Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 111 / 2 / a / Solution

The Cayley graph ${\mathrm{Cay}}_S(W)$ has vertex set $W$ and an unoriented edge $\{w,ws\}$ for every $w\in W$ and $s\in S$.

For a reflection of a Coxeter group $r\in R$, its wall isEquivalently, these are the edges fixed setwise and reversed by left multiplication by $r$. Removing their interiors separates the Cayley graph into two half-spaces.