kk-tree connectivity of line graphs

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, an $S$-Steiner tree $T$ is a subgraph of $G$ that is a tree with $S\subseteq V(T)$. Two $S$-Steiner trees $T$ and $T'$ are internally disjoint (resp. edge-disjoint) if $E(T)\cap E(T')=\emptyset$ and $V(T)\cap V(T')=S$ (resp. if $E(T)\cap E(T')=\emptyset$). Let $\kappa_G (S)$ (resp. $\lambda_G (S)$) denote the maximum number of internally disjoint (resp. edge-disjoint) $S$-Steiner trees in $G$. The $k$-tree connectivity $\kappa_k(G)$ (resp. $k$-tree edge-connectivity $\lambda_k(G)$) of $G$ is then defined as the minimum $\kappa_G (S)$ (resp. $\lambda_G (S)$), where $S$ ranges over all $k$-subsets of $V(G)$. In [H. Li, B. Wu, J. Meng, Y. Ma, Steiner tree packing number and tree connectivity, Discrete Math. 341(2018), 1945--1951], the authors conjectured that if a connected graph $G$ has at least $k$ vertices and at least $k$ edges, then $\kappa_k(L(G))\geq \lambda_k(G)$ for any $k\geq 2$, where $L(G)$ is the line graph of $G$. In this paper, we confirm this conjecture and prove that the bound is sharp

The generalized connectivity of some regular graphs

The generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$ is a parameter that can measure the reliability of a network $G$ to connect any $k$ vertices in $G$, which is proved to be NP-complete for a general graph $G$. Let $S\subseteq V(G)$ and $\kappa_{G}(S)$ denote the maximum number $r$ of edge-disjoint trees $T_{1}, T_{2}, \cdots, T_{r}$ in $G$ such that $V(T_{i})\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots, r\}$ and $i\neq j$. For an integer $k$ with $2\leq k\leq n$, the {\em generalized $k$-connectivity} of a graph $G$ is defined as $\kappa_{k}(G)= min\{\kappa_{G}(S)|S\subseteq V(G)$ and $|S|=k\}$. In this paper, we study the generalized $3$-connectivity of some general $m$-regular and $m$-connected graphs $G_{n}$ constructed recursively and obtain that $\kappa_{3}(G_{n})=m-1$, which attains the upper bound of $\kappa_{3}(G)$ [Discrete Mathematics 310 (2010) 2147-2163] given by Li {\em et al.} for $G=G_{n}$. As applications of the main result, the generalized $3$-connectivity of many famous networks such as the alternating group graph $AG_{n}$, the $k$-ary $n$-cube $Q_{n}^{k}$, the split-star network $S_{n}^{2}$ and the bubble-sort-star graph $BS_{n}$ etc. can be obtained directly.Comment: 19 pages, 6 figure