Hamiltonicity in generalized quasi-dihedral groups

Witte Morris showed in [Discrete Math, 38.1 (1982)] that every connected Cayley graph of a finite (generalized) dihedral group has a Hamiltonian path. The infinite dihedral group is defined as the free product [Formula: see text]. We show that every connected Cayley graph of the free product with amalgamation [Formula: see text] has a Hamiltonian double ray, where [Formula: see text] is a generalized quasi-dihedral group on [Formula: see text]. Additionally, this leads to the conclusion that each connected Cayley graph of the infinite dihedral group also contains a Hamiltonian double ray.