In this paper we defined a new fuzzy graph named Double layered complete fuzzy graph. (DLCFG). The double layered complete fuzzy graph gives a 3-D structure. Further we introduced vertex colouring of the double layered complete fuzzy graph using ?-cut.
Fuzzy graph theory was introduced by Azriel Rosenfeld in 1975.Graph theory is proved to be useful in modelling the essential features of systems with finite components. Graph theory plays a major role in colouring of various graphs. If the relation among accounts is to be measured as good or bad according to the frequency of contacts among the accounts, fuzziness should be added to representation. This and many another problem motivated to define fuzzy graphs Rosenfeld first introduced the concept of fuzzy graph. Fuzzy graph is a part of our life of graph theory which is involved in colouring, fuzzy logic, and some features of graphical concepts. After that fuzzy graph theory becomes a vast researched area. A fuzzy set is defined mathematically by assigning to each possible individual in the universe of discourse a value, representing its grade of membership, which corresponds to the degree, to which that individual is similar or compatible with the concept represented by the fuzzy set. Fuzzy graphs have many more applications in modelling real time systems where the level of information inherent in the systems varies with different level of precision. In this paper ,The notion of fuzzy set which is characterized by a membership function which assigns to each object a grade of membership which ranges from 0 to 1. The fuzzy colouring problem consists of determining the chromatic number of a fuzzy graph and an associated colouring function. For any level Alpha, the minimum number of colours needed to colour the crisp graph G α will be computed. In this way the fuzzy chromatic number is defined as fuzzy number through it’s α– cuts.
In this paper we have briefly discussed about the vertex colouring of the double layered complete fuzzy graph using alpha cut. We conclude that the chromatic number is decrease when the value of alpha cut is increase. This concept will help not only in vertex colouring also in edge colouring of DLCFG using alpha cut. Also we have found a double layered complete fuzzy graph, and theoretical concepts based on double layered fuzzy graph and proved double layered fuzzy graph as strong fuzzy graph and illustrated with some examples and have given a relation between complete fuzzy graph and a double layered complete fuzzy graph. Further structures can be developed by increasing number of cycles. This structural pattern with the cycles gives further information into different patterns in networking models.