All graphs considered in this paper are connected, simple, finite and undirected graphs. A coloring of a graph G is a function from V(G) to a set of colors which assigns different colors to adjacent vertices. The minimum number of colors needed for a proper coloring of G is called the chromatic number of G and is denoted by

A subset U of V(G) is said to be a total dominating set of G if every vertex of V(G) has an adjacent vertex in U. The minimum cardinality of a total dominating set of G is called the total domination number of G and is denoted by

Let C be a proper coloring of a graph G. A subset U of V(G) is said to be C-colorful if every vertex of U receives different color under the coloring C. A subset U of V(G) is said to be color transversal with respect to the coloring C if U intersects every color class of the coloring C. The study of a graph theoretical parameter in Mycielskian construction of graphs is one of the interesting research fields in graph theory. Some of such studies are Dominator coloring of Mycielskian graphs

In this section, we introduce the notion of gamma coloring of a graph along with an example.

Suppose C is a

In this section we discuss about the gamma coloring of Mycielskian graph

_{4} is given in Figure 2.

It has been proved in

Let us now prove that,

Let

In view of Theorem 3.2, the set of all connected graphs can be classified into two groups namely Class-1 graphs and Class-2 graphs. Class-1 graphs consist of all connected graphs G with

(i) There exists a graph G such that

(ii) There exists a graph H such that

Consider a star graph on

Clearly

If

Now, we prove that

Complete graphs on k vertices serve the purpose as proved in Theorem 3 7.

The following theorem provides a necessary and sufficient condition for a graph G in terms of

Conversely, let us assume that

In this case,

Let

Clearly

In this case

and

Then

By a similar argument we can prove the following theorem for cycle graphs.

Theorem 3.4 provides a condition for a graph G to be of Class-2 graph in terms of minimum gamma coloring of

(i) There exists a positive integer m such that

(ii) There exists a colorful total dominating set D such that

Then

Let

We have initiated a study on Gamma coloring for Mycielskian graph

(1) Theorem 3.4 provides a necessary and sufficient condition under which a graph falls in Class-2; however, it does not infer about the structure of those graphs. So, it is worthy finding a structural characterization of Class-1/ Class-2 graphs.

(2) Characterize trees which are of Class-1/ Class-2.