Connectivity keeping pendant extensions of paths in $k$-connected triangle-free graphs
Fuzhou University（福州大学）；Center for Discrete Mathematics, Fuzhou University（福州大学离散数学中心）；School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）；Key Laboratory for Operations Research and Cybernetics of Fujian Universities（福建省高校运筹与控制论重点实验室）
Menghan Ma, Qinghai Liu, Liping Zhang, Yanmei Hong
Motivated by Mader's conjecture on connectivity keeping trees, we study the corresponding problem in triangle-free graphs.
For positive integers $k,m$, we first prove that every $k$-connected triangle-free graph $G$ with $δ(G)\geq k+\left \lceil \frac{m}{2} \right \rceil $ contains a path $P$ of order $m$ such that $G-V(P)$ remains $k$-connected. Moreover, if $κ(G)=k$, then for every end $F$ of $G$ and every vertex $v\in F$, the path $P$ can be chosen in $G[F]$ with $v$ as an endvertex. If $κ(G)\geq k+1$, then any vertex of $G$ can be chosen as an endvertex of $P$. This result confirms the conjecture of Luo, Tian and Wu on connectivity keeping trees in bipartite graphs for all paths, and it also sharpens Fujita's path theorem to the optimal minimum degree bound.
For an integer $i$ with $1\leq i\leq m-1$, 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. Let $(X,Y)$ be the bipartition of $P_m^+(i)$. We further prove that every $k$-connected triangle-free graph $G$ with $δ(G)\geq k+\max\{|X|,|Y|\}$ contains a subgraph $T\cong P_m^+(i)$ such that $G-V(T)$ remains $k$-connected. The minimum degree bound is sharp, and the result confirms the bipartite conjecture for all pendant extensions of paths.
Connectivity keeping pendant extensions of paths in $k$-connected graphs and triangle-free graphs
Center for Discrete Mathematics, Fuzhou University（福州大学离散数学中心）；Key Laboratory for Operations Research and Cybernetics of Fujian Universities（福建省高校运筹与控制论重点实验室）；School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）
Excluding induced subdivisions of the bull and related graphs
对于任何图$H$，设${\rm Forb}^*(H)$是不具有$H$的诱导细分的图类。在[A. D. Scott，Induced trees in graphs of large色数，{ \em Journal of图论}，24：297- 311，1997]中阐明，对于每个图$H$，存在函数$f_H:\mathbb{N} \rightarrow \mathbb{R}$，使得对于每个图$G \in {\rm Forb}^*(H)$，$χ(G) \leq f_H(ω(G))$。我们证明了这个猜想的几个图$H$，即爪（有下垂边的三角形），公牛（具有两个顶点不相交的悬垂边的三角形），以及我们称之为“项链”的东西，即，通过选择匹配使得匹配的任何边都不与路径的端点相关联而从路径获得的图，并且对于匹配的每个边，添加与该边的端点相邻的顶点。
Connectivity keeping trees in triangle-free graphs
Korea Institute for Advanced Study（韩国高等科学研究院）；School of Data Science, Yokohama City University（横滨市立大学数据科学学院）；Department of Mathematics Education, Seoul National University（首尔国立大学数学教育系）；Department of Mathematics, Ajou University（阿丘大学数学系）
London School of Economics and Political Science (LSE)（伦敦政治经济学院）；Colorado State University（科罗拉多州立大学）；King’s College London（伦敦国王学院）；Universitat Politècnica de Catalunya (UPC)（加泰罗尼亚理工大学）
