# Fault-free Hamiltonian cycles in balanced hypercube with conditional edge faults

## TL;DR

This paper proves that in balanced hypercubes, a fault-free Hamiltonian cycle exists despite up to 4n-5 edge faults, provided each vertex remains incident to at least two fault-free edges, demonstrating optimal fault tolerance.

## Contribution

It establishes the maximum number of edge faults that a balanced hypercube can tolerate while still containing a fault-free Hamiltonian cycle, improving fault tolerance understanding.

## Key findings

- Fault-free Hamiltonian cycles exist with up to 4n-5 edge faults.
- Each vertex remains incident to at least two fault-free edges under these conditions.
- Result is proven to be optimal regarding maximum tolerated edge faults.

## Abstract

The balanced hypercube, $BH_n$, is a variant of hypercube $Q_n$. Zhou et al. [Inform. Sci. 300 (2015) 20-27] proposed an interesting problem that whether there is a fault-free Hamiltonian cycle in $BH_n$ with each vertex incident to at least two fault-free edges. In this paper, we consider this problem and show that each fault-free edge lies on a fault-free Hamiltonian cycle in $BH_n$ after no more than $4n-5$ faulty edges occur if each vertex is incident with at least two fault-free edges for all $n\ge 2$. Our result is optimal with respect to the maximum number of tolerated edge faults.