Huang proved that every set of more than half the vertices of the d-dimensional hypercube $Q_d$ induces a subgraph of maximum degree at least $\sqrt{d}$, which is tight by a result of Chung, Füredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs.

First, we present three infinite families of Cayley graphs of unbounded degree that contain induced subgraphs of maximum degree 1 on more than half the vertices. In particular, this refutes a conjecture of Potechin and Tsang, for which first counterexamples were shown recently by Lehner and Verret. The first family consists of dihedrants and contains a sporadic counterexample encountered earlier by Lehner and Verret. The second family are star graphs, these are edge-transitive Cayley graphs of the symmetric group. All members of the third family are d-regular containing an induced matching on a $\frac{d}{2d-1}$-fraction of the vertices. This is largest possible and answers a question of Lehner and Verret.

Second, we consider Huang's lower bound for graphs with subcubes and show that the corresponding lower bound is tight for products of Coxeter groups of type ${\mathbf{A}}_{\mathbf{n}}$, ${\mathbf{I}}_{\mathbf{2}}(2k+1)$, and most exceptional cases. We believe that Coxeter groups are a suitable generalization of the hypercube with respect to Huang's question.

Finally, we show that induced subgraphs on more than half the vertices of Levi graphs of projective planes and of the Ramanujan graphs of Lubotzky, Phillips, and Sarnak have unbounded degree. This gives classes of Cayley graphs with properties similar to the ones provided by Huang's results. However, in contrast to Coxeter groups these graphs have no subcubes.

Huang 证明了d维超立方体的每组一半以上的顶点${问}_d$ 至少引出一个最大度数的子图 $\sqrt{d}$，这是由 Chung、Füredi、Graham 和 Seymour 的结果造成的。黄问其他高度对称的图是否也能得到类似的结果。

首先，我们提出了三个无限度的凯莱图家族，其中包含在一半以上的顶点上的最大度数为 1 的诱导子图。这尤其驳斥了 Potechin 和 Tsang 的猜想，Lehner 和 Verret 最近为此给出了第一个反例。第一个家族由二面体组成，包含 Lehner 和 Verret 之前遇到的零星反例。第二类是星图，它们是对称群的边传递凯莱图。第三个家族的所有成员都是d正则的，包含在 a 上的诱导匹配$\frac{d}{2d-1}$- 顶点的分数。这是最大的可能，并回答了 Lehner 和 Verret 的问题。

其次，我们考虑 Huang 的子立方体图的下界，并表明相应的下界对于 Coxeter 类型的产品是紧的 ${\mathbf{一个}}_{\mathbf{n}}$, ${\mathbf{一世}}_{\mathbf{2}}(2ķ+1)$，以及最特殊的情况。我们相信 Coxeter 群是超立方体对于 Huang 问题的适当概括。

最后，我们证明了在射影平面的 Levi 图和Lubotzky、Phillips 和 Sarnak的 Ramanujan图的一半以上顶点上的诱导子图具有无界度。这给出了与黄的结果提供的属性相似的凯莱图类。然而，与 Coxeter 组相比，这些图没有子立方体。