Minimum degree of minimal (\emph{n}-10)-factor-critical graphs

Abstract

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. 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 minimal $k$-factor-critical graph of order $n$ has the minimum degree $k+1$ and confirmed it for $k=1, n-2, n-4$ and $n-6$. By using a novel approach, we have confirmed it for $k = n - 8$ in a previous paper. Continuing this method, we prove the conjecture to be true for $k=n-10$ in this paper.