Directed Hamilton cycles in digraphs and matching alternating Hamilton cycles in bipartite graphs

In 1972, Woodall raised the following Ore type condition for directed Hamilton cycles in digraphs: Let $D$ be a digraph. If for every vertex pair $u$ and $v$, where there is no arc from $u$ to $v$, we have $d^{+}u)+d^{-}(v)\ge \mathrm{\mid }D\mathrm{\mid }$, then $D$ has a directed Hamilton cycle. By a correspondence between bipartite graphs and digraphs, the above result is equivalent to the following result of Las Vergnas: Let $G=(B,W)$ be a balanced bipartite graph. If for any $b \in B$and$w \in W$,where$b$and$w$arenonadjacent,wehave$d(w)+d(b) \geq |G|/2 + 1$,theneveryperfectmatchingof$G$ is contained in a Hamilton cycle. The lower bounds in both results are tight. In this paper, we reduce both bounds by $1$, and prove that the conclusions still hold, with only a few exceptional cases that can be clearly characterized.