A theorem on fractional ID-(g,f)-factor-critical graphs

Abstract

Let $a$, $b$ and $r$ be three nonnegative integers with $2\leq a\leq b-r$, let $G$ be a graph of order $p$ satisfying the inequality $p(a+r) \geq (a+b-3)(2a+b+r)+1$, and let $g$ and $f$ be two integer-valued functions defined on $V(G)$ satisfying $a\leq g(x)\leq f(x)-r\leq b-r$ for every $x\in V(G)$. A graph $G$ is said to be fractional ID-$(g,f)$-factor-critical if $G-I$ contains a fractional $(g,f)$-factor for every independent set $I$ of $G$. In this paper, we prove that $G$ is fractional ID-$(g,f)$-factor-critical if $\operatorname{bind}(G)((a+r)p - (a+b-2)) > (2a+b+r-1)(p-1)$, which is a generalization of a previous result of Zhou.