1. Introduction

Graphs in this paper are finite, simple and undirected graphs without loops. The total coloring was introduced by Behzad and Vizing in 1964. A total coloring of a graph G is a coloring of all elements (i.e,vertices and edges) of G , such that no two adjacent or incident elements receive the same color. The minimum number of colors is called the total chromatic number of G and is denoted by X”(G). In 1973, Meyer[7] presented the concept of equitable coloring and conjectured that the equitable chromatic number of a connected graph G, is atmost In 1994, Hung-lin Fu first introduced the concepts of equitable total coloring and equitable total chromatic number of a graph. Furthermore Fu presented a conjecture concerning the equitable total chromatic number, Le G = (V, E) be a graph with vertex set V(G) and edge set E(G). Clearly , is the maximum degree of 𝐺. In 1989, Sanchez Arroyo ⁸ proved that the problem of determining the total chromatic number of an arbitrary graph is NP-hard. It is also NP - Hard to decide Graphs with are said to be of Type 1, and graphs with are said to be of Type 2. The problem of deciding whether a graph is Type 1 has been shown NP-Complete in this paper for

2. Preliminaries

Definition 2.1. For any integer n ≥ 4 , the wheel grap Wn h is the n vertex graph obtained by joining a vertex 𝑣 0 to each of the n - 1 vertices of the cycle graph Cn-1.

Definition 2.2. The Helm graph Hn is the graph obtained from a Wheel graph Wn by adjoining a pendant edge to each vertex of the n - 1 cycle in Wn.

Definition 2.3. The Gear graph Gn is the graph obtained from a Wheel graph Wn by adding a vertex to each edge of the n - 1 cycle in Wn.

Definition 2.4. The n- sunlet graph on 2n vertices is obtained by attaching 𝑛 pendant edges to the cycle Cn and is denoted by Sn.

Definition 2.5. (⁶)For a simple graph G(V,E), let f be a proper K - total coloring of G . The partition is called a K - equitable total coloring (K - ETC of 𝐺 in brief), and

Is called the equitable total chromatic number of G, where

Following (⁴), let us denote the Total Coloring Conjecture by TCC.

Conjecture 2.6. TCC For any graph 2.

Conjecture 2.7. (⁴) (¹⁰)For every graph G, G has an equitable total K - coloring for each

Conjecture 2.8. (⁴)[ETCC]For every graph G, X”=(G)≤ Δ (G) + 2.

Lemma 2.9. (⁶) For complete graph Kp with order p,

Lemma 2.10. (¹⁰) Let G be a graph consisting of two components G1 and G2. If G1 and G2 are equitably total K - colorable, then so is G.

Proof. Let be equitable total K - colorings of G1 and G2 repectively, satisfying . It is easy to see that is an equitable total k - coloring of G.

In the following section, we determine the equitable total chromatic number of .

3. Main Results

Theorem 3.1. For Sunlet graph

Proof. Let Sn be the sunlet graph on 2n vertices and 2n edges.

We define an equitable total coloring f , such that f : S → C where and . The order of coloring is followed by coloring the pendant vertices first followed by pendant edges, rim vertices and rim edges respectively. In this total coloration, C(u _i ) means the color of the i ^th pendant vertex u _{i ,} C (e _i ) means the color of the i ^th rim edge e _i and C(u _i ) means the color of the i ^th pendant edge e’_I . While coloring, when the value mod4 is equal to 0 it should be replaced by 4.

Case 1:

Case 2:

Case 3:

Case 4:

Based on the above mehod of coloring, we observe that S _n is equitably total colorable with 4 colors, such that its color classes are are independent sets of Sn with no vertices and edges in common and satisfies For example consider the case (See Figure 1), in this which implies , for and so it is equitably total colorable with 4 colors. Hence .

Algorithm : Equitable total coloring of Sunlet graph

Input: n , the number of vertices of Sn

Output: Equitably total colored Sn

Theorem 3.2. For Wheel graph

Proof. The Wheel graph Wn consists of 𝑛 vertices and 2(n - 1) edges.

means the color of the ith edge ei and when the value 𝑛 is equal to 0 it is replaced by n. The equitable total coloring is obtianed by coloring the vertices and edges as follows:

It is clear from the above rule of coloring Wn is equitably total colorable with n colors. The color class of Wn are grouped as , which are independent sets with no vertices and edges in common and . For example consider the case n = 7 (See Figure 2), for which

Algorithm: Equitable total coloring of Wheel graph

Input: n, the number of vertices of Wn

Output: Equitably total colored Wn

Initialize Wn with 𝑛 vertices, the center vertices by and rim vertices by

Initialize the adjacent edges on the center by and adjacent edges on the rim by

Let f be the coloring of vertices and edges in Wn such that f : S → .

Apply the coloring rules of Theorem 3.2 for each of the following cases

Theorem 3.3. For Helm graph .

Proof. The Helm graph H_n consists of 2 n - 1 vertices and 3(n - 1) edges.

With this pattern we can equitably total color the graph H_n with 𝑛 colors. The color classes of H_n are grouped as which are independent sets and satisfies the condition . For example consider the case n = 7 (See Figure 3), for Which .

Algorithm : Equitable total coloring of Helm graph

Input: n, the number of vertices of H_n

Output: Equitably total colored H_n

Initialize H_n with 2n - 1 vertices, the center vertices by v_o, the rim vertices by

Initialize the3 (n - 1) edges, the adjacent edges on the center by

f : S → Let 𝑓 be the coloring of vertices and edges in 𝐻 𝑛 such that f : S →

Apply the coloring rules of Theorem 3.3 for each of the following cases

Theorem 3.4. For Gear graph

Proof. The Gear graph G_n consists of 2n - 1 vertices and 3(n - 1) edges.

Define a function f : S → C where

The coloring pattern is as follows:

Based on the above procedure, the graph G_n is equitably total colored with n colors and by sustituting differnet values for n, it is inferred that no adjacent vertices and edges receives the same color. The color classes can be classified as

For example consider the case n = 7 (See Figure 4), for Which This implies and so it is equitably total colorable with n colors. Hence , we have

Algorithm : Equitable edge coloring of Gear graph

Input: n , the number of vertices of G_n

Output: Equitably edge colored G_n

Initialize the 3(n - 1) edges, the adjacent edges on the center by

Apply the coloring rules of Theorem 3.4 for each of the following cases