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

Abstract

Let $G$ be a simple graph and let $n,m$ be two integers with $0<m<n$. We prove that $iso(G-S)\leq \frac{n}{m}|S|$ for every $S \subset V(G)$ if and only if $G$ has a $\{C_{2i+1},T \colon 1 \leq i < \frac{m}{n-m}, T\in\mathcal{T}_{\frac{n}{m}}\}$-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.