The Rabin numbers of enhanced hypercubes

Abstract

The ω-Rabin number rω(G) and strong ω-Rabin number rω⁎(G) are two effective parameters to assess transmission latency and fault tolerance of an interconnection network G. As determining the Rabin number of a general graph is NP-complete, we consider the Rabin number of the enhanced hypercube Qn,k which is a variant of the hypercube Qn. For n≥k≥5, we prove that rω(Qn,k)=rω⁎(Qn,k)=d(Qn,k) for 1≤ω<n−⌊k2⌋; rω(Qn,k)=rω⁎(Qn,k)=d(Qn,k)+1 for n−⌊k2⌋≤ω≤n+1, where d(Qn,k) is the diameter of Qn,k. In addition, we present algorithms to construct internally disjoint paths of length at most rω⁎(Qn,k) from a source vertex to other ω (1≤ω≤n+1) destination vertices (not necessarily distinct) in Qn,k.