The 4-set tree connectivity of hierarchical folded hypercube

Abstract

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 $S\subseteq V$, 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 $E(T)\cap E(T')=\emptyset$ 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 denoted by $\kappa _{k}(G)=\min \{\kappa _{G}(S)|S\subseteq V(G)\ \text {and} \ |S|=k\}$. In this note, we determine the 4-set tree connectivity of $HFQ_{n}$. That is, $\kappa _{4}(HFQ_{n})=n+1$ for $n\ge 7$, where $HFQ_{n}$ is hierarchical folded hypercube.