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

The \textit{Distinguishing Chromatic Number} of a graph $G$, denoted $χ_D(G)$, was first defined in \cite{collins} as the minimum number of colors needed to properly color $G$ such that no non-trivial automorphism $ϕ$ of the graph $G$ fixes each color class of $G$. In this paper, 1. We prove 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 $χ_D(G)=χ(G)+1$. 2.We give an example of families of graphs that admit large automorphism groups in which every proper coloring is distinguishing. We also describe families of graphs with (relatively) very small automorphism groups which satisfy $χ_D(G)=χ(G)+1$, for arbitrarily large values of $χ(G)$. 3. We describe non-trivial families of bipartite graphs that satisfy $χ_D(G)>r$ for any positive integer $r$.

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