If a graph $G$ has distinguishing number 2, then there exists a partition of its vertex set into two parts, such that no nontrivial automorphism of $G$ fixes setwise the two parts. Such a partition is called a 2-distinguishing coloring of $G$, and the parts are called its color classes. If $G$ admits such a coloring, it is often possible to find another in which one of the color classes is sparse in a certain sense. In this case we say that $G$ has 2-distinguishing density zero. An extreme example of this would be an infinite graph admitting a 2-distinguishing coloring in which one of the color classes is finite.

The Infinite Motion Conjecture is a well-known open conjecture about 2-distinguishability. A graph $G$ is said to have infinite motion if every nontrivial automorphism of $G$ moves infinitely many vertices, and the conjecture states that every connected, locally finite graph with infinite motion is 2-distinguishable. In this paper we show that for many classes of graphs for which the Infinite Motion Conjecture is known to hold, the graphs have 2-distinguishing density zero.

如果是图 $G$ 具有可区分的数字2，则存在其顶点集分为两个部分的分区，因此没有非平凡的自同构 $G$固定修复这两个部分。这样的分区被称为2-区分着色的$G$，这些零件称为其颜色类别。如果$G$承认这种颜色，通常可以找到一种在某种意义上稀疏的颜色类别。在这种情况下，我们说$G$具有2个可区分的密度零。一个极端的例子是一个无限图，该图允许2色区分，其中一种颜色类别是有限的。

无限运动猜想是众所周知的关于2可区分性的开放猜想。图$G$ 如果每个非平凡自同构都被称为具有无限运动 $G$移动无限多个顶点，并且猜想表明，每个具有无限运动的局部有限连通图都是2可区分的。在本文中，我们表明，对于许多类别的已知有无限运动猜想的图，这些图具有2的可区分密度0。