Paired many-to-many 2-disjoint path cover of balanced hypercubes with faulty edges

TL;DR

This paper proves that balanced hypercubes can still be fully covered by two disjoint paths connecting specified vertices even with up to 2n-3 faulty edges, demonstrating optimal fault tolerance.

Contribution

It establishes the maximum number of faulty edges (2n-3) that balanced hypercubes can tolerate while maintaining disjoint path coverage, which is proven to be optimal.

Findings

Balanced hypercubes tolerate up to 2n-3 faulty edges.

Existence of two disjoint paths covering all vertices with faults.

Optimal fault tolerance bound established.

Abstract

As a variant of the well-known hypercube, the balanced hypercube $BH_n$ was proposed as a novel interconnection network topology for parallel computing. It is known that $BH_n$ is bipartite. Assume that $S=\{s_1,s_2\}$ and $T=\{t_1,t_2\}$ are any two sets of two vertices in different partite sets of $BH_n$ ($n\geq1$). It has been proved that there exist two vertex-disjoint $s_1,t_1$-path and $s_2,t_2$-path of $BH_n$ covering all vertices of it. In this paper, we prove that there always exist two vertex-disjoint $s_1,t_1$-path and $s_2,t_2$-path covering all vertices of $BH_n$ with at most $2n-3$ faulty edges. The upper bound $2n-3$ of edge faults tolerated is optimal.