Degree conditions for the existence of a {P2 ,P5}-factor in a graph

A subgraph of a graph $G$ is spanning if the subgraph covers all vertices of $G$. A path-factor of a graph $G$ is a spanning subgraph $H$ of $G$ such that every component of $H$ is a path. In this article, we prove that (\romannumeral1) a connected graph $G$ with $\delta(G)\geq5$ admits a $\{P_2,P_5\}$-factor if $G$ satisfies $\delta(G)>\frac{3\alpha(G)-1}{4}$; (\romannumeral2) a connected graph $G$ of order $n$ with $n\geq7$ has a $\{P_2,P_5\}$-factor if $G$ satisfies $\max\{d_G(x),d_G(y)\}\geq\frac{3n}{7}$ for any two nonadjacent vertices $x$ and $y$ of $G$.