Pancyclicity in strong k-quasi-transitive digraphs of large diameter

•A digraph in this family admits a partition into two semicomplete digraphs.•Different cycles can be obtained by combining cycles in the parts of D.•A hamiltonian cycle in this family can be constructed in polynomial time. A digraph is k-quasi-transitive if the existence of a directed path of length k from vertex u to vertex v implies that u and v are adjacent; this family is a generalization of both tournaments and transitive digraphs. Wang and Zhang proved that, when k is even, every strong k-quasi-transitive digraph with diameter at least k+2 contains a hamiltonian path, and asked whether under the same assumptions, a hamiltonian cycle could be found. In this work we answer this question in the affirmative, and moreover, we prove that not only hamiltonicity, but pancyclicity can be proved. As a byproduct of our proof, we derive a polynomial time algorithm to find a hamiltonian cycle in a k-quasi-transitive digraph with diameter at least k+2, for even values of k.