The Bounds of Generalized 4-Connectivity of Folded Divide-and-Swap Cubes

Abstract

Connectivity along with its extensions are important metrices to estimate the fault-tolerance of interconnection networks. The classic connectivity [Formula: see text] of a graph [Formula: see text] is the minimum cardinality of a vertex set [Formula: see text] such that [Formula: see text] is connected or a single vertex. For any subset [Formula: see text] with [Formula: see text], a tree [Formula: see text] in [Formula: see text] is called an [Formula: see text]-tree if [Formula: see text]. Furthermore, any two [Formula: see text]-tree [Formula: see text] and [Formula: see text] are internally disjoint if [Formula: see text] and [Formula: see text]. We denote by [Formula: see text] the maximum number of pairwise internally disjoint [Formula: see text]-trees in [Formula: see text]. For an integer [Formula: see text], the generalized [Formula: see text]-connectivity of a graph [Formula: see text] is defined as [Formula: see text] and [Formula: see text]. For the [Formula: see text]-dimensional folded divide-and-swap cubes, [Formula: see text], we show the upper bound and the lower bound of [Formula: see text], that is [Formula: see text], where [Formula: see text] and [Formula: see text] in this paper.