A sufficient condition for pre-Hamiltonian cycles in bipartite digraphs

Abstract: Let $D$ be a strongly connected balanced bipartite directed graph of order $2a\geq 10$ other than a directed cycle. Let $x,y$ be distinct vertices in $D$. ${x,y}$ dominates a vertex $z$ if $x\rightarrow z$ and $y\rightarrow z$; in this case, we call the pair ${x,y}$ dominating. In this paper we prove: If $ max{d(x), d(y)}\geq 2a-2$ for every dominating pair of vertices ${x,y}$, then $D$ contains cycles of all lengths $2,4, \ldots , 2a-2$ or $D$ is isomorphic to a certain digraph of order ten which we specify.