,j=4,5,0.1,fori,j=6,1,0,Otherwise.
To justify that the family of fuzzy setsΓ=γ1,γ2 defined as above satisfies the definition of the total coloring of the fuzzy graph and determines its total chromatic number, χTfG, we use Table 1 to check for the three conditions of the total coloring of a fuzzy graph.
From Table1, we observe that the family of the fuzzy set Γ=γ1,γ2 satisfies the definition of the total coloring of a fuzzy graph G. Hence, χTfG=2.
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 Example3.
Now, to construct a 2-quasitotal fuzzy graphQQ2TfG=V∪E,σQ2Tf,μQ2Tf, where. V∪E=v1,v2,v3,v4,v5,v6,v1v2,v2v3,v3v4,v4v5,v5v6,v6v1. The fuzzy subset of Q2TfG will be as follows: (37)σQ2Tfvi=0.2,fori=1,0.7,fori=2,0.5,fori=3,0.4,fori=4,0.6,fori=5,0.3,fori=6,σQ2Tfvi,vj=0.2,fori,j=1,2,0.5,fori,j=2,3,0.1,fori,j=3,4,0.4,fori,j=4,5,0.3,fori,j=5,6,0.1,fori,j=6,1.
The fuzzy relation will be as follows:(38)μQ2Tfvi,vj=0.2,fori,j=1,2,0.5,fori,j=2,3,0.1,fori,j=3,4,0.4,fori,j=4,5,0.3,fori,j=5,6,0.1,fori,j=6,1,μQ2Tfvi,vivj=0.2,fori,ij=1,12,0.1,fori,ij=1,61,0.5,fori,ij=2,23,0.2,fori,ij=2,12,0.1,fori,ij=3,34,0.5,fori,ij=3,23,0.4,fori,ij=4,45,0.4,fori,ij=4,34,0.3,fori,ij=5,56,0.4,fori,ij=5,45,0.1,fori,ij=6,61,0.3,fori,ij=6,56.
Hence, the 2-quasitotal fuzzy graphQQ2TfG=V∪E,σQ2Tf,μQ2Tf of the fuzzy graph G in Example 3 is as shown in Figure 6.
LetΓ=γ1,γ2 be a family of fuzzy subset defined on V∪E as follows:(i)
For the vertex set:(39)γ1vi=0.2,fori=1,0.5,fori=3,0.6,fori=5,0,Otherwise,γ1vivj=0.2,forij=12,0.1,forij=34,0.3,forij=56,0,Otherwise,γ2vi=0.7,fori=2,0.4,fori=4,0.3,fori=6,0,Otherwise,γ2vivj=0.5,fori,j=2,3,0.4,fori,j=4,5,0.1,fori,j=6,1,0,Otherwise.(ii)
For the edge set:(40)γ1vi,vj=0.2,fori,j=1,2,0.1,fori,j=3,4,0.3,fori,j=5,6,0,Otherwise,γ1vi,vivj=0.2,fori,ij=1,12,0.5,fori,ij=2,23,0.1,fori,ij=3,34,0.4,fori,ij=4,45,0.3,fori,ij=5,56,0.1,fori,ij=6,61,0,otherwise,γ2vivj=0.5,fori,j=2,3,0.4,fori,j=4,5,0.1,fori,j=6,1,0,Otherwise,γ2vi,vivj=0.1,fori,ij=1,61,0.2,fori,ij=2,12,0.5,fori,ij=3,23,0.4,fori,ij=4,34,0.4,fori,ij=5,45,0.3,fori,ij=6,56,0,otherwise.
Using Table2 below, we can check whether Γ satisfies the definition of 2-quasitotal fuzzy coloring of G.
As shown in Table1, Γ=γ1,γ2 satisfies the definition of 2-quasitotal fuzzy coloring of a fuzzy graph G.
Therefore,χQ1TfG=2.Figure 5
Fuzzy graph ofG.Table 1
Example of the total coloring of a fuzzy graphG=V,σ,μ.
v and eγ1γ2Max.γ1Λγ2Min.v10.200.20Minγiv1,γiv2=0v200.70.70Minγiv2,γiv3=0v30.500.50Minγiv3,γiv4=0v400.40.40Minγiv4,γiv5=0v50.600.60Minγiv5,γiv6=0v600.30.30Minγiv6,γiv1=0v1v20.200.20v2v300.50.50v3v40.100.10v4v500.40.40v5v60.300.30v6v100.10.10Figure 6
Total fuzzy graph of fuzzy graphG.Table 2
Example of 2-quasitotal coloring of a fuzzy graphG=V,σ,μ
v and eγ1γ2Max.γ1Λγ2Min.v10.200.20Minγiv1,γiv2=0v200.70.70Minγiv2,γiv3=0v30.500.50Minγiv3,γiv4=0v400.40.40Minγiv4,γiv5=0v50.600.60Minγiv5,γiv6=0v60.30.30Minγiv6,γiv1=0v1v20.200.20Minγiv1v2,γiv2v3=0v2v300.50.50Minγiv2v3,γiv3v4=0v3v40.100.10Minγiv3v4,γiv4v5=0v4v500.40.40Minγiv4v5,γiv5v6=0v5v60.300.30Minγiv5v6,γiv6v1=0v6v100.10.10Minγiv6v1,γiv1v2=0v1,v20.200.20v2,v300.50.50v3,v40.100.10v4,v500.40.40v5,v60.300.30v6,v100.10.10v1,v1v20.200.20v1,v6v100.10.10v2,v2v30.500.50v2,v1v200.20.20v3,v3v40.400.40v3,v2v300.50.50v4,v4v50.400.40v4,v3v400.40.40v5,v5v60.300.30v5,v4v500.40.40v6,v6v10.100.10v6,v5v600.30.30Note 4.
Unfortunately,χQ1TfG=χQ2TfG=2 for this example and no evidence that it is always true in this manuscript.


## 6. Conclusion

This article has introduced the new concept of 2-quasitotal fuzzy graph for a given fuzzy graph. The concept is clearly explained with the particle examples by giving a fuzzy graph and its 2-quasitotal fuzzy graph. Some properties of the 2-quasitotal fuzzy graphs have been proposed and proved. Further, the theorems and results obtained for 2-quasitotal fuzzy graphs are compared with the existing properties of total fuzzy graphs and 1-quasitotal fuzzy graphs. Lastly, it has been defined 2-quasitotal coloring for a fuzzy graph and its total coloring is exemplified.



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