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# upper bound for fuzzy chromatic number

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### upper bound for fuzzy chromatic number

1. 1. AKCE Int. J. Graphs Comb., X, No. X (XXXX), pp. 1-12 AN UPPER BOUND FOR FUZZY CHROMATIC NUMBER OF FUZZY GRAPHS AND THEIR COMPLEMENT Isnaini Rosyida Ph.D Student at Department of Mathematics, GadjahMada University, Indonesia Department of Mathematics, Semarang State University, Indonesia e-mail: iisisnaini@gmail.com S.Lavanya Department of Mathematics, D.G. Vaishnav College, Chennai, Tamilnadu, India e-mail: lavanyaprasad1@gmail.com Widodo, Ch.R.Indrati Department of Mathematics, GadjahMada University, Indonesia K.A.Sugeng Department of Mathematics, University of Indonesia, Indonesia Abstract An upper bound for the sum and product of chromatic number of a crisp graph and chromatic number of its complement has been given by Nordhaus and Gaddum. On the other hand, Sattanathan and Lavanya gave a theorem about un upper bound for the sum and product of fuzzy chromatic number of a fuzzy graph and its complement. In this paper, we give an example that the bounds given by Sattanathan and Lavanya do not hold for certain fuzzy graphs. Further, we add conditions to a fuzzy graph so it satisﬁes the theorem of Sattanathan and LavanyaKeywords: Chromatic Number, Fuzzy Coloring, Fuzzy graph, Fuzzy Chromatic Num-ber, Complement Fuzzy graph 2000 Mathematics Subject Classiﬁcation: 1. Introduction Let G be a graph with vertex set V(G) and edge set E(G). Vertex coloring of G is amapping C : V (G) → N with N is a set of natural numbers, such that C(x) = C(y)if (x, y) ∈ E(G) . Given an integer k , a k -coloring of G is a mapping C : V (G) →{1, 2, ..., k} such that C(x) = C(y) if (x, y) ∈ E(G) . The chromatic number of G ,denoted by χ(G) is the smallest integer k such that there is a k -coloring of G . Forsimplicity, we will use symbol V for V (G) and E for E(G) .
2. 2. 2 An Upper of fuzzy chromatic number of Fuzzy graphs and their Complement Vertex coloring of graph G can be interpreted as a problem of special kind of partitionof the vertex set as in [1]. Therefore there is another deﬁnition of vertex coloring as follows.A color partition of general graph G(V, E) is a partition of V into subsets, called color-classes, such that V = V1 ∪ V2 ∪ ... ∪ Vk where the subsets Vi ( 1 ≤ i ≤ k ) are non-emptyand mutually disjoint, and each Vi contains no pair of adjacent vertices. The chromaticnumber of G is the smallest natural number k for which such a partition is possible. The Fuzzy set theory has been introduced by Zadeh in 1965 [10]. Ideas of fuzzy settheory have been introduced into graph theory by Rosenfeld in 1975 as in [5]. We calla graph G(V, E) by crisp graph. Rosenfeld has expanded the crisp graph G(V, E) into ˜fuzzy graph G(V, σ, µ) that is a graph which has fuzzy vertex set with the membershipfunction σ : V → [0, 1] and fuzzy edge set with the membership function µ : E → [0, 1] . ˜While Kaufman has introduced a fuzzy graph G(V, σ, µ) that is a graph with crisp vertexset and fuzzy edge set as in [6]. ˜ The vertex coloring of a fuzzy graph G(V, σ, µ) was introduced by Munoz et al [6].Munoz expanded the concept of coloring function C : V → N of a crisp graph into coloringfunction Cd,f : V → S of a fuzzy graph, where S is the available color set, d is a distancefunction deﬁned between the colors on S, and f is a real scale function deﬁned on image( µ ). Pourpasha and Soheilifar [7] expanded the coloring function C d,f : V → S of fuzzy ˜ ˜graph G(V, µ) into the coloring function Cd,f,g : V → S of a fuzzy graph G(V, σ, µ)where g is a real scale function deﬁned on image ( σ ). Further, the fuzzy vertex coloring ˜of a fuzzy graph G(V, σ, µ) was deﬁned by Eslahchi and Onagh [2]. They deﬁned fuzzyvertex coloring of a fuzzy graph through fuzzy color-classes of V . Several authors have studied the problem of obtaining an upper bound for the chromaticnumber of complementary crisp graphs and complementary fuzzy graphs. According toSattanathan and Lavanya as in [8], Nordhaus and Gaddum gave the ﬁrst theorem aboutan upper bound for crisp graph and its complement in 1956. While Sattanathan and La-vanya [8] gave an upper bound for the sum and product of the fuzzy chromatic number ofcomplementary fuzzy graphs. Moreover, Lavanya and Sattanathan [4] gave the deﬁnitionof fuzzy vertex coloring of fuzzy graph with slight modiﬁcations. In this paper we give anexample of fuzzy graphs which do not satisfy the theorem of Sattanathan and Lavanya.We also give some theorems on fuzzy chromatic number of fuzzy cycle and its comple-ment. Further, we add conditions to a fuzzy graph such that it satisﬁes the theorem ofSattanathan and Lavanya. 2. Preliminaries We review brieﬂy some deﬁnitions in fuzzy sets as in [10]. Let X be a space of objects.A fuzzy set A on X is the set of the form {(x, µ A (x)): x ∈ X }, where µA is a mapping:X → [0, 1] . We call µA is a membership function of the fuzzy set A , and µ A (x) at xrepresenting the grade of membership of x in A . Other deﬁnition said that a fuzzy set A on X is a mapping µ : X → [0, 1] , as in [3].According to the ﬁrst notation, the symbol of the fuzzy set A is distinguished from the
3. 3. I.Rosyida, S.Lavanya, Widodo, Ch.R.Indrati, K.A.Sugeng 3symbol of its membership function (µ A ) . According to the second notation, there is nodistinction between the two symbols. In this paper we use the second notation. A fuzzy set µ on X is empty if and only if µ(x) = 0 for all x ∈ X . Let µ and σbe fuzzy sets on X. The union µ ∪ σ is the fuzzy set on X deﬁned by (µ ∪ σ)(x) =max{µ(x), σ(x)} for all x ∈ X . The intersection µ ∩ σ is the fuzzy set on X deﬁned by(µ ∩ σ)(x) = min{µ(x), σ(x)} for all x ∈ X . We review brieﬂy some deﬁnitions in fuzzy graphs as in [5], and [9].Deﬁnition 2.1. ([5]) Let V be a ﬁnite nonempty set. Let E ⊆ V × V . A fuzzy graph˜G(V, σ, µ) is a graph consisting of a pair of functions (σ, µ) where σ is a fuzzy set onV and µ is a fuzzy set on E, i.e. σ : V → [0, 1] and µ : E → [0, 1] such thatµ(u, v) ≤ min{σ(u), σ(v)} for all u, v ∈ V .Note that a crisp graph G(V, E) is a special case of a fuzzy graph with each vertex andedge of V and E having degree of membership 1. In this paper we refer the deﬁnition of complement of fuzzy graph as in Sunitha andVijayakumar’s paper [9]. ˜ ¯ ˜Deﬁnition 2.2. ([9]) The complement of a fuzzy graph G(V, σ, µ) is a fuzzy graph G =(¯ , µ) where σ = σ and µ(u, v) = min{σ(u), σ(v)} − µ(u, v) for all u, v ∈ V. σ ¯ ¯ ¯ ˜ ˜ ¯ ˜Deﬁnition 2.3. ([9]) A fuzzy graph G(V, σ, µ) is self complementary if G = G. The deﬁnition of adjacency is classify as strong and weak adjacency as in [4]. ˜Deﬁnition 2.4. ([4]) Two vertices u and v of fuzzy graph G(V, σ, µ) are called strongly 1adjacent if µ(u, v) ≥ 2 min{σ(u), σ(v)} , otherwise weakly adjacent. Sunitha and Vijayakumar in [9] gave the condition for a fuzzy graph to be self comple-mentary as follows. ˜Theorem 2.5. ([9]) Let G(V, σ, µ) be a fuzzy graph. If µ(u, v) = 1 min{σ(u), σ(v)} for 2all u, v ∈ V then G is self complementary. We use the deﬁnition of fuzzy vertex coloring and fuzzy chromatic number as in [4].Deﬁnition 2.6. ([4]) A family Γ = {γ1 , γ2 , ..., γk } of fuzzy subsets on V is called a ˜k -fuzzy coloring of G(V, σ, µ) if a. γ1 ∪ γ2 ∪ ... ∪ γk = σ b. min {γi (u), γj (u) | 1 ≤ i = j ≤ k} = 0 for all u ∈ V ˜ c. for every strongly adjacent vertices u, v of G , min{γi (u), γi (v)} = 0 (1 ≤ i ≤ k).