Edge-Disjoint Hamiltonian Cycles in Balanced Hypercubes with Applications to Fault-Tolerant Data Broadcasting

The existence of multiple edge-disjoint Hamiltonian cycles (EDHCs for short) is a desirable property of interconnection networks. These parallel cycles can provide an advantage for algorithms that require a ring structure. Additionally, EDHCs can enhance all-to-all data broadcasting and edge fault tolerance in network communications. In this paper, we investigate the construction of EDHCs in the balanced hypercube, which is a variant of the hypercube with many attractive properties, such as strong connectivity, regularity, and symmetry. In particular, each processor in the balanced hypercube has a backup processor that shares the common neighbors, enabling fault tolerance and efficient system reconfiguration. In 2019, Lü et al. provided an algorithm to construct two EDHCs in an $n$-dimensional balanced hypercube $B{H}_n$ for $n\ge 2$. We further study this topic and give some construction schemes to construct $2^{\lfloor {log}_{2}n\rfloor }$ EDHCs in $B{H}_n$ for $n\ge 2$. Since $B{H}_n$ is $2n$-regular, our result is optimal for $n=2^r$ ($r\ge 1$). In addition, we simulate the fault-tolerant data broadcasting through these parallel cycles as transmission channels.