Fractional Factors, Component Factors and Isolated Vertex Conditions in Graphs

For a graph $G = (V, E)$, a {\em fractional $[a, b]$-factor} is a real valued function $h:E(G)\to [0,1]$ that satisfies $a \le ~ \sum_{e\in E_G(v)} h(e) ~ \le b$ for all $v\in V(G)$, where $a$ and $b$ are real numbers and $E_G(v)$ denotes the set of edges incident with $v$. In this paper, we prove that the condition $\mathit{iso}(G-S) \le (k+\frac{1}{2})|S|$ is equivalent to the existence of fractional $[1,k+ \frac{1}{2}]$-factors, where ${\mathit{iso}}(G-S)$ denotes the number of isolated vertices in $G-S$. Using fractional factors as a tool, we construct component factors under the given isolated conditions. Namely, (i) a graph $G$ has a $\{P_2,C_3,P_5, \mathcal{T}(3)\}$-factor if and only if $\mathit{iso}(G-S) \le \frac{3}{2}|S|$ for all $S\subset V(G)$; (ii) a graph $G$ has a $\{K_{1,1}, K_{1,2}, \ldots,$ $K_{1,k}, \mathcal{T}(2k+1)\}$-factor ($k\ge 2$) if and only if $\mathit{iso}(G-S) \le (k+\frac{1}{2})|S|$ for all $S\subset V(G)$, where $\mathcal{T}(3)$ and $\mathcal{T}(2k+1)$ are two special families of trees.