The 4-Set Tree Connectivity of Hierarchical Folded Hypercube

The $k$-set tree connectivity, as a natural extension of classical connectivity, is a very important index to evaluate the fault-tolerance of interconnection networks. Let $G=(V, E)$ be a connected graph and a subset $S\subseteq V$, an $S$-tree of graph $G$ is a tree $T=(V',E')$ that contains all the vertices of $S$. Two $S$-trees $T$ and $T'$ are internally disjoint if and only if $E(T)\cap E(T')=\varnothing$ and $V(T)\cap V(T')=S$. The cardinality of maximum internally disjoint $S$-trees is defined as $\kappa_{G}(S)$, and the $k$-set tree connectivity is defined by $\kappa_{k}(G)=\min\{\kappa_{G}(S)|S\subseteq V(G)\ \text{and} \ |S|=k\}$. In this paper, we show that the $k$-set tree connectivity of hierarchical folded hypercube when $k=4$, that is, $\kappa_{4}(HFQ_{n})=n+1$, where $HFQ_{n}$ is hierarchical folded hypercube for $n\geq 7$.