Halin可去边定理向匹配的一个严格推广
A sharp extension of Halin's removable-edge theorem to matchings
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中文总结 AI 辅助
该研究将Halin可去边定理推广到匹配，证明满足特定最小度条件的k-连通图存在指定大小的k-可去匹配，证实相关猜想，明确严格最小度阈值并给出证明方法。
中文摘要 AI 辅助
k-连通图G的子图H称为k-可去的，若移除H的边集后G仍保持k-连通。Halin证明了每个满足δ(G)≥k+1的k-连通图都存在一条k-可去边。本文将该结论从单条边推广到任意指定大小的匹配，证明对正整数k和m，每个满足δ(G)≥max{k+1,2m-2}的k-连通图G，都存在大小为m的k-可去匹配，除非G同构于K_{2m-1}，或(k,m)=(1,2)且G是一个环，这证实了Li、Zhou、Fujita和Mao的一个猜想。该最小度界是严格的，两个例外均不可避免。由此，max{k+1,2m-1}是保证无例外此类匹配的严格最小度阈值。证明结合了Halin可去边定理的指定集强化形式与最大k-可去匹配的极值分析。
英文摘要
A subgraph $H$ of a $k$-connected graph $G$ is called \emph{$k$-removable} if $G-E(H)$ remains $k$-connected. Halin proved that every $k$-connected graph $G$ with $δ(G)\ge k+1$ has a $k$-removable edge. We extend this result from a single edge to matchings of any prescribed size by showing that, for positive integers $k$ and $m$, every $k$-connected graph $G$ with $δ(G)\ge\max\{k+1,2m-2\}$ contains a $k$-removable matching of size $m$, unless $G\cong K_{2m-1}$, or $(k,m)=(1,2)$ and $G$ is a cycle. This confirms a conjecture of Li, Zhou, Fujita, and Mao. The minimum degree bound is sharp, and both exceptions are unavoidable. Consequently, $\max\{k+1,2m-1\}$ is the sharp minimum degree threshold guaranteeing such a matching without exceptions. The proof combines a prescribed-set strengthening of Halin's removable-edge theorem with an extremal analysis of maximum $k$-removable matchings.
