Minimally k-factor-critical graphs for some large k

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 $n-4\leq \Delta(G)\leq n-1$.