  A graph $G$ of order $n$ is said to be $k$-factor-critical for integers
$1\leq k < n$, if the removal of any $k$ vertices results in a graph with a
perfect matching. $1$- and $2$-factor-critical graphs are the well-known
factor-critical and bicritical graphs, respectively. A $k$-factor-critical
graph $G$ is called minimal if for any edge $e\in E(G)$, $G-e$ is not
$k$-factor-critical. In 1998, O. Favaron and M. Shi conjectured that every
minimally $k$-factor-critical graph of order $n$ has the minimum degree $k+1$
and confirmed it for $k=1, n-2, n-4$ and $n-6$. In this paper, we use a simple
method to reprove the above result. As a main result, the further use of this
method enables ones to prove the conjecture to be true for $k=n-8$. We also
obtain that every minimally $(n-6)$-factor-critical graph of order $n$ has at
most $n-\Delta(G)$ vertices with the maximum degree $\Delta(G)$ for $n-4\leq
\Delta(G)\leq n-1$.
