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 dHamilton 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 ∈ B and w ∈ W , where b and w are nonadjacent, we have d(w) + d(b) ≥ |G|/2 + 1, then every perfect matching of 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.

A spanning subgraph of a graph $G$ is called a path-factor of $G$ if its each component is a path. A path-factor is called a $\mathcal{P}_{\geq k}$-factor of $G$ if its each component admits at least $k$ vertices, where $k\geq2$. Zhang and Zhou [\emph{Discrete Mathematics}, \textbf{309}, 2067-2076 (2009)] defined the concept of $\mathcal{P}_{\geq k}$-factor covered graphs, i.e., $G$ is called a $\mathcal{P}_{\geq k}$-factor covered graph if it has a $\mathcal{P}_{\geq k}$-factor covering $e$ for any $e\in E(G)$. In this paper, we firstly obtain a minimum degree condition for a planar graph being a $\mathcal{P}_{\geq 2}$-factor and $\mathcal{P}_{\geq 3}$-factor covered graph, respectively. Secondly, we investigate the relationship between the maximum degree of any pairs of non-adjacent vertices and $\mathcal{P}_{\geq k}$-factor covered graphs, and obtain a sufficient condition for the existence of $\mathcal{P}_{\geq2}$-factor and $\mathcal{P}_{\geq 3}$-factor covered graphs, respectively.

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