# An Extensive Study on Graph Colourings and Dominator Chromatic Number of Sugeno-Type Fuzzy Graphs

Authors: A. Sangeetha Devi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The concept of graph colouring has become a very active field of research that enhances many practical applications and theoretical challenges. An accurate (vertex) colouring through the condition that each vertex is whichever unaccompanied in its colour class or adjoining to all vertices of at least one colour class is known as dominator colouring. A graph is dominator colouring is a suitable colouring in which at least one vertex dominates each colour class. Many difficult and global issues are solved using fuzzy graph colouring approaches. In this paper, we introduce a new Sugeno-Type Fuzzy graph of groups, which is found using several fuzzy graph operations such as dominant path-colouring number, cycle, union, join, and products. A number that is representative based on those vertices in the formations of all paths with those vertices as their starts and ends to compare with other paths is the minimum number of shared edges based on those vertices in the formations of all paths with those vertices as their starts and ends to compare with other paths. The sugeno dominant path-colouring number is the minimum Sugeno number of shared edges among the Sugeno cardinality of all sets of shared edges, which allows for various techniques. These results are used in studying various and recently introduced chromatic numbers of graphs.


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## Body

## 1. Introduction

A graph is a practical method to express information relating to relationships between things and is a straightforward model of relation. Vertices indicate the object, while edges show relationships [1]. We need to construct a fuzzy network model when there is ambiguity in the description of the items, their connection, or both. Fuzzy graph colouring, which is used to tackle combinatorial optimisation issues like traffic light control, exam scheduling, and register allocation is one of the most crucial characteristics of fuzzy graph models. Fuzzy dominator colouring is essential in these applications. Frequently, in addition to the requirement that neighbouring vertices receive distinct colours, some extra colouring condition(s) must be met [2, 3]. A graph is correct colouring is when colours are assigned to its vertices in such a way that no two neighbouring vertices accept the matching colour. These separate sets are referred to as colour classes in this context [4–6]. A valid colouring of a graph G using Q colours is termed a Q-colouring of G and the chromatic number of G is the minimal number of colours necessary in a suitable colouring. The difficulty of determining a graph is chromatic number is a typical NP-hard problem [7]. In the fields of precise algorithms, approximation algorithms, and parameterized algorithms, this subject has received a lot of attention. Graph colouring variations such as Edge-Chromatic Number, Achromatic Number, P-Chromatic Number (PCN), Total Chromatic Number (TCN), Dominator Colouring (DC), and Class Domination Colouring (CDC) have also been thoroughly investigated [8]. Remember that in a graph G, a collection Set of vertices is a dominant set if every vertex outside Set has a neighbour in Set. Furthermore, if a vertex/subset dominates every vertex S′ in a set S′, it is said to be main [9, 10]. The G minimal number of colour classes in a dominator colouring G of is the dominator chromatic number ΨdcG.Here is further research on chromatic number-based graph colouring in fuzzy sets. The dominator chromatic number ofG, as described by Henning and Michael [11], is the smallest number of colours among all dominator colourings of G. A suitable colouring of the vertices of a graph G is a total dominator colouring in which each vertex of the graph dominates every vertex of a colour class other than its own. G represents the total dominator chromatic number is the smallest number of colours in all of G is entire dominator colouring. Klavar and Tavakoli [12] introduced dominated and dominator colouring. The dominating chromatic number of edge corona products has clear lower and upper boundaries. Planar graphs of small diameter have a dominator chromatic number and a total dominator chromatic number, according to Goddard and Henning [13]. A planar graph with a dimension of 2 has a dominator chromatic number of at most 5. The total dominator chromatic number of Kneser graphs has been created by Jalilolghadr and Behtoei [14]. A complete dominator colouring (or TDC) of a graph G is a valid colouring (i.e., a division of the vertex set V(G) into independent subsets called colour classes) in which each vertex of the graph is adjacent to all vertices of some colour class. Chen and Qin [15] studied the dominator and total dominator colouring of the circulant graph Cn(1,2). The minimal number of colours necessary for a dominator colouring. This and many another problem motivated to define fuzzy graphs Rosenfeld first introduced the concept of fuzzy graph. The study of fuzzy graph theory expands significantly after that. A fuzzy set is mathematically defined by giving each potential member of the discourse universe a grade of membership that corresponds to how much that person resembles or is compatible with the notion represented by the fuzzy set. In modelling real-time systems where the degree of information inherent in the systems vary with various levels of accuracy, fuzzy graphs have many additional uses. In this study, we expression at these factors from a different perspective. They also demonstrated the NP-completeness of the total dominator chromatic number. A survey of total dominator chromatic number in graphs. Consider the graph G, which has suitably coloured vertices. A Sugeno-type Fuzzy (STF-graph) is a collection of coloured vertices with one of each colour. Let d signify the best of vertices subsets: this best might mean that the set is dominant, forms a set, or is connected. This is what we say:If every STF-graph ofG has property d, then G has a correct colouring.If such a colouring exists, the dominator chromatic number of the graphG is defined as the least number of colours in a valid colouring that satisfies condition d. This is denoted by ΨdG the symbol. Consider the 5-cycle as an example. There ΨdG is a three-color unique colouring up to symmetry. As a result, each ΨdG is dominant, although they are not all related. There is also a four-color colouring that is essentially unique; the resultant d is connected. As a result, the d-compelling chromatic number for CN5 is 3, where d stands ΨdG for dominating, but 4 for presence connected. We expression at the Best-compelling chromatic number for a variety of features in this research. We begin by demonstrating that the dominator chromatic number corresponds to convinced chromatic selections ΨdG. Then we give some wide limitations on interesting numbers.


## 2. Preliminaries

The basic material utilise in this essay is supplied in this subsection. New ideas and explanations are also prompted [16]. The basic concept is based on the model that is being used. The first definition introduces the fundamental model.Definition 1.
(Graph). If is a collection of objects and is a subset ofv×v (is a set of 2-subsets) where v is called vertex set and e is called edge set, then G=v,e is termed a graph. At most one edge has been assigned to every two vertices.
The STF-graph, which is defined as follows, is the foundation of the results in this study. Some related concepts are also illustrated.Definition 2.
(STF-Graph and Its Special Case).STFG=v,e,α=α1,α2,α3,β=β1,β2,β3 if it’s a graph, it’s termed an STF-graph, αx:v⟶0,1 and βx:e⟶0,1. We add one condition to it and utilise a variant of the STF graph with the same name. The additional condition is as follows: for each vx′vy′∈e.(1)βvx′vy′≤αvx′∧αvy′,(i)
α is known as the STF vertex set(ii)
β is referred to as STF edge set(iii)
v It is known as the STF array and is denoted by ASTF(iv)
∑v′∈vαxv′ It is known as the STF order of STF and is denoted by AnSTF(v)
e The size of the STF e is specified by the symbol MSTF(vi)
The size of the STFSenSTF is represented by and ∑e′∈e∑i=13βxe′ is termed STF size
Some well-known STF graph classes are defined. This study is based on these types of STF graphs, and the majority of the results are about them.Definition 3.
LetSTFG=v,e,α,β be a Sugeno-type Fuzzy graph. Then,(i)
a sequence of consecutive verticesPT:i0,i1,…,iASTF is called path where ixix+1∈e,x=0,1,…ASTF−1(ii)
a path is a series of consecutive verticesPT:i0,i1,…,iASTF where; ixix+1∈e,x=0,1,…ASTF−1(iii)
path strengthPT:i0,i1,…,iASTF is Λx=0,…,ASTF−1βixix+1(iv)
connectedness among verticesi0 and ik is(2)β∞i0,ik=∨PT:i0,i1,…,ikΛx=0,…,k−1βixix+1.(v)
a cycle is a sequence of successive verticesPT:i0,i1,…,iASTF where ixix+1∈e,x=0,1,…ASTF−1,iASTFio∈e and there are two edges ij and uv such that βij=βuv=Λx=0,1,…,n−1βvx′vi+1′
I bring preliminaries of this article in two upcoming definitions in alternative methods to make them concrete.Definition 4.
(Sugeno-type Fuzzy graph and its special case).STFG=v,e,α=α1,α2,α3,β=β1,β2,β3 is termed a Sugeno-type Fuzzy graph if it is a graph, αx:v⟶0,1 and βx:e⟶0,1; we add one condition to it and utilise a Sugeno-type Fuzzy graph special case with the same name [17]. The additional condition is as follows: for each vx′vy′∈e,(3)βvx′vy′≤αvx′∧αvy′.
v is known as the STF array and is denoted by ASTF, Sugeno-type Fuzzy graph order ∑v′∈vαxv′ is indicated by the symbol AnSTF.Definition 5.
(path-colouring numbers that are dominant). Consider a Fuzzy graph of the Sugeno kindSTF:v,e,α,β, Then,(i)
There are several pathways fromi to j given two vertices and i and j. When two pathways share a single edge, they are allocated i to j distinct hues. In this approach, Se the set of various colours is termed the dominant path-colouring set from i to j if there is at least one edge within with a common vertex for every edge outside. The dominant path-colouring number, indicated by, MSTF is the minimal cardinality between all dominating path-colouring number from two specified vertices [18].(ii)
There exist certain pathways fromi to j given two vertices and i and j, When two pathways share a single edge, they are allocated i to j distinct hues. i to j if every edge outside has at least one edge inside with which they share a common vertex, the set Se of distinct colours in this procedure is termed dominant path-colouring set. MnSTF is the Sugeno-type Fuzzy graph dominating path-colouring number is the least Sugeno-type Fuzzy graph cardinality ∑e′∈e∑i=13βxe′ between all dominating path-colouring sets, and it is symbolised by Se.
The term Sugeno-type Fuzzy graph, which was used in the preceding definition, will not be utilised in most cases. Clarifications on the main definition are provided in the next section. Examples are frequently supplied after each portion to minimise misunderstanding and for convenient usages, and names are utilised in such a way that brevity, simplicity, and summarising are effects of concern.Example 1.
A working Sugeno-type Fuzzy graph is shown in Figure1. The following are some of the points reproduced in the followup items. To gain additional examination and discrimination, new perspectives need a variety of definitions.(i)
Consider the case of two verticesR1 and R2. The following are all possible paths:PT1: R1, R2 ⟶ PinkPT2: R1, R3, R2 ⟶ PinkPT3: R1, R4, R2 ⟶ PinkPT4: R1, R3, R4, R2 ⟶ RedPT5: R1, R4, R3, R2⟶ BlueBecause the pathways PT1, PT2, and PT3 do not share an edge, they have been coloured pink. Because the path PT4 shares shared edge R1R3 with PT2 and shared edge R4R2 with PT3, it is coloured red in contrast to them. Because the path PT5 shares shared edge R1R4 with PT3 and shared edge R3R4 with PT4, it has been coloured blue in comparison to other pathways. Thus Set = {pink, red, blue} is the most common path-colouring set, while 3 is the most common path-colouring number. To summarise, there exist certain pathways from i to j for two vertices, i and j. If two pathways from i to j have the same edge, they are coloured differently. In this approach, the set of colours Set = {pink, red, blue} is referred to as the dominating path-colouring set from i to j. The dominant path-colouring number, which is 3, MnSTF=3 is the lowest cardinality between all dominating path-colouring sets from two specified vertices.(ii)
When it comes to designing pathways, all vertices have the same locations. As a result, the number of pathways and their behaviours are the same for every two vertices;(iii)
There are three alternative pathways, none of which share any edges. As a result, they have been given the same colour;(iv)
Colour representatives are formed by shared edges. Each colour is associated with an edge with the lowest STF graph.(v)
Every colour in the Set has the same minimal STF graph as an edge. This is one technique to get minimal STF graph, but another is to utilise shared edges to construct S and then get minimum STF graph that way.(vi)
Two of the edges,R1R3 and R4R2, are shared PT4 by PT3, and PT2. The smallest STF cardinality is 0.4, which corresponds to R4R2. The other hue has only one common edge R3R4, and the STF cardinality minimum is 0.6. As a result, the minimal STF cardinality is 1. The set that corresponds to it is Se=R4R2,R3R4. To summarise, there exist certain pathways from i to j for two vertices, i and j. If two pathways from i to j have the same edge, they are coloured differently. The dominating path-colouring set from i to j is the set of shared edges in this procedure. The STF dominating path-colouring number, represented as MnSTF=1.0, is the smallest number STF ∑e′∈e∑i=13βxe′ between all dominating path-colouring sets Se.Figure 1
In terms of its dominant path-coloring number and its sugeno dominating path-coloring number, a Sugeno-type fuzzy graph.


## 3. Coloring Theorems: Sugeno-Type Fuzzy Graph Theory

Samanta et al. [19] developed the idea of dominator chromatic number of graph (Definition 6). The authors observed at fuzzy graphs with a Sugeno type vertex set, that is, fuzzy graphs for which αi=1∀i∈v and edges membership functions of [0,1]. The fuzzy dominator chromatic number idea was defined by Parvathi and Karunambigai [20].Definition 6.
(see [21]). Two nodes m and n in a fuzzy graph Gγ=αγ,βγ are strong adjacent if m,n is strong arc. Otherwise, they are said to be weak adjacent.Definition 7.
(see [22]). A Sugeno-Type Fuzzy (STF) graph of a fuzzy graph G is a proper fuzzy colouring in which each node of G dominates every node in some colour class.Definition 8.
(see [22]). A Sugeno type-fuzzy colouring of a fuzzy graph G is an assignment of ΨSTF colours, usually denoted by 1,2,3,....,m, to the nodes of G. A fuzzy colouring is proper if no two distinct strong adjacent nodes have the same colour. A fuzzy graph G is Sugeno type -fuzzy colourable if G has a proper Sugeno type fuzzy coloring. The fuzzy chromatic number Gγ, of a fuzzy graph G is the minimum ΨSTF for which G is Sugeno type fuzzy colourable.Definition 9.
(see [23]). Consider the case where R=γ1<γ2<⋯<γm is the basic set (level set) of G. For each γ∈Q,Gγ, designate the Sugeno type graph Gγ=v,eγ, eγ=xy|1≤x≤y≤n,λx,y≥γ and Ψγ=ΨGγ for each, indicate the Sugeno type graph is Gγ dominator chromatic number.
According to this definition, the Sugeno fuzzy numberΨG=x,v′x|x∈I is the dominator chromatic number of a Sugeno fuzzy graph G, where v′x=Maxγ∈Q|x∈Rγ and Rγ=1,2,3,…,Ψγ,Definition 10.
(see [23]). A m-Sugeno fuzzy colouring of G=v,α,β if is a family Ω=η1,η2,⋯,ηm of Sugeno type fuzzy sets on a set v.(i)
∨Ψ=α(ii)
ηx∧ηj=0(iii)
for each strong edget,wi.e.,λt,w>0 of G, Minηxt,ηxw=0, 1≤x≤m
The Sugeno type fuzzy dominator chromatic number ifG, denoted as ΨSTFG is the smallest number m for which an m-fuzzy colouring exists [24].
Rosyida et al. [25] modify the dominator chromatic number of ΨG Sugeno type fuzzy graph as follows, using the properties of the aforementioned Definitions 6 and 7.Definition 11.
(see [25]). For each, indicate the Sugeno type graph Gγ=αγ,βγ and ΨG=ΨGγ the Sugeno type graph is Gγ dominator chromatic number. The number of fuzzy graphs is dominator chromatic number. The dominator chromatic number of fuzzy graphs G is the number ΨG=MaxΨGγ|γ∈η,Theorem 1.
(see [25]). For dominator chromatic number ΨG=ΨSTFG of a fuzzy graph G:v,α,β,Remark 1.
(see [25]). For some, the basic set Q of G is the set of Q=γ/eitherαη=γ; for others u∈v; or βu,v=γ, it is the set of u,v∈v×v. We make a decision Q=γ1<γ2…<γm,Remark 2.
Ifγx≤γy, then ΨGγx≥ΨGγy.Theorem 2.
For a fuzzy graphG:v,α,β except for m-strong Sugeno fuzzy graphs ΨG¯>ΨG and ΨG¯=ΨG if G they are self-complementary.Proof.
For all graphs m-strong Sugeno fuzzy graphs, for all graphsβ¯u,v=αu∧αv−βu,v>0∀u,v. If the fuzzy graph is m-strong, we know βu,v≤αu∧αv∀u,v∈v that equality holds. As a result, the related Sugeno graph will contain more adjacencies than G¯γ of G. The equality is self-evident.Theorem 3.
When it comes to fuzzy graphsG1=v1,α1,β1 and G2=v2,α2,β2,ΨG1+G2≤n, If G1∗ and G2∗ are complete graphs, equality holds.Proof.
By definition, the join provision becomes a full graph if and only if bothG1∗ and G2∗ is complete graphs.
The dominator chromatic number of products of Sugeno type fuzzy graphs is now discovered.Definition 12.
IfG1=v1,α1,β1 and G2=v2,α2,β2, then the direct product G1∩G2 is the Sugeno type fuzzy graph G=v1×v2,α,β, where e=u1v1,u2v2|u1,u2∈e1,v1,v2∈e2,(4)αuv=Minα1u,α2v∀uv∈v1×v2βu1v1,u2v2=β1u1,u2∧β2v1,v2,Theorem 4.
For Sugeno fuzzy graphsG1=v1,α1,β1 and G2=v2,α2,β2, the dominator chromatic number of the direct product, ΨG1∩G2=MinΨG1,ΨG2,Proof.
Any two nodes are nearby and only if the corresponding nodes inG1 and G2 are adjacent, according to the concept of direct product. Because dominator chromaticity is determined by adjacency, the least number necessary to colour the nodes of G1∩G2 is ΨG1 and ΨG2.Definition 13.
(see [26]). If G1=v1,α1,β1 and G2=v2,α2,β2, then the strong product G1⊗G2 is the Sugeno type fuzzy graph G=v1×v2,α,β, where(5)e=uv1,uv2u∈v1,v1,v2∈e2∪u1w,u2wu1,u2∈e1,w∈v2∪u1v1,u2v2u1,u2∈e1,v1,v2∈e2,αuv=Minα1u,α2v∀uv∈v1×v2,βuv1,uv2=αu∧β2v1,v2,βu1w,u2w=β1u1,u2∧α2w.Theorem 5.
Ifβ1u1,u2>0, β1v1,v2>0∀u1u2∈v1 and v1,v2∈v2, the dominator chromatic number of the strong product ΨG1⊗G2=d1d2, where d1=v1 and d2=v2,Proof.
By the definition of strong product, ifβ1u1,u2>0 and β1v1,v2>0∀u1u2∈v1 and v1,v2∈v2 , then G1⊗G2∗ is a complete graph with d1d2 nodes.
HenceΨG1⊗G2=d1d2,Example 2.
Consider a routing issue in which locations are represented as nodesf1f2,f3,f4 in a fuzzy graph of the Sugeno type. Allow arcs to represent approachability between pairs of nodes, with strengths indicating journey difficulty. The nodes are rated as High (0.4), Medium (0.3), or Low (0.2) depending on visitor demand (0.1). Let’s look at Figure 2(a), which is a Sugeno type fuzzy graph G that represents the circumstance.
The levels are {0.1, 0.2, 0.3, 0.4}. Consider the arc strengthsG of Gγ for γ=0.1,0.2,0.3and0.4 a moment. Figure 2(b) is a graph that depicts G0.1. Then, according to Definition 11, ΨG=ΨG0.1. The numbers inside the brackets are the colours of the nodes in G0.1 as a result ΨG=ΨG0.1=3. The dominator chromatic number 3 denotes that there are a maximum of three places that may be accessed from each other. That is, the maximum number of sites that are directly linked by nodes f1,f2 and f4 are next to each other is three, represented as nodes.Figure 2
Sugeno type fuzzy coloring graphG.
(a)(b)Theorem 6.
According toG a simple graph, and let G−e and G/e be the graphs obtained by removing and obtaining an edge e from G. After that,(6)LG,m=LG−e,m−LGe,m.Theorem 7.
LeaveG an STF-graph alone, and the full colouring of G is the Ω0=γ1,γ2,…,γm least fuzzy. When it happens,(7)∑x∈cr1μ1x≥∑x∈cr2μ1x≥⋯∑x∈crmμmx.Theorem 8.
For a fuzzy graphG∗=v,e and Sugeno type of fuzzy graph ΨGλω∗,Proof.
ConsiderG∗=v,e the following STF-graph with n vertices v1,v2,⋯,vn:
LetΨSTFG∗=m:(i)
Ω=γ1,γ2⋯γm this is an m-fuzzy colouring, with the Cy colour deported to the vertices in Ay∗; y=1,2,....m(ii)
γ1,γ2⋯γm it is a perceptive STF-group in the sense that(8)γyix=iy,β1iy,v1iyStix,β1ix,v1ixβ2ixiy=0,x6=y.(iii)
y=1,2,⋯,m each vertex has γyix its own arrangement of vertices. ΨGλω∗ is the hidden fresh graph G∗. Now(9)ΨGλω∗−ΨGt∗=mwheret=λ,ω∈0,1=MaxΨλ,ωλ,ω∈0,1.
Also, henceforth the confirmation.Theorem 9.
ConsiderG=v,β an STF-graph with fresh vertices and STF-edges. The level of L∞STFG,m=v; for all ∞∈R; for all was reached at that moment.Proof.
ConsiderG=v,β an STF-graph with fresh vertices and STF-edges. Currently, G∞=v,e∞, where e∞=βu,v≥∞, We realize that the level of the dominator chromatic polynomial of a new graph G∞ is the vertices quantity of G∞ that is, the level of LG∞,m=v, L∞STFG,m=LG∞,m for ∞∈R: from this it follows that the level of L∞STFG,m is equivalent to the level of L∞STFG,m, Along these lines, L∞STFG,m=v.Theorem 10.
Prove that STF vertex dominator chromatic number isΨG=3.Proof.
Using Eslahchi, Changiz, and Bi NaimehOnagh’s concept [27].
We colour this Fuzzy graph. They defined fuzzy dominator chromatic number as the smallest estimate ofm for which the fuzzy graph G has m–STF-colouring, with m–STF-colouring defined as follows. A v known as an m-colouring of STF-graph G=v,e,u,T is a group Ω=γ1,γ2⋯γm of F-sets on a graph [28].(i)
γ1∩γ2=0(ii)
∨Ω=α(iii)
For each solid edgeij of G, γ1i,γ2=01≤x≤m
LetΩ=γ1,γ2,γ3,γ4 be a Sugeno type fuzzy sets assembly differentiate on v by, Mwf=Aγ1vx=Mwf=D0OtherwiseHwf=Bγ2vx=Hwf=E0OtherwiseLwf=Aγ3vx=Lwf=E0OtherwiseMwf=Cγ4vx=Mwf=F0Otherwise.
We can observe that the familyΩ=γ1,γ2,γ3,γ4 satisfies the vertex colouring states of meaning. As a result, the fuzzy network in our example has 3-STF vertex colouring, which is the insignificant vertex STF colouring since any family with fewer than three members does not meet the definition’s conditions. Figure 3 depicts the STF vertex dominator chromatic number ΨG=3 in this method.Figure 3
Graph coloring of 3 colors.Theorem 11.
If there aren vertices G in an STF-enchantment path, then ΨSTFm′=2.Proof.
Let’sG an STF-enchantment network with the purpose of creating G∗ a fuzzy enchantment path with n vertices. Every edge of G is clearly not solid. On the other hand, we may offer the boundaries of the colours Red and Blue to the edges of G. This is a fuzzy edge colouring done automatically. As a result, ΨSTFm′=2, Figure 4 was illustrated.Figure 4
STF-enchantment graph.


## 4. Conclusion

Fuzzy graph theory has substantial applications for real-world life in different domains, such as in the fields of biological science, neural networks, decision making, physics, and chemistry. At present, the graph colouring problem can be applied in sequencing, timetabling, scheduling, electronic bandwidth allocation, computer register allocation, and printed circuit board testing. Also, the domination is also one of the fundamental concepts in graph theory and it has been widely used to distributed computing, biological networks, resource allocation, and social networks. The dominator chromatic number and perfectness on Sugeno Type Fuzzy graph are studied in depth in this research. The interrelationship between a Sugeno Type Fuzzy graph is dominator chromatic number and its complement graph has been widely explored, yielding a few of the most important conclusions. The chromaticity of the product formed by joining two Sugeno Type Fuzzy graphs and other sorts of products is also determined. A vital role for STF graph colouring is assumed in both the practical and theoretical applications. In this case, a network consists of nodes, vertices, or efforts connected by edges, bends, or lines. A graph is a chromatic number and associated colouring capacity have to be determined in order to address the colouring issue. The edges of a graph may be directed, starting with one vertex and moving on to the next, or they may be undirected, suggesting that there is no distinction between the two vertices associated with each edge. The limits of the dominator chromatic numbers of additional Sugeno Type Fuzzy graph types, as well as their complements, are a topic we intend to explore more in the future. The solution proposed in this technique shows the number of sites that are directly linked by route. Distinct colouring techniques can be used to discover the most efficient route between different locations by considering arc strengths. A level of contrariness associated with the assessment of the enrolment work on each edge is taken into account by the new concept of colouring capability.



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