  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, F\"uredi, 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 $\mathbf{A_n}$, $\mathbf{I_2}(2k+1)$, 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.
