The generalized 3-connectivity of the folded hypercube $FQ_n$

TL;DR

This paper investigates the generalized 3-connectivity of the folded hypercube network, establishing that for any three vertices, there are exactly n internally disjoint connecting trees, highlighting its fault tolerance.

Contribution

It proves that the generalized 3-connectivity of the folded hypercube $FQ_n$ equals n for all n ≥ 2, providing new insights into its network reliability.

Findings

$oxed{ ext{For } n ext{ ≥ 2, } \, oxed{ ext{kappa}_3(FQ_n) = n}}$

Existence of n internally disjoint trees connecting any three vertices in $FQ_n$

Enhanced understanding of fault tolerance in folded hypercube networks

Abstract

The generalized $k$-connectivity of a graph $G$, denoted by $\kappa_k(G)$, is a generalization of the traditional connectivity. It is well known that the generalized $k$-connectivity is an important indicator for measuring the fault tolerance and reliability of interconnection networks. The $n$-dimensional folded hypercube $FQ_n$ is obtained from the $n$-dimensional hypercube $Q_n$ by adding an edge between any pair of vertices with complementary addresses. In this paper, we show that $\kappa_3(FQ_n)=n$ for $n\ge 2$, that is, for any three vertices in $FQ_n$, there exist $n$ internally disjoint trees connecting them.