$2$-连通图中的可移除匹配
Removable matchings in $2$-connected graphs
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中文总结 AI 辅助
本文证明每个最小度至少为$d\ge5$且顶点数至少$2d$的$2$-连通图必存在可移除的$d$-匹配，该结果最优且完整回答了相关公开问题，证明基于极小非可移除匹配的分析。
中文摘要 AI 辅助
对于$2$-连通图$G$，若$G-M$是$2$-连通的，则称匹配$M$是可移除的，这推广了Halin关于可移除边的经典概念。我们证明：对每个整数$d\ge 5$，每个满足$\delta(G)\ge d$且$|V(G)|\ge 2d$的$2$-连通图$G$都有一个可移除的$d$-匹配。这是最优的，因为具有$n\\,(\ge 2d)$个顶点的完全二部图$K_{d,\\,n-d}$没有$(d+1)$-匹配。因此，我们的结果对每个$d\ge 5$完整回答了Li、Zhou、Fujita和Mao提出的关于最小度保证的可移除匹配最大规模的问题。此前由Li、Zhou、Fujita和Mao以及Chu、Kim和Park给出的最佳已知界在相同假设下保证存在可移除的$(d-2)$-匹配。我们的证明基于对\emph{极小非可移除匹配}的分析，即那些本身不可移除但其所有真子匹配都可移除的匹配。
英文摘要
A matching $M$ of a $2$-connected graph $G$ is removable if $G-M$ is $2$-connected, extending Halin's classical notion of a removable edge. We prove that for every integer $d\ge 5$, every $2$-connected graph $G$ with $δ(G)\ge d$ and $|V(G)|\ge 2d$ has a removable $d$-matching. This is best possible, since the complete bipartite graph $K_{d,\,n-d}$ on $n\,(\ge 2d)$ vertices has no $(d+1)$-matching. Consequently, our result gives a complete answer, for every $d\ge 5$, to a question of Li, Zhou, Fujita, and Mao on the maximum size of a removable matching guaranteed by the minimum degree. The previously best known bound, due to Li, Zhou, Fujita, and Mao and to Chu, Kim, and Park, guaranteed a removable $(d-2)$-matching under the same assumptions. Our proof is based on an analysis of \emph{minimal non-removable matchings}, matchings that are not removable although all of their proper submatchings are.
发表机构
- Inha University（仁荷大学）
机构由 AI 辅助整理，请以论文原文为准。
