On Hamiltonian Property of Cayley Digraphs

Let G be a finite group generated by S and C ( G, S ) the Cayley digraphs of G with connection set S . In this paper, we give some sufficient conditions for the existence of hamiltonian circuit in C ( G, S ), where G = Z m ⋊ H is a semiproduct of Z m by a subgroup H of G . In particular, if m is a prime, then the Cayley digraph of G has a hamiltonian circuit unless G = Z m × H . In addition, we introduce a new digraph operation, called φ -semiproduct of Γ 1 by Γ 2 and denoted by Γ 1 ⋊ φ Γ 2 , in terms of mapping φ : V (Γ 2 ) → {1, −1}. Furthermore we prove that C ( Z m , { a }) ⋊ φ C ( H, S ) is also a Cayley digraph if φ is a homomorphism from H to { 1 , − 1 } ≤ Z m ∗ , which produces some classes of Cayley digraphs that have hamiltonian circuits.