发表机构
Center for Discrete Mathematics, Fuzhou University; Key Laboratory for Operations Research and Cybernetics of Fujian Universities; School of Mathematics and Statistics, Fuzhou University（福州大学离散数学中心; 福建省高校运筹与控制论重点实验室; 福州大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明在$k$-连通图（满足最小度条件）和无三角形图中，路径的悬挂扩展$P_m^+(i)$可作为子图被移除后保持$k$-连通性，从而证实Mader猜想对路径悬挂扩展成立。
AI 中文摘要
受Mader关于保持连通性的树的猜想的启发，我们研究了由路径添加一个悬挂顶点得到的树，以及无三角形图中的相关问题。对于整数$m$和$1\leq i\leq m-1$，令$P_m^+(i)$表示从阶为$m-1$的路径通过添加一个悬挂顶点与其第$i$个顶点相邻而得到的树。我们证明，对于正整数$k,m,1\leq i\leq m-1$，每个满足$\delta(G)\geq \lfloor \frac{3k}{2}\rfloor+m-1$的$k$-连通图$G$包含一个子图$T\cong P_m^+(i)$，使得$\kappa(G-V(T))\geq k$。这证实了Mader猜想对所有路径的悬挂扩展成立。对于高度连通的无三角形图，在[J. Combin. Theory Ser. B, 174 (2025), 190-206]中获得了路径的连通性保持结果。令$(X,Y)$为$P_m^+(i)$的二部划分。我们进一步证明，每个满足$\delta(G)\geq k+\max\{|X|,|Y|\}+[P_m^+(i)\text{ is bad}]$的$k$-连通无三角形图$G$包含一个子图$T\cong P_m^+(i)$，使得$\kappa(G-V(T))\geq k$，其中我们使用Iverson约定表示$[P_m^+(i)\text{ is bad}]$。这将路径的相应结果扩展到路径的悬挂扩展。
英文摘要
Motivated by Mader's conjecture on connectivity keeping trees, we study trees obtained from paths by adding one pendant vertex, as well as related problems in triangle-free graphs. For an integer $m$ and $1\leq i\leq m-1$, let $P_m^+(i)$ denote the tree obtained from a path of order $m-1$ by adding one pendant vertex adjacent to its $i$th vertex. We prove that, for positive integers $k,m,1\leq i\leq m-1$, every $k$-connected graph $G$ with $δ(G)\geq \lfloor \frac{3k}{2}\rfloor+m-1$ contains a subgraph $T\cong P_m^+(i)$ such that $κ(G-V(T))\geq k$. This confirms Mader's conjecture for all pendant extensions of paths. For highly connected triangle-free graphs, a connectivity keeping result for paths was obtained in [J. Combin. Theory Ser. B, 174 (2025), 190-206]. Let $(X,Y)$ be the bipartition of $P_m^+(i)$. We further prove that every $k$-connected triangle-free graph $G$ with $δ(G)\geq k+\max\{|X|,|Y|\}+[P_m^+(i)\text{ is bad}]$ contains a subgraph $T\cong P_m^+(i)$ such that $κ(G-V(T))\geq k$, where we use Iverson's convention for $[P_m^+(i)\text{ is bad}]$. This extends the corresponding result for paths to pendant extensions of paths.
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