1-Quasi Total Fuzzy Graph and Its Total Coloring

: The fuzzy graph theory, its properties, total coloring and applications are currently climbing up. With this concept of fuzzy graph, total fuzzy graph is defined and its properties as well as fuzzy total colorings have been well discussed and studied. Similarly the theory of crisp graph, its properties, applications and colorings are well considered. Moreover, 1-quasi total graphs for crisp graphs, their properties and colorings were discussed by some researchers and the bounds for its total coloring have been established. In this manuscript, from the concept of fuzzy graph we introduced the definition of 1-quasi total graph for fuzzy graphs. To elaborate the definition we provide practical example of fuzzy graph and from this graph we construct the 1-quasi total fuzzy graph of the given fuzzy graph


Introduction
After its emerging, the graph theory rapidly moved in to the main stream of mathematics as it has application in diverse fields of science and computer [1][2]. The total coloring of graphs were introduced by Behazad (1965) after Harry (1972) made chops on the concept of total graphs [3][4]. D. Muthuramakrishnan and G. Jayaraman (2018) studied the total chromatic number of total graphs [5]. The definition of quasi-total graph is given on the paper by D. V. S. Sastry (1984) and R. V. N. SirnivasaRao, etal has introduced 1-quasi total graphs and bounds for its total chromatic number [6][7]. After Zadeh's paper on fuzzy sets, Rosenfeld (1975) introduced fuzzy graphs [8][9].
Later on, Bhattacharya [10] gave some remarks on fuzzy graphs, and some operations on fuzzy graphs were introduced by Mordeson J. N. and Peng C. S. [11]. As an advancement fuzzy coloring of fuzzy graph was defined by Eslahchi and Onagh in 2004, and later developed by them as Fuzzy vertex coloring in 2006 [12]. Lavanay. S and Sattanathan extended the concept of fuzzy vertex coloring in to a family of fuzzy sets [13]. S. Kavitha and S. Lavanya defined the total fuzzy graph and studied total chromatic number of total graphs of fuzzy graphs [14].
Total coloring of 1-quasi total graph for crisp graph was studied and total fuzzy graphs and its total chromatic number has been already established. This article addresses the following: i. Define 1-quasi total fuzzy graph and elaborate it by examples. ii. Discuss and proof some properties of 1-quasitotal fuzzy graphs and compare the result with the properties of total fuzzy graphs. iii. Determine the total chromatic number of 1-quasi total fuzzy graph and justify it by example.
is a special fuzzy graph with each vertex and each edges of has the same degree of membership equal to 1. Definition 2.3: Let = , , be a fuzzy graph with the underlying set . Then, the order of denoted by !"#$" is defined as: and size of denoted by ( ) * $ ( ) and defined as: If , ∈ , then the strength of connectedness between and is, > , = sup { 9 , : 8 = 1,2, … } Definition 2.8: Let = , , be a fuzzy graph. Then, is said to be connected if > , > 0 @A" BCC , ∈ * . An arc , is said to be a strong arc if , ≥ > , and a node , is said to be an isolated node, if , = 0 @A" BCC ≠ .

1-Quasi Fuzzy Total Graph
In this section we introduce the definition of 1-quasi fuzzy total graph and draw the 1-quasi fuzzy total graph of a given fuzzy graph.
Definition 3.1: Let = , , be a fuzzy graph with its underlying set and crisp graph * = * , * . The pair T U S = V W R X , V W R X of the fuzzy graph is defined as follows: Let the node set of T U S be ∪ , where is the vertex set and is the edge set of the underlying crisp graph. The fuzzy subset V W R X is defined on ∪ as: called edges of T U S as: By definition; V W R X , ≤ V W R X Λ V W R X for all , ∈ ∪ . Hence, V W R X is a fuzzy relation on the fuzzy subset V W R X . Thus, the pair T U S [ V W R X , V W R X \ is a fuzzy graph, and it is termed as 1-Quasi total fuzzy graph of . Since, Y 0 , P Z ≤ 0 Λ Y P Z for all 0 , P ∈ , the graph = , , is a fuzzy graph and its graph is as shown in the figure 1. Now, let us construct the 1-quasi total fuzzy graph of the fuzzy graph in the example as follows: That is, We define the fuzzy subset `V W R X as follows: Hence, we have the following fuzzy subsets V W R : The fuzzy relations V W R X will be as follows: if$ 0 and $ P have a node in common between them 0, otherwise Hence,  Since, Y 0 , P Z 0 Λ Y P Z for all 0 , P , the graph , , is a fuzzy graph and its graph is as shown in the figure 3 below. Now, from the fuzzy graph in example 3 the 1-quasi total fuzzy graph of , T U S [ V W R X , V W R X \ will be defined as follows: i. Fuzzy vertex set V W R X is as follows: Hence; ii. The fuzzy edge set µ e W f g is as follows: if$ 0 and $ P have a node in common between them 0, Otherwise Hence, Clearly, V W R X Y 0 , P Z V W R X 0 Λ V W R X Y P Z for all 0 , P and hence the graph T U S [`V W R X , V W R X \ is a fuzzy graph.
Th e graph T U S [ V W R X , V W R X \ is a 1-quasi total fuzzy graph and its graph is as shown in figure 4.

Properties of 1-Quasi Fuzzy Total Graph
Theorem 4.1: Let , , be a fuzzy graph.
Proof: By the definition of 1-quasi total fuzzy graph, the node set of T U S is k and the fuzzy subset V W R X , )@ and V W R X $ $ , )@ $ . Now, , mG #$@)5)J)A5 A@ T U S !"#$" j ()*$ .
Note 1: For any fuzzy graph , , , !"#$"YU Z !"#$" j ()*$ . Where U is fuzzy total graph. Proof: Let, = , , be a fuzzy graph. The fuzzy vertex set of T U S consists of ∪ of and the fuzzy vertex relation is defined only between , ∈ and $ 0 , $ P ∈ . Since there is no fuzzy relation between ∈ and $ ∈ of elements in the vertex set of T U S , then there is no path that connects and $ in T U S and > , = 0. Hence, T U S is disconnected graph.

1-Quasi Fuzzy Total Coloring
In this section we introduce the concept of 1-quasi fuzzy total coloring and discuss some of its properties. When we come to our point of concern, we need to determine the chromatic number of 1-quasi total fuzzy graph of the fuzzy graph in the example 5. The fuzzy relation will be:

Conclusion
In this article we have defined 1-quasi total fuzzy graph for a given fuzzy graph. For this concept to be clear we constructed a fuzzy graph and from this graph 1-quasi total fuzzy graph is developed and its graph is sketched. For easy understanding the graph of a given fuzzy graph and its 1quasi total fuzzy graph is given. For the sake of making the study to be complete, theorems regarding properties of total fuzzy graph are established for 1-quasi total fuzzy graph and the results obtained from the proof of the theorem are compared against the results for total fuzzy graphs. Moreover, we have defined 1-quasi total coloring for fuzzy graph and its total coloring is exemplified.