The 4-set tree connectivity of hierarchical folded hypercube

The k-set tree connectivity, as a natural extension of classical connectivity, is an extremely significant index to measure the fault-tolerance of interconnection networks. For a connected graph \(G=(V, E)\) $G=(V,E)$ and a subset \(S\subseteq V\) $S\subseteq V$ , an S-tree \(T=(V',E')\) $T=(V^{\prime },E^{\prime })$ of graph G is a tree that contains the subset S. Any two S-trees T and \(T'\) $T^{\prime }$ are internally disjoint if and only if \(E(T)\cap E(T')=\emptyset \) $E(T)\cap E(T^{\prime })=\mathrm{\varnothing }$ and \(V(T)\cap V(T')=S\) $V(T)\cap V(T^{\prime })=S$ . The cardinality of maximum internally disjoint S-trees is defined as \(\kappa _{G}(S)\) ${\kappa }_G(S)$ , and the k-set tree connectivity is denoted by \(\kappa _{k}(G)=\min \{\kappa _{G}(S)|S\subseteq V(G)\ \text {and} \ |S|=k\}\) ${\kappa }_k(G)=min\{{\kappa }_G(S)|S\subseteq V(G)\text{and}|S|=k\}$ . In this note, we determine the 4-set tree connectivity of \(HFQ_{n}\) $HF{Q}_n$ . That is, \(\kappa _{4}(HFQ_{n})=n+1\) ${\kappa }_4(HF{Q}_n)=n+1$ for \(n\ge 7\) $n\ge 7$ , where \(HFQ_{n}\) $HF{Q}_n$ is hierarchical folded hypercube.