Sufficient conditions for the existence of a path-factor which are related to odd components

In this paper, we are concerned with sufficient conditions for the existence of a $\{P_{2},P_{2k+1}\}$-factor. We prove that for $k\geq 3$, there exists $\varepsilon_{k}>0$ such that if a graph $G$ satisfies $\sum_{0\leq j\leq k-1}c_{2j+1}(G-X)\leq \varepsilon_{k}|X|$ for all $X\subseteq V(G)$, then $G$ has a $\{P_{2},P_{2k+1}\}$-factor, where $c_{i}(G-X)$ is the number of components $C$ of $G-X$ with $|V(C)|=i$. On the other hand, we construct infinitely many graphs $G$ having no $\{P_{2},P_{2k+1}\}$-factor such that $\sum_{0\leq j\leq k-1}c_{2j+1}(G-X)\leq \frac{32k+141}{72k-78}|X|$ for all $X\subseteq V(G)$.