Let $G$ be a connected graph, and $S\subseteq V(G)$ with $\left|S\right|\ge 2$. ${\kappa }_G(S)$ refers to the maximum number $k$ of edge-disjoint trees $T_1,T_2,\dots ,T_k$ in $G$ such that $V(T_i)\cap V(T_j)=S$ for distinct $i,j\in \{1,2,\dots ,k\}$. The generalized $k$-connectivity of $G$, denoted by ${\kappa }_k(G)$, is defined as the minimum value of ${\kappa }_G(S)$ over all $S\subseteq V(G)$ with $\left|S\right|=k$. In fact, ${\kappa }_2(G)$ is exactly the traditional connectivity of $G$. Exchanged crossed cube $ECQ(s,t)$, a variation of hypercube, has better properties than other variations of hypercube. In this work we obtain that ${\kappa }_3(ECQ(s,t))=s$ with $2\le s\le t$.

中文翻译：

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交换交叉立方体的广义3连通性

让$G$是一个连通图，并且$S\subseteq V（G）$和$\left|S\right|\ge 2$。${\kappa }_G（S）$指最大数量$k$边不相交的树${\mathrm{时间}}_1,{\mathrm{时间}}_2,\dots \dots ,{\mathrm{时间}}_k$在$G$这样$V（{\mathrm{时间}}_{我}）\cap V（{\mathrm{时间}}_j）=S$对于不同的$我,j\epsilon \{1,2,\dots \dots ,k\}$。广义的$k$-连接性$G$，表示为${\kappa }_k（G）$，定义为最小值${\kappa }_G（S）$全面的$S\subseteq V（G）$和$\left|S\right|=k$。实际上，${\kappa }_2（G）$正是传统的连接方式$G$。交换交叉立方体$乙C问（s,t）$是超立方体的一种变体，比超立方体的其他变体具有更好的属性。在这项工作中我们得到了${\kappa }_3（乙C问（s,t））=s$和$2\le s\le t$。