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A star coloring of a graph $G$ is a proper vertex coloring in which every path on four vertices in $G$ is not bicolored. The star chromatic number $\chi_{s}\left(G\right)$ of $G$ is the least number of colors needed to star color $G$. In this paper, we find the exact values of the star chromatic number of modular product of complete graph with complete graph $K_m \diamond K_n$, path with complete graph $P_m \diamond K_n$ and star graph with complete graph $K_{1,m}\diamond K_n$.\par All graphs in this paper are finite, simple, connected and undirected graph and we follow \cite{bm, cla, f} for terminology and notation that are not defined here. We denote the vertex set and the edge set of $G$ by $V(G)$ and $E(G)$, respectively. Branko Gr\"{u}nbaum introduced the concept of star chromatic number in 1973. A star coloring \cite{alberton, fertin, bg} of a graph $G$ is a proper vertex coloring in which every path on four vertices uses at least three distinct colors. The star chromatic number $\chi_{s}\left(G\right)$ of $G$ is the least number of colors needed to star color $G$. \par During the years star coloring of graphs has been studied extensively by several authors, for instance see \cite{alberton, col, fertin}.

A proper vertex colouring is called a 2-dynamic colouring, if for every vertex v with degree at least 2, the neighbours of v receive at least two colours. The smallest integer k such that G has a dynamic colouring with k colours denoted by $\chi _2(G) $. We denote the cartesian product of G and H by $G\Box H $. In this paper, we find the 2-dynamic chromatic number of cartesian product of complete graph with complete graph $K_{r} \Box K_{s} $, complete graph with complete bipartite graph $K_n \Box K_{1,s} $ and wheel graph with complete graph $W_l \Box K_n $.

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals χ(G) + 1. In this paper, we find the star chromatic number for the Mycielskian graph of complete graphs, paths, cycles and complete bipartite graphs.

The notion of equitable colorability was introduced by Meyer in $1973$ \cite{meyer}. In this paper we obtain interesting results regarding the equitable chromatic number $\chi_{=}$ for the corona graph of a simple graph with a wheel graph $G\circ W_n$. Some extensions into $l$-corona products are also determined.

A star edge coloring of a graph G is a proper edge coloring without bichromatic paths and cycles of length four. In this paper, we obtain the star edge chromatic number of the corona product of path with cycle, path with wheel, path with helm and path with gear graphs, denoted by Pm ◦ Cn, Pm ◦ Wn, Pm ◦ Hn, Pm ◦ Gn respectively.

The notion of equitable colouring was introduced by Meyer in 1973. In this paper we find the equitable chromatic number χ= for the line graph of Knodel graphs L(WΔ, n), central graph of Knodel graphs C(WΔ, n) and corona graph of Knodel graphs WΔ, m° WΔ, n. As a by-product we obtain a new class of graphs that confirm Equitable Colouring Conjecture.

The notion of equitable coloring was introduced by Meyer in 1973. In this paper we discuss the equitable coloring of certain well known graph families of central graphs and total graphs. We obtain interesting results regarding the equitable chromatic number χ = for the above said graph families.

In this present paper, we have proved for the line graph of double star graph, the harmonious chromatic number and the achromatic number are equal. As a motivation this work can be extended by classifying the different families of graphs for which these two numbers are equal.

The notion of equitable coloring was introduced by Meyer in 1973. In this paper we obtain interesting results regarding the equitable chromatic number 𝜒 = for the total graph of complete bigraphs 𝑇 ( 𝐾 𝑚 , 𝑛 ) , the central graph of cycles 𝐶 ( 𝐶 𝑛 ) and the central graph of paths 𝐶 ( 𝑃 𝑛 ) .