Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
23
The categories of this paper are: solv-int
24
The categories of this paper are: dg-ga