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The \textit{Distinguishing Chromatic Number} of a graph $G$, denoted$\chi_D(G)$, was first defined in \cite{collins} as the minimum number ofcolors needed to properly color $G$ such that no non-trivial automorphism$\phi$ of the graph $G$ fixes each color class of $G$. In this paper, 1. Weprove a lemma that may be considered a variant of the Motion lemma of \cite{RS}and use this to give examples of several families of graphs which satisfy$\chi_D(G)=\chi(G)+1$. 2.We give an example of families of graphs that admitlarge automorphism groups in which every proper coloring is distinguishing. Wealso describe families of graphs with (relatively) very small automorphismgroups which satisfy $\chi_D(G)=\chi(G)+1$, for arbitrarily large values of$\chi(G)$. 3. We describe non-trivial families of bipartite graphs that satisfy$\chi_D(G)>r$ for any positive integer $r$.

$\chi_D(G)$, $|Aut(G)|$, and a variant of the Motion Lemma