A dominator coloring of a graph 𝐺 is a proper coloring of 𝐺 in which every vertex dominates every vertex of at least one color class. The minimum number of colors required for a dominator coloring of 𝐺 is called the dominator chromatic number of 𝐺 and is denoted by $\chi d(G)$. In this paper we present several results on graphs with $\chi d(G)=\chi(G)$ and $\chi d(G)=\gamma(G)$ where $\chi(G)$ and $\gamma(G)$ denote respectively the chromatic number and the domination number of a graph 𝐺. We also prove that if $\mu(G)$ is the Mycielskian of 𝐺, then $\chi d(G)+1\leq\chi d(\mu(G))\leq\chi d(G)+2$.
