Reliability assessment of the divide-and-swap cube in terms of generalized connectivity

Abstract

The generalized connectivity of a graph G, denoted as κk(G), is a generalization of the traditional connectivity and is a parameter to measure the capability of connecting multiple vertices in G. The divide-and-swap cube DSCn is a variant of the hypercube with a nice hierarchical structure and plentiful properties. This paper investigates the generalized connectivity on DSCn. We obtain the result κ4(DSCn)=d, where d=log2⁡n≥1, by the construction showing that there are d internally disjoint trees connecting any four arbitrary vertices of DSCn. As a corollary, one can directly obtain κ3(DSCn)=d.