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) and a subset S⊆ V , an S-tree T=(V',E') of graph G is a tree that contains the subset S. Any two S-trees T and T' are internally disjoint if and only if E(T)∩ E(T')=∅ and V(T)∩ V(T')=S . The cardinality of maximum internally disjoint S-trees is defined as κ _G(S) , and the k-set tree connectivity is denoted by κ _k(G)=min{κ _G(S)|S⊆ V(G) and |S|=k} . In this note, we determine the 4-set tree connectivity of HFQ_n . That is, κ _4(HFQ_n)=n+1 for n≥ 7 , where HFQ_n is hierarchical folded hypercube.