Independence Number, Connectivity and All Fractional (a, b, k)-Critical Graphs

Let $G$ be a graph and $a$, $b$ and $k$ be nonnegative integers with $ 1 \le a \le b $. A graph $G$ is defined as all fractional $(a, b, k)$-critical if after deleting any $k$ vertices of $G$, the remaining graph has all fractional $[a, b]$-factors. In this paper, we prove that if \( \kappa(G) \ge \text{max} \{ \tfrac{(b+1)^2+2k}{2}, \tfrac{(b+1)^2 \alpha(G)+4ak}{4a} \} \), then $G$ is all fractional $(a, b, k)$-critical. If $k = 0$, we improve the result given in [Filomat 29 (2015) 757-761]. Moreover, we show that this result is best possible in some sense.