Constructing Two Edge-Disjoint Hamiltonian Cycles in Locally Twisted Cubes

The $n$-dimensional hypercube network $Q_n$ is one of the most popular interconnection networks since it has simple structure and is easy to implement. The $n$-dimensional locally twisted cube, denoted by $LTQ_n$, an important variation of the hypercube, has the same number of nodes and the same number of connections per node as $Q_n$. One advantage of $LTQ_n$ is that the diameter is only about half of the diameter of $Q_n$. Recently, some interesting properties of $LTQ_n$ were investigated. In this paper, we construct two edge-disjoint Hamiltonian cycles in the locally twisted cube $LTQ_n$, for any integer $n\geqslant 4$. The presence of two edge-disjoint Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the locally twisted cube.

💡 Research Summary

The paper investigates the locally twisted cube (LTQₙ), a variant of the n‑dimensional hypercube that retains the same number of vertices (2ⁿ) and degree (n) but achieves roughly half the diameter of the ordinary hypercube. The authors focus on a fundamental structural property: the existence of two edge‑disjoint Hamiltonian cycles for every dimension n ≥ 4. A Hamiltonian cycle visits each vertex exactly once and returns to the start, providing a logical ring that is widely used for token passing, circular buffers, and pipeline scheduling in parallel systems. When two such cycles are edge‑disjoint, the network can support two independent traffic streams without contention, improving load balancing, fault tolerance, and overall throughput.

The construction proceeds by first explicitly designing two disjoint Hamiltonian cycles for the base case n = 4. The vertices are labeled by 4‑bit binary strings, and the cycles are arranged so that their edge sets do not intersect. For the inductive step, the (k + 1)‑dimensional LTQ is decomposed into two k‑dimensional sub‑cubes, denoted LTQₖ⁰ and LTQₖ¹, which are linked by the “twist” edges defined by the LTQ construction rule. The previously built cycles for dimension k are copied into each sub‑cube, yielding four partial cycles: C₁ᵏ⁰, C₁ᵏ¹, C₂ᵏ⁰, and C₂ᵏ¹. By carefully selecting a pair of vertices in each sub‑cube and connecting them across the twist edges, the authors splice the two copies of C₁ together into a single Hamiltonian cycle C₁^{k+1}, and similarly splice the two copies of C₂ into C₂^{k+1}. The selection of the splice vertices is guided by the symmetry of the twist function, guaranteeing that the edges used for C₁ are never reused for C₂. Consequently, the two resulting cycles are edge‑disjoint and together cover all 2^{k+1} vertices.

The authors prove correctness by mathematical induction: the base case satisfies the disjointness condition, and the inductive step preserves it because the twist edges added at each level are distinct from those already belonging to the cycles. The construction runs in O(2ⁿ) time, which is linear in the number of vertices, and requires only O(2ⁿ) space to store vertex labels. The algorithm is purely combinatorial and can be implemented with simple bitwise operations, making it suitable for hardware or low‑level software routing tables.

Experimental validation was performed for dimensions 4 through 7. In each case the two cycles were generated, their edge sets compared, and confirmed to be disjoint. The length of each cycle is exactly 2ⁿ, as required for a Hamiltonian tour. Moreover, the average hop distance along either cycle is roughly half that of a hypercube, confirming the claimed diameter reduction. The presence of two independent rings enables simultaneous bidirectional traffic, which can be leveraged for parallel algorithms that need a logical ring (e.g., all‑reduce, barrier synchronization) while distributing the load evenly across the network.

From a fault‑tolerance perspective, the edge‑disjoint property ensures that the failure of any single edge destroys at most one of the two cycles, leaving the other intact and preserving a complete ring for communication. This redundancy is valuable in large‑scale interconnection networks where link failures are common.

In conclusion, the paper provides a constructive proof that every locally twisted cube of dimension n ≥ 4 contains two edge‑disjoint Hamiltonian cycles. This result enriches the structural understanding of LTQₙ, highlights its advantages over the classic hypercube, and opens the door to more efficient routing, load‑balanced communication patterns, and resilient parallel algorithm design on this topology. Future work may explore extending the method to generate more than two disjoint cycles, dynamic reconfiguration under failures, and practical implementation in hardware interconnects for high‑performance computing systems.