Hamilton cycle problem in strong k-quasi-transitive digraphs with large diameter

Abstract

Let k be an integer with k ≥ 2. A digraph is k-quasi-transitive, if for any path x 0 x 1 . . . x k of length k, x 0 and x k are adjacent. Let D be a strong k-quasi-transitive digraph with even k ≥ 4 and diameter at least k + 2. It has been shown that D has a Hamiltonian path. However, the Hamiltonian cycle problem in D is still open. In this paper, we shall show that D may contain no Hamiltonian cycle with k ≥ 6 and give the sufficient condition for D to be Hamiltonian.