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 $X\subseteq V(G)$.
Categories: math.CO