On a Problem of Wang Concerning the Hamiltonicity of Bipartite Digraphs

R. Wang (Discrete Mathematics and Theoretical Computer Science, vol. 19(3), 2017) proposed the following problem. Problem. Let D be a strongly connected balanced bipartite directed graph of order 2a ≥8. Suppose that d(x) ≥ 2a - k, d(y) ≥a + k or d(y) ≥2a - k, d(x) ≥a + k for every pair of vertices {x; y}with a common out-neighbour, where 2 • k • a=2. Is D Hamiltonian? In this paper, we prove that if a digraph D satis¯es the conditions of this problem, then (i) D contains a cycle factor, (ii) for every vertex x ∈ V (D) there exists a vertex y ∈ V (D) such that x and y have a common out-neighbour.