### A minimum degree condition for fractional ID-[a,b]-factor-critical graphs

#### Abstract

Let $G$ be a graph of order $n$, and let $a$

and $b$ be two integers with $1\leq a\leq b$. Let $h:

E(G)\rightarrow [0,1]$ be a function. If $a\leq\sum_{e\ni

x}h(e)\leq b$ holds for any $x\in V(G)$, then we call $G[F_h]$ a

fractional $[a,b]$-factor of $G$ with indicator function $h$ where

$F_h=\{e\in E(G): h(e)>0\}$. A graph $G$ is fractional

independent-set-deletable $[a,b]$-factor-critical (in short,

fractional ID-$[a,b]$-factor-critical) if $G-I$ has a fractional

$[a,b]$-factor for every independent set $I$ of $G$. In this

paper, it is proved that if $n\geq\frac{(a+2b)(a+b-2)+1}{b}$ and

$\delta(G)\geq\frac{(a+b)n}{a+2b}$, then $G$ is fractional

ID-$[a,b]$-factor-critical. This result is best possible in some

sense, and it is an extension of Chang and Liu's

previous result.

DOI: 10.1017/S0004972711003467

and $b$ be two integers with $1\leq a\leq b$. Let $h:

E(G)\rightarrow [0,1]$ be a function. If $a\leq\sum_{e\ni

x}h(e)\leq b$ holds for any $x\in V(G)$, then we call $G[F_h]$ a

fractional $[a,b]$-factor of $G$ with indicator function $h$ where

$F_h=\{e\in E(G): h(e)>0\}$. A graph $G$ is fractional

independent-set-deletable $[a,b]$-factor-critical (in short,

fractional ID-$[a,b]$-factor-critical) if $G-I$ has a fractional

$[a,b]$-factor for every independent set $I$ of $G$. In this

paper, it is proved that if $n\geq\frac{(a+2b)(a+b-2)+1}{b}$ and

$\delta(G)\geq\frac{(a+b)n}{a+2b}$, then $G$ is fractional

ID-$[a,b]$-factor-critical. This result is best possible in some

sense, and it is an extension of Chang and Liu's

previous result.

DOI: 10.1017/S0004972711003467

#### Keywords

graph, minimum degree, independent set, fractional $[a,b]$-factor, fractional ID-$[a,b]$-factor-critical graph.

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Bulletin of the Aust. Math. Soc., copyright Australian Mathematical Publishing Association Inc.