Let $c$ and $c'$ be edge or vertex colourings of a graph $G$. We say that $c'$ is less symmetric than $c$ if the stabiliser (in $\operatorname{Aut} G$) of $c'$ is contained in the stabiliser of $c$. We show that if $G$ is not a bicentred tree, then for every vertex colouring of $G$ there is a less symmetric edge colouring with the same number of colours. On the other hand, if $T$ is a tree, then for every edge colouring there is a less symmetric vertex colouring with the same number of edges. Our results can be used to characterise those graphs whose distinguishing index is larger than their distinguishing number.

中文翻译：

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关于图的边和顶点着色的对称性

令 $c$ 和 $c'$ 是图 $G$ 的边或顶点着色。如果 $c'$ 的稳定器（在 $\operatorname{Aut} G$ 中）包含在 $c$ 的稳定器中，我们说 $c'$ 比 $c$ 对称。我们表明，如果 $G$ 不是双中心树，那么对于 $G$ 的每个顶点着色，都有一个具有相同颜色数量的不太对称的边缘着色。另一方面，如果 $T$ 是一棵树，那么对于每个边着色，都有一个具有相同边数的不太对称的顶点着色。我们的结果可用于表征那些区分指数大于区分数的图。