The generalized 4-connectivity of locally twisted cubes

Abstract

The hypercube is one of the most famous interconnection networks as the topology structure for parallel and distributed systems because of its many good properties. The locally twisted cube, an important variant of hypercube, has lower diameter compared with the same dimensional hypercube, and thus attracts much interesting of many researchers. For $L \subseteq V(G)$, the L-tree means a tree joining each node of L. The L-trees $T_1, T_2, \ldots , T_{l}$ are internally disjoint if $V(T_a) \cap V(T_b)=L$ and $E(T_a) \cap E(T_b)=\emptyset$ for any two different integers $a, b \in \{1, 2, \ldots , l\}$. Let $\kappa (L)=\max \{l \mid T_1, T_2, \ldots , T_l$ are internally disjoint L-trees $\}$ and $\kappa _l(G)=$min$\{\kappa (L) \mid L \subseteq V(G)$ and $\vert L \vert =l\}$. For a graph G, the generalized l-connectivity, denoted by $\kappa _l(G)$, is an extension of connectivity to better evaluate the fault-tolerance of networks. For the n-dimensional locally twisted cube, in this paper, we show that $\kappa _4(LTQ_n)=n-1$, where $n \ge 2$. Since $\kappa _4(LTQ_n) \le \delta (LTQ_n)-1$ and $LTQ_n$ is n-regular, our result is optimal.