AI 中文总结
本研究针对Teng和田提出的关于k-连通图中保持连通性的生成(u,v)-路径的最小度条件猜想，给出了肯定证明，同时改进了图的阶数下界要求。
AI 中文摘要
Teng和田证明了如下结果：设$k\ge 2$且$t\ge 3$，令$G$为一个阶数为$n$的$k$-连通图。若当$t=3$时，$n\ge 6k+1$且$δ(G)\ge \lceil(n+6)/2\rceil$；当$t\ge 4$时，$n\ge 6k+7t-17$且$δ(G)\ge \lceil(n+t+2)/2\rceil$，则对任意两个不同顶点$u,v$以及每个满足$1\le s\le t$的整数$s$，存在$s$条内部顶点不交的$(u,v)$-路径$P_1,\dots,P_s$，它们的并生成$G$，且$G-E(P_1\cup\cdots\cup P_s)$是$k$-连通的。他们提出疑问：对所有$t\ge 3$，最小度条件是否可以降低到$δ(G)\ge \lceil(n+t)/2\rceil$。我们对此问题给出了肯定回答，并进一步将所需阶数降至$n\ge \max\{6k+9-3t,\\,2k+t+3\}$。
英文摘要
Teng and Tian proved the following result. Let $k\ge 2$ and $t\ge 3$, and let $G$ be a $k$-connected graph of order $n$. If $n\ge 6k+1$ and $δ(G)\ge \lceil(n+6)/2\rceil$ when $t=3$, while $n\ge 6k+7t-17$ and $δ(G)\ge \lceil(n+t+2)/2\rceil$ when $t\ge 4$, then, for any two distinct vertices $u,v$ and every integer $s$ with $1\le s\le t$, there exist $s$ internally vertex-disjoint $(u,v)$-paths $P_1,\dots,P_s$ whose union spans $G$ and such that $G-E(P_1\cup\cdots\cup P_s)$ is $k$-connected. They asked whether the minimum-degree condition could be lowered to $δ(G)\ge \lceil(n+t)/2\rceil$ for every $t\ge 3$. We answer this question affirmatively and further reduce the required order to $n\ge \max\{6k+9-3t,\,2k+t+3\}$.
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