[ VOLUME 5 I ISSUE 3 I JULY– SEPT 2018] E ISSN 2348 –1269, PRINT ISSN 2349-5138 734 IJRAR- International Journal of Research and Analytical Reviews Research Paper IntuitionisticFuzzy Graph Coloring M.A. Rifayathali 1 , A. Prasanna 2 and S. Ismail Mohideen 3 1 Research Scholar, PG and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli-620020, Tamilnadu, India. 2 Assistant Professor, PG and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli-620020, Tamilnadu, India. 3 Principal and Head, PG and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli-620020, Tamilnadu, India. Received: June 06, 2018 Accepted: July 23, 2018 ABSTRACT In this paper, the concept of coloring the intuitionistic fuzzy graph introduced with illustrative examples and also introducing a novel concept called chromatic excellence inintuitionistic fuzzy graph and its properties. Keywords: Chromatic excellence,Chromatic number,Intuitionistic fuzzy graph,Vertex coloring, Edge coloring, Total coloring. AMS Subject Classification (2010): 05C72, 05C15. 1. Introduction: Graph coloring dates back to 1852, when Francis Guthrie come up with the four color conjecture. Gary Chartrand and Ping Zhang [4] discussed various colorings of graph and its properties in their book entitled Chromatic Graph Theory. A graph coloring is the assignment of a color to each of the vertices or edges or both in such a way that no two adjacent vertices and incident edges share the same color. Graph coloring has been applied to many real world problems like scheduling, allocation, telecommunications and bioinformatics, etc. The concept of fuzzy sets and fuzzy relations were introduced by L.A.Zadeh in 1965 [15]. A. Rosenfeld who considered fuzzy relations on fuzzy sets and developed the theory of fuzzy graphs in 1975 [13]. The concept of chromatic number of fuzzy graph was introduced by Munoz et.al. [14]. Later C. Eslahchi and B.N. Onagh introduced fuzzy graph coloring of fuzzy graph [2]. S. Lavanya and R. Sattanathan discussed total fuzzy coloring [8]. Anjaly Kishore and M.S. Sunitha discussed chromatic number of fuzzy graph [1]. A. Nagoor Gani and B.Fathima Kani deliberated about Fuzzy vertex order colouring [9].K.M. Dharmalingam and R. Udaya Suriya conferred chromatic excellence in fuzzy graph in 2017 [3]. Intuitionistic fuzzy sets [6] and Intuitionistic fuzzy graph [7] were introduced by Krassimir T. Atanassov in 1986 and 1999 respectively. R. Parvathi et.al. discussed the intuitionistic fuzzy graph and its properties [10, 11]. S. Ismail Mohideen et.al. introduced coloring of intuitionistic fuzzy graph using , - cuts [5] and strong intuitionistic fuzzy graph coloring [12]. This paper proposed to define the vertex coloring, edge coloring and total coloring of intuitionistic fuzzy graphs interms of a family of intuitionistic fuzzy sets satisfying certain conditions and the chromatic number is the least value of k such that k-coloring exists.And also introducing a novel concept called chromatic excellence inintuitionistic fuzzy graph and its properties. 2. Preliminaries 2.1. Definition(L.A. Zadeh [15]) Let X be a non-empty set. Then a fuzzy set A in X (i.e., a fuzzy subset A of X) is characterized by a function of the form : →0,1, such a function is called the membership function and for each ∈, () is the degree of membership of (membership grade of ) in the fuzzy set A. In otherwords, = , ()/ ∈ where : →0,1. 2.2. Definition(A. Rosenfeld [13]) A fuzzy graph =(, ) is a pair of functions : → [0, 1] and µ: × → [0,1] , where for all , ∈, we have (, ) ≤()˄(). 2.3. Definition(Krassimir T. Atanassov [6]) An Intuitionistic Fuzzy set A in a set X is defined as an object of the form