Binding numbers for all fractional (a,b,k)-critical graphs

Abstract

Let $G$ be a graph of order $n$, and let $a,b$ and
$k$ nonnegative integers with $2\leq a\leq b$. A graph $G$ is
called all fractional $(a,b,k)$-critical if after deleting any $k$
vertices of $G$ the remaining graph of $G$ has all fractional
$[a,b]$-factors. In this paper, it is proved that $G$ is all
fractional $(a,b,k)$-critical if
$n\geq\frac{(a+b-1)(a+b-3)+a}{a}+\frac{ak}{a-1}$ and
$bind(G)>\frac{(a+b-1)(n-1)}{an-ak-(a+b)+2}$. Furthermore, it is
shown that this result is best possible in some sense.