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 Γ₁ by Γ₂ and denoted by Γ₁ ⋊_φ Γ₂, in terms of mapping φ: V(Γ₂) → {1, −1}. Furthermore we prove that C(Z_m, {a}) ⋊_φC(H, S) is also a Cayley digraph if φ is a homomorphism from H to \(\{ 1, - 1\} \le Z_m^ * \) $$\{1,-1\}\le Z_m^{\ast }$$ , which produces some classes of Cayley digraphs that have hamiltonian circuits.