$χ_D(G)$、$|Aut(G)|$和运动引理的一个变体
$χ_D(G)$, $|Aut(G)|$, and a variant of the Motion Lemma
摘要：
图$G$的\textit{Distinguishing Chromatic Number}，表示为$χ_D(G)$，首先在\cite{collins}中定义为正确着色$G$所需的最小颜色数，使得图$G$的非平凡自同构$ϕ$不固定$G$的每个颜色类。在本文中，1。我们证明了一个引理，可以被认为是一个变种的运动引理的\cite{RS}和使用这几个家庭的图形，满足$χ_D(G)=χ(G)+1$的例子。2.我们给出了一类图族的例子，这些图族中的每一个真染色都是可区别的。我们还描述了家庭的图（相对）非常小的自同构群满足$χ_D(G)=χ(G)+1$，任意大值的$χ(G)$。3.我们描述了非平凡族的二部图，满足$χ_D(G)>r$的任何正整数$r$。
英文摘要：
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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