Exercise 9, § 2

So let $(W,S)$ be a Coxeter system (LIE, IV, §1, p. 11, def. 3); suppose that the group W is finite.

a) Suppose that $(W,S)$ is irreducible and has at most two vertices. Prove that the Cayley graph $\mathscr{G}(W,S)$ is Hamiltonian.

b) Suppose that $(W,S)$ is irreducible and has at least three vertices. Let $s\in S$ be a terminal vertex (LIE IV, appendix) of the Coxeter graph associated with $(W,S)$. Put $S'= S-\{s\}$ and denote by $W'$ the subgroup of W generated by $S'$. Let Γ be the graph whose set of vertices is the set of left classes modulo $W'$ in W and such that two distinct vertices A and B are joined by an edge if and only if $As\cap B$ is not empty. Using a maximal subtree of the graph Γ, prove that if the Cayley graph $\mathscr{G}(W',S')$ is Hamiltonian, the same is true of the graph $\mathscr{G}(W,S)$.

c) Prove that the Cayley graph $\mathscr{G}(W,S)$ is Hamiltonian (Conway–Sloane–Wilks theorem).