A neighborhood 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 $1\le a\le b$. Let $h:E(G)\to [0,1]$ be a function. If $a\le {\sum }_{e\ni x}h(e)\le b$ 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 $n\ge \frac{(a+2b)(2a+2b-3)+1}{b}$, $\delta (G)\ge \frac{bn}{a+2b}+a$ and $|N_G(x)\cup N_G(y)|\ge \frac{(a+b)n}{a+2b}$ for any two nonadjacent vertices $x,y\in V(G)$, then $G$ is fractional ID-$[a,b]$-factor-critical. Furthermore, it is shown that this result is best possible in some sense.

7 pages