Fractional factors and component factors in graphs with isolated toughness smaller than 1

Let $G$ be a simple graph and let $n,m$ be two integers with $0<m<n$. We prove that $iso(G-S)\le \frac{n}{m}\mathrm{\mid }S\mathrm{\mid }$ for every $S\subset V(G)$ if and only if $G$ has a $\{C_{2i+1},T\text{\hspace{0.17em}⁣}:1\le i<\frac{m}{n-m},T\in {\mathcal{T}}_{\frac{n}{m}}\}$-factor, where $iso(G-S)$ denotes the number of isolated vertices of $G-S$ and ${\mathcal{T}}_{\frac{n}{m}}$ is a special family of trees. Furthermore, we characterize the trees in ${\mathcal{T}}_{\frac{n}{m}}$ in terms of their bipartition.