Two kinds of generalized connectivity of dual cubes

Abstract

Let $S\subseteq V(G)$ and $\kappa_{G}(S)$ denote the maximum number $k$ of edge-disjoint trees $T_{1}, T_{2}, \cdots, T_{k}$ in $G$ such that $V(T_{i})\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots, k\}$ and $i\neq j$. For an integer $r$ with $2\leq r\leq n$, the {\em generalized $r$-connectivity} of a graph $G$ is defined as $\kappa_{r}(G)= min\{\kappa_{G}(S)|S\subseteq V(G)$ and $|S|=r\}$. The $r$-component connectivity $c\kappa_{r}(G)$ of a non-complete graph $G$ is the minimum number of vertices whose deletion results in a graph with at least $r$ components. These two parameters are both generalizations of traditional connectivity. Except hypercubes and complete bipartite graphs, almost all known $\kappa_{r}(G)$ are about $r=3$. In this paper, we focus on $\kappa_{4}(D_{n})$ of dual cube $D_{n}$. We first show that $\kappa_{4}(D_{n})=n-1$ for $n\geq 4$. As a corollary, we obtain $\kappa_{3}(D_{n})=n-1$ for $n\geq 4$. Furthermore, we show that $c\kappa_{r+1}(D_{n})=rn-\frac{r(r+1)}{2}+1$ for $n\geq 2$ and $1\leq r \leq n-1$.