AI 中文总结
研究k - 连通图中可移除匹配的最小度条件，证明对于满足特定条件的k - 连通图G，存在规定大小的k - 可移除匹配，验证了相关猜想，主要工具是强化哈林结果产生避开规定顶点集的k - 可移除边。
AI 中文摘要
1969年，哈林证明每个最小度至少为k + 1的k - 连通图G包含一条边e使得G - e是k - 连通的。由于边是大小为1的匹配，自然会问哈林的结果是否能扩展到更大尺寸的匹配，李、周、藤田和毛最近研究了这个问题。k - 连通图G的匹配M若G - M是k - 连通的，则称M为k - 可移除的。本文研究保证存在规定大小的k - 可移除匹配的最小度条件。具体证明了对于所有正整数k和m，每个至少有2m个顶点的k - 连通图G，若满足特定最小度条件，则包含大小为m的k - 可移除匹配。结果验证了李、周、藤田和毛在δ(G)≥2k + 1范围内的一个猜想。主要工具是强化哈林的结果以产生避开规定顶点集的k - 可移除边。
英文摘要
In 1969, Halin proved that every $k$-connected graph $G$ with minimum degree at least $k+1$ contains an edge $e$ such that $G-e$ is $k$-connected. As an edge is a matching of size one, it is natural to ask whether Halin's result extends to matchings of larger size, a question recently investigated by Li, Zhou, Fujita, and Mao. A matching $M$ of a $k$-connected graph $G$ is called \emph{$k$-removable} if $G-M$ is $k$-connected. In this paper, we study minimum degree conditions that guarantee the existence of a $k$-removable matching of prescribed size. Specifically, we prove that for all positive integers $k$ and $m$, every $k$-connected graph $G$ with at least $2m$ vertices contains a $k$-removable matching of size $m$ if \[δ(G)\ \ge\ \begin{cases} \max\bigl\{k+\bigl\lceil\tfrac m2\bigr\rceil,\ 2m\bigr\} & \text{if } k\ge m,\\[2pt] k+m & \text{if } k<m. \end{cases}\] As a consequence, every $k$-connected graph $G$ with $δ(G)\ge2k+1$ contains a $k$-removable matching of size $\bigl\lceil(δ(G)+1)/2\bigr\rceil$, unless $δ(G)$ is even and $G\cong K_{δ(G)+1}$. This verifies a conjecture of Li, Zhou, Fujita, and Mao in the range $δ(G)\ge2k+1$. Our main tool, of independent interest, is a strengthening of Halin's result producing a $k$-removable edge that avoids a prescribed set of vertices.
