Sufficient conditions for Hamiltonian cycles in bipartite digraphs

Abstract: We prove two sharp sufficient conditions for hamiltonian cycles in balanced bipartite directed graph. Let $D$ be a strongly connected balanced bipartite directed graph of order $2a$. 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. (i) {\it If $a\geq 4$ and $max {d(x), d(y)}\geq 2a-1$ for every dominating pair of vertices ${x,y}$, then either $D$ is hamiltonian or $D$ is isomorphic to one exceptional digraph of order eight.} (ii) {\it If $a\geq 5$ and $d(x)+d(y)\geq 4a-3$ for every dominating pair of vertices ${x,y}$, then $D$ is hamiltonian.} The first result improves a theorem of R. Wang (arXiv:1506.07949 [math.CO]), the second result, in particular, establishes a conjecture due to Bang-Jensen, Gutin and Li (J. Graph Theory , 22(2), 1996) for strongly connected balanced bipartite digraphs of order at least ten.