Hamiltonicity of random Cayley graphs with O(log |G|) generators (Pak–Radoičić conjecture)

Show that for a finite group G, if S is a uniformly random subset of G of size d = C log |G| for a sufficiently large constant C, then the random Cayley graph Γ(G,S) is almost surely Hamiltonian.

Background

Alon–Roichman showed connectivity of random Cayley graphs with Θ(log |G|) generators. Pak and Radoičić conjectured that such random Cayley graphs are almost surely Hamiltonian, a random analogue of Strasser’s conjecture.

Using Theorem 1.5 on spectral expanders, the paper proves Theorem 1.8, settling this conjecture affirmatively.