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OutlineIntroductionPreliminariesMain ResultsConclusionReferences

A NEW APPROACH IN CONSTRUCTINGFUZZY CHROMATIC NUMBER OF A FUZZY

GRAPH

Isnaini Rosyida

Widodo, Ch.Rini Indrati, Kiki Ariyanti Sugeng

IICMA 2013

UGM, Yogyakarta

November 6, 2013

Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC NUMBER OF A FUZZY GRAPH


1 Introduction

2 Preliminaries

3 Main ResultsA new Approach for fuzzy chromatic Set of a fuzzy GraphSome properties of fuzzy chromatic Set

4 Conclusion

5 References



Introduction

Abstract

A δ-chromatic number of a fuzzy graph (δ ∈ [0, 1]) has beenintroduced by Cioban [1]. However, he did not define a fuzzychromatic set of a fuzzy graph yet. In this paper we have a newapproach in constructing a fuzzy chromatic set of a fuzzy graph.The fuzzy chromatic set is constructed through the δ-chromaticnumber. Further, we show that the fuzzy chromatic set satisfiesproperties of a discrete fuzzy number and then it is called fuzzychromatic number.



Introduction



Introduction



Preliminaries

Fuzzy Set (Zadeh, 1965)

Let X be a space of objects. A fuzzy set A in X is a set of the form

{(x , µA(x)) : x ∈ X},

where µA : X → [0, 1] is a membership function of the fuzzy set A.

Fuzzy Set (Zadeh, 1965)

The support of fuzzy set A, denoted by S(A), is the crisp set givenby S(A) = {x ∈ X |µA(x) > 0}. Let α ∈ (0, 1], the α-cut of the

fuzzy set A is the crisp set Aα = {x ∈ X |µA(x) ≥ α}. While

A0 = {x ∈ X |µA(x) > 0} is the closure of support A.



Preliminaries

Fuzzy number

A fuzzy set A, defined on the set of real numbers R, is said to be afuzzy number if it satisfies the following conditions (Bector andChandra, 2005):

i A is normal, i.e.∃x0 ∈ R such that µA(x0) = 1.

ii Aα is a closed interval for every α ∈ (0, 1].

iii The support of A is bounded .



Preliminaries

Discrete fuzzy number (Wang, 2008)

Let C ⊆ R be a countable set. A fuzzy set A is called discretefuzzy number in C if it satisfies the following conditions [2]:

i the set A0 ⊂ C and it is finite;

ii there exists x0 ∈ C such that µA(x0) = 1;

iii µA(xs) ≤ µA(xt) for any xs , xt ∈ C with xs ≤ xt ≤ x0;

iv µA(xs) ≥ µA(xt) for any xs , xt ∈ C with xs ≥ xt ≥ x0.



Preliminaries

Fuzzy Graph

Kaufmann (1973): A fuzzy graph G (V , E ) is a graph having acrisp vertex set V and a fuzzy edge set E with a membershipfunction µ : V × V → [0, 1] (Sunitha, 2001).A graph G (V ,E ) will be called a crisp graph.



Preliminaries

Fuzzy independent vertex set (Bershtein and Bozhenuk,1999)

Given a fuzzy graph G (V , E ). Let G (S ,ES) be a fuzzy subgraphof G (V , E ). A set S ⊆ V will be called fuzzy independent vertexset (fuzzy internal stable set) with the degree of independenceα(S) = 1−max{µ(x , y)|x , y ∈ S}.The separation degree of a fuzzy graph G with k colors is definedas L = min{α(Vi )|i = 1, ..., k}.

Maximal fuzzy independent vertex set

Let G (V , E ) be a fuzzy graph with n vertices, A subset S ′ ⊆ V iscalled a maximal fuzzy independent vertex set, with the degreeα(S ′), if the condition α(S

′′) < α(S ′) is true for any subset

S ′ ⊂ S′′

.Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC NUMBER OF A FUZZY GRAPH


Preliminaries

Fuzzy chromatic set (Bershtein and Bozhenuk,2001)

A fuzzy set γ ={

(k , Lγ(k))|k = 1, ..., n}

, where Lγ(k) denotes the

separation degree of G with k colors, is called a fuzzy chromaticset of G if and only if there is not more than k ′-maximal fuzzyindependent vertex sets X1,X2, . . . ,Xk ′ (k ′ ≤ k) with the degreesof independence respectively α1, α2, . . . , αk ′ , that satisfy:

1) min{α1, α2, . . . , αk ′} = Lγ(k)

2) ∪j=1,...,k ′Xj = V ;

3) there do not exist another family {X ′1,X ′2, . . . ,X ′k ′′} withk ′′ ≤ k for whichmin{α′1, α′2, . . . , α′k ′′} > min{α1, α2, . . . , αk ′} and thecondition 2) is true.



Preliminaries

δ-Fuzzy independent vertex set

Given δ ∈ [0, 1], a δ-fuzzy independent vertex set A is a set whereµ(u, v) ≤ δ for all u, v ∈ A. The δ-fuzzy independent vertex set Awill be denoted as Sδ.

δ-Coloring

The δ-coloring of a fuzzy graph G (V , E ) is a partition of V into ksubsets {Sδ1 , ...,Sδk} such that Sδi ∩ Sδj = ∅ for all i 6= j and

Sδ1 ∪ ... ∪ Sδk = V .The δ-chromatic number of G , denoted by χδ(G ), is the smallestnatural number k so that such partition is possible.



Preliminaries

Example: δ-coloring.

For δ = 0: χδ = 3; For δ = 0, 3: χδ = 2



A new Approach for fuzzy chromatic Set of a fuzzy GraphSome properties of fuzzy chromatic Set

A new approach

New construction for fuzzy chromatic Set

Let G (V , E ) be a fuzzy graph with n vertices. Let δ ∈ [0, 1]. Thefuzzy chromatic set of G , denoted by χ(G ), is a fuzzy setχ(G ) = {(k , L(k))} where

L(k) = max{1− δ|χδ(G ) = k , k = 1, . . . , n}. (1)

The value L(k) represents the membership grade of the chromaticnumber k in the fuzzy chromatic set χ.




Example: Fuzzy chromatic set

Figure : A fuzzy graph G with µ(V × V ) = {0, 0.2, 0.3, 0.5, 0.6, 1}.




A new approach

Fuzzy chromatic set: the new approach

Considered a fuzzy graph G (V , E ) in Figure 1.

For δ = 0, the partition of V is {{B,D}, {A}, {C}} andχδ(G ) = 3. For 0 ≤ δ < 0.3, the δ-chromatic number is 3.

For δ = 0.3, the partition of V is {{B,C ,D}, {A}} andχδ(G ) = 2. For 0.3 ≤ δ < 1, the δ-chromatic number is 2.

For δ = 1, the partition is {A,B,C ,D} = V and χδ(G ) = 1.

The fuzzy chromatic set of G is

χ(G ) = {(1, 0), (2, 0.7), (3, 1), (4, 1)}




Example: Bershtein’s method

Table : Maximal fuzzy independent vertex sets of G

x1 x2 x3 x4 x5 x6 x7D 1 0 0 0,8 0,7 0,5 0,4

A 0 1 0 0 0 0,5 0,4

B 1 0 0 0 0,7 0 0,4

C 0 0 1 0.8 0,7 0 0

If k = 3 then the covering of all rows by 3 columns gives the setX1,X2,X3 with the separation degree L(3) = min{1, 1, 1} = 1.If k = 2, we have the set X2,X5 with the separation degreeL(2) = min{1, 0.7} = 0.7.The fuzzy chromatic set of G isχ(G ) = {(1, 0), (2, 0.7), (3, 1), (4, 1)}




Some properties‘of fuzzy chromatic Set

Theorem 1

Given a fuzzy graph G (V , E ) with n vertices. Let δt ∈ [0, 1], thereis a partition {Sδt1 , . . . ,S

δtk } which gives χδt (G ) = k and the

greatest value of L(k) = 1− δt if and only if there is k-maximalfuzzy independent vertex sets X1,X2, . . . ,Xk with the degrees ofindependence respectively α1, α2, . . . , αk , that satisfy:

1) min{α1, α2, . . . , αk} = L(k).

2) ∪j=1,...,kXj = V .

3) There do not exist another family {X ′1,X ′2, . . . ,X ′k ′} withk ′ ≤ k for whichmin{α′1, α′2, . . . , α′k ′} > min{α1, α2, . . . , αk} and thecondition 2) is true.




Outline proof of Theorem 1

(⇒). we consider two cases.for k = n. We have Sδti = {vi} and α(Sδti ) = 1 for all i = 1, . . . , n.

While α(Sδ) < 1 for all Sδti ⊂ Sδ. Thus, we have k-maximal fuzzy

independent vertex sets: X1 = Sδt1 , . . . ,Xk = Sδtk .





for k < n. There does not exist v ∈ V such thatmax{µ(x , y)|x , y ∈ Sδt1 ∪ {v}} = δt . Since otherwise, we will havea set Sδt1′ = Sδt1 ∪ {v} with α(Sδt1′ ) = 1− δt and a partition

{Sδt1′ , . . . ,Sδtk ′} such that χδt (G ) = k ′ and k ′ < k . Thus, we have a

maximal fuzzy independent vertex set X1 = Sδt1 .If there is v ∈ V such thatmax{µ(v , x)|∀x ∈ Sδtj , j = 2, . . . , k} = δt , then we construct

Xj = Sδtj ∪ {v} with α(Xj) = 1− δt . The process is continueduntil there does not exist w ∈ V such thatmax{µ(w , x)|∀x ∈ Sδtj } = δt . Thus, Xj , j = 2, . . . , k are maximal

fuzzy independent vertex set in G .We can prove that the maximal fuzzy independent vertex setsX1 = Sδt1 , . . . ,Xk = Sδtk satisfy the properties 1)-3).





(⇐). We will prove that there exists δt ∈ [0, 1] such that thepartition {Sδt1 , . . . ,S

δtk } gives χδt (G ) = k and the greatest value of

L(k) = 1− δt .Let Xt ∈ {X1,X2, . . . ,Xk} such thatL(k) = min{α(X1), . . . , α(Xk)} = α(Xt).By choosing δt = max{µ(u, v)|u, v ∈ Xt}, we have µ(u, v) ≤ δtfor all u, v ∈ Xi , i = 1, . . . , k andL(k) = α(Xt) = 1−max{µ(u, v)|u, v ∈ Xt} = 1− δt .





we can construct δt-fuzzy independent vertex sets:Sδt1 = X1,Sδt2 = X2 − (X1 ∩ X2),Sδt3 = X3 − {(X1 ∩ X3) ∪ (X2 ∩ X3)},Sδt4 = X4 − {(X1 ∩ X4) ∪ (X2 ∩ X4) ∪ (X3 ∩ X4)},...Sδtk = Xk − {(X1 ∩ Xk) ∪ (X2 ∩ Xk) ∪ . . . ∪ (Xk−1 ∩ Xk)}.Then the δt-fuzzy independent vertex sets Sδt1 , S

δt2 , . . . ,S

δtk satisfy:

∪j=1,...,kSδtj = V and Sδti ∩ Sδtj = ∅ for all i 6= j .




Some properties of fuzzy chromatic Set

Theorem 2

Let G (V , E ) be a fuzzy graph. Given δ1, δ2 ∈ [0, 1]. The valueδ1 ≥ δ2 if and only if χδ1(G ) ≤ χδ2(G ).

Theorem 3

Let G (V , E ) be a fuzzy graph. L(1) = 0 if and only if ∃u, v ∈ Vsuch that µ(u, v) = 1.





Lemma 4

If δ = 0 then χ0(G ) = χ(G ) = k and L(k) = 1.

Lemma 5

Let i and j be δ-chromatic numbers of G . If i ≥ j then L(i) ≥ L(j).

Lemma 6

If χδ(G ) = i has the degree L(i) = 1 and i 6= n then L(k) = 1 forall k ∈ {i + 1, ..., n}.





Theorem 7

Given a fuzzy graph G (V , E ) with n vertices. Let G (V ,E ) be anunderlying crisp graph of G . Let C be a set of the δ chromaticnumber of G . The fuzzy chromatic set χ(G ) has the followingproperties:

i the support: Supp(χ) ⊂ C and it is finite;

ii there exists k0 ∈ C such that L(k0) = 1;

iii L(ks) ≤ L(kt) for any ks , kt ∈ C with ks ≤ kt < k0;

iv L(ks) = L(kt) = 1 for any ks , kt ∈ C with ks ≥ kt ≥ k0.



Conclusion

In this paper, a new approach for finding a fuzzy chromatic set of afuzzy graph G (V , E ) is introduced. The fuzzy chromatic set isconstructed through the δ-chromatic number. We show that thenew construction in this paper is equivalent with the fuzzychromatic set resulted by Bershtein and Bozhenuk’s method ([1]).The proposed method is very easy to apply for solving the fuzzychromatic set problems of a fuzzy graph. Further, we show thatthe fuzzy chromatic set of a fuzzy graph is a discrete fuzzy numberand then it is called fuzzy chromatic number.



References

Bershtein, L. and Bozhenuk, A., ”A Colour Problem for FuzzyGraph”, Proceedings of International Conference 7th FuzzyDays, LNCS 2206, Dortmund. (2001), 500 - 505.

Bershtein,L. and Bozhenuk, A, ”Maghout Method forDetermination of Fuzzy Independent, Dominating Vertex Setsand Fuzzy Graph Kernels”, International Journal of GeneralSystems. 30(1) (1999), 45-52.

Bector, C.R. and Chandra, S., Fuzzy MathematicalProgramming and Fuzzy Matrix Games, Springer Verlag, 2005.

Biggs, N., Algebraic Graph Theory, Cambridge UniversityPress, 1993.



References

Cioban, V., ”On Independent Sets of Vertices of Graph”,Studia Univ. Babes-Bolyai Informatica L.II. 1 (2007), 97 - 100.

Cioban, V. and Prejmerean, V., ”Algorithms for ComputeIndependent Sets of Vertices in Graphs and Fuzzy Graphs”,Proceedings of IEEE International Conference onAutomation,Quality and Testing,Robotics (ATQR), ClujNapoca Romania. (2010), 1 - 5.

Munoz, S., Ortuno, M.T., Javier, R., Yanez, J., ”ColouringFuzzy Graph”, Omega: The Journal of Management Science.33 (2005), 211 - 221.



References

Sunitha, M.S., Studies on Fuzzy Graphs, Ph.D. Dissertation:Cochin University of Science and Technology, 2001.

Wang, G., Zhang, Q., Cui, X., ”The Discrete Fuzzy Numberson a Fixed Set With Finite Support Set”, Proceedings of IEEEConference on Cybernetics and Intelligent Systems, Chengdu.(2008), 812 - 817.

Zadeh, L.A, ”Fuzzy Sets”, Information and Control., 33(1965), 338-353.


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