A Study On Distinct Fuzzy Colourings And Domination Parameters In Double Layered Fuzzy Graph

In Double Layered Fuzzy Graph, it is discussed that, the Fuzzy Vertex Colouring, Fuzzy Edge Colouring, Fuzzy Total Colouring and determined bounds for Fuzzy Chromatic Number, Fuzzy Edge Chromatic Number and Fuzzy Total Chromatic number. Also established some of the Domination parameters such as Fuzzy Dominator Chromatic Number, Inverse Domination, Connected Inverse Domination.


Introduction:
Here we have taken a graph to be a simple FG, n-order & m-size. Zadeh.L.A [9] was insinuated Fuzzy relation conceit in the year 1965. Rosenfeld was interpose the theoretical conception like cycle and connectedness and also intercalate the fuzzy graph. Fuzzy graphs colouring was interpolate by Mu˜noz et al in [6].Fuzzy Chromatic number as a Fuzzy number same the -cut's. The FVC same by Onagh and Eslahchi [3] Developing Fuzzy Vertex Colouring in 2006. Term of Family of Fuzzy sets extended to Fuzzy Total Colouring in by Lavanya. S and Sattanathan. R [4]. Dominator colouring and dominator chromatic number was introduced by Raluca Gera et al.

2.2.Definition
A DLF(G) is a PDLFC which that each vertex to DLF(G) Dominate every vertex of atleast some colour class.

3.1.Theorem
If G = ( , )   is a simple FG of Fuzzy Cycle of order n, then

Proof :
This theorem can be proved in 2 cases, 1. Case When n -even, vertex set V 1 consisting of even numbers, it is coloured by C 1 and C 2 preferentially without strike the concept of Fuzzy Proper Colouring and hence it is desired 2 colours. Thus the Fuzzy Chromatic , ( k=1,2,3,…).

Case
When n -odd, vertex set V 2 has odd numbers, it is also coloured by two distinct colours C 1 and C 2 possibility and end vertex of V 2 must be coloured by colour C 3 . Hence it required 3 colours, therefore

Case 1:
For n = 10 for C 10 is 

3.3.Theorem
If G DL = ( DL , DL ) is a DLFG of Fuzzy Cycle of order n,then

Proof :
This theorem can be proved in 2 cases, 1. Case When n -even, svertex set V 1 consisting of even numbers, it is coloured by C 1 and C 2 preferentially without strike the concept of Fuzzy Proper Colouring and hence it is desired 2 colours. Thus the Double Layered Fuzzy , ( k=1,2,3,…).

Case
When n -odd, vertex set V 2 has odd numbers, it is also coloured by two distinct colours C 1 and C 2 possibility and end vertex of V 2 must be coloured by colour C 3 . Hence it required 3 colours, therefore

4.1.Theorem
If G DL = ( DL , DL ) is a FG of a Fuzzy Cycle of size m, Then

Proof :
This theorem can be proved in 2 cases, 1. Case When m -even, Edge set E 1 consisting of even numbers, it is coloured by C 1 and C 2 preferentially without strike the concept of Fuzzy Proper Colouring and hence it is desired 2 colours. Thus the Fuzzy Chromatic

Case
When m -odd, Edge set E 2 has odd numbers, it is also coloured by two distinct colours C 1 and C 2 possibility and end edge of E 2 must be coloured by colour C 3 . Hence it required 3 colours, therefore

Proof :
It is proved in two cases:

Case 1:
Let us colour the set V ′ 1 = V 1  E 1 by using the colours C 1 and C 2 alternatively without affecting the concept of Fuzzy Total Colouring and the either edge or vertex must be coloured by C 3 and hence it required 3 colours, therefore  F Let us colour the set V ′ 2 = V 2  E 2 by using the colours C 1 , C 2 and C 3 alternatively without affecting the concept of Fuzzy Total Colouring and the final edge or vertex must be coloured by C 4 .

Proof :
Let DLF(G) has 2 n vertices.The vertex set and edge sets are divided V 1 and V 2 and also E 1 and

Case 1:
colour the vertices and also edges V 1 = V 1  E 1 by using the colours C 1 and C 2 preferentially without strike the concept of Double Layered Fuzzy Total Colouring and the final edge or vertex must be coloured by C 3 and C 4 hence it is desired 4 colours, therefore  DLF

Case 2:
Similarly colour the vertices and edges V 2 = V 2  E 2 by using the colours C 1 and C 2 possibility without rather than the concept of Double Layered Fuzzy Total Colouring and the final edge or vertex must be coloured by C 3 and C 4 and using the colour C 5 . Hence it is required 5 colours, therefore  DLF , (  k =1,2,3,..).      is a FG of "Fuzzy Cycle" of order m, Then F d ( ) = 3.

7.3.Theorem
If G DL = ( DL , DL ) is a DLFG of "Fuzzy Cycle" of order n, Then