大k的极小k-因子临界图
Minimally k-factor-critical graphs for some large k
摘要：
一个$n$阶的图$G$被称为是整数$1\leq k < n$的$k$ -因子关键图，如果除去任何$k$顶点都能得到一个完美匹配的图。$1$ -因子临界图和$2$ -因子临界图分别是著名的因子临界图和双临界图。一个$k$ -因子临界图$G$称为极小图，如果对任何边$e\in E(G)$，$G-e$不是$k$ -因子临界图. 1998年，O. Favaron和M.证明了$n$阶的最小$k$ -因子临界图都有最小度$k+1$，并对$k=1, n-2, n-4$和$n-6$证明了这一点.本文用一种简单的方法重新证明了上述结果。作为一个主要的结果，进一步使用这种方法，使人们能够证明猜想是正确的$k=n-8$。我们还得到了对于$n-4\leq Δ(G)\leq n-1$，每一个$n$阶极小$(n-6)$ -因子临界图至多有$n-Δ(G)$个顶点的最大度为$Δ(G)$.
英文摘要：
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 $n-Δ(G)$ vertices with the maximum degree $Δ(G)$ for $n-4\leq Δ(G)\leq n-1$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
