Theorem 1.1. If |G| = 27p, where p is prime, then every connected Cayley graph on G has a Hamiltonian cycle. 

Combining this with results in [1–3] establishes that 

Every Cayley graph on G has a hamiltonian cycle if |G| = kp,  where p is prime,  1 ? k < 32,  and k = / 24. 

 

The remainder of the paper provides a proof of the theorem. Here is an outline. Section 2 recalls known results on hamiltonian cycles in Cayley graphs; Section 3 presents the proof under the assumption that the Sylow p-subgroup of G is normal; Section 4 presents the proof under the assumption that the Sylow p-subgroups of G are not normal. 

(1.1)