[Paper Review] Edge-fault-tolerant edge-bipancyclicity of balanced hypercubes

Pingshan Li, Min Xu|arXiv (Cornell University)|Jun 16, 2016

Interconnection Networks and Systems10 references3 citations

TL;DR

This paper proves that the balanced hypercube $BH_n$ remains 6-edge-bipancyclic even with up to $(2n-2)$ faulty edges, meaning every fault-free edge lies on a fault-free cycle of every even length from 6 to $2^{2n}$. The result improves upon prior work by extending the fault tolerance threshold to the optimal limit of $2n-2$ edge faults, using recursive decomposition and cycle construction techniques in a recursive structure.

ABSTRACT

The balanced hypercube, $BH_n$, is a variant of hypercube $Q_n$. R.X. Hao et al. $(2014)$ \cite{R.X.Hao} showed that there exists a fault-free Hamiltonian path between any two adjacent vertices in $BH_n$ with $(2n-2)$ faulty edges. D.Q. Cheng et al. $(2015)$ \cite{Dongqincheng2} proved that $BH_n$ is $6$-edge-bipancyclic after $(2n-3)$ faulty edges occur for all $n\ge2$. In this paper, we improve these two results by demonstrating that $BH_n$ is $6$-edge-bipancyclic even when there exist $(2n-2)$ faulty edges for all $n\ge2$. Our result is optimal with respect to the maximum number of tolerated edge faults.

Motivation & Objective

• To determine the maximum number of edge faults that $BH_n$ can tolerate while preserving edge-bipancyclicity.
• To improve upon prior results showing $BH_n$ remains 6-edge-bipancyclic under $(2n-3)$ faulty edges.
• To establish optimality of the fault tolerance threshold for edge-bipancyclicity in $BH_n$.
• To provide a constructive proof that every fault-free edge lies on cycles of all even lengths from 6 to $2^{2n}$, even with up to $(2n-2)$ faulty edges.

Proposed method

• Recursive decomposition of $BH_n$ into $n-1$-dimensional balanced hypercubes $BH_{n-1}$ to analyze fault resilience.
• Use of edge-disjoint cycle constructions and path extensions within subgraphs to maintain cycle diversity.
• Application of case analysis based on fault distribution across subcube boundaries ($∂D_0$, $∂D_1$) and edge types.
• Construction of fault-free cycles of length $\ell$ for $6 \leq \ell \leq 2^{2n}$ by combining paths of length 1 or 3 in subcubes.
• Leveraging known properties of $BH_n$, such as bipancyclicity and Hamiltonian laceability, in the fault-tolerant setting.
• Proof by induction and exhaustive case analysis on fault sets of size $2n-2$, ensuring at least one valid cycle of each required even length exists.

Experimental results

Research questions

• RQ1 What is the maximum number of edge faults that $BH_n$ can tolerate while still maintaining 6-edge-bipancyclicity?
• RQ2 Can the fault tolerance threshold for edge-bipancyclicity in $BH_n$ be improved beyond $(2n-3)$ faulty edges?
• RQ3 Is the bound of $(2n-2)$ faulty edges optimal for $BH_n$ to remain 6-edge-bipancyclic?
• RQ4 How can fault-free cycles of all even lengths from 6 to $2^{2n}$ be systematically constructed in $BH_n$ under edge faults?

Key findings

• The balanced hypercube $BH_n$ is $6$-edge-bipancyclic under up to $(2n-2)$ faulty edges for all $n \geq 2$, which improves upon the prior bound of $(2n-3)$ faulty edges.
• The result is optimal, as $BH_n$ cannot tolerate more than $(2n-2)$ faulty edges while preserving 6-edge-bipancyclicity.
• For any fault-free edge $e$, there exists a fault-free cycle of every even length $\ell$ with $6 \leq \ell \leq 2^{2n}$ containing $e$, even with $(2n-2)$ faulty edges.
• The proof establishes that such cycles can be constructed via recursive path extensions and cycle combinations in subcubes, regardless of fault distribution across boundaries.
• The method confirms that $BH_n$ maintains full edge-bipancyclicity under the maximum possible edge fault count, demonstrating robustness in fault-tolerant network design.
• The work resolves an open problem by showing that the fault tolerance limit for edge-bipancyclicity in $BH_n$ is tight at $(2n-2)$ edges.