Embedding two edge-disjoint Hamiltonian cycles into locally twisted cubes

Abstract

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 $LT{Q}_n$, 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 $LT{Q}_n$ is that the diameter is only about half of the diameter of $Q_n$. Recently, some interesting properties of $LT{Q}_n$ have been investigated in the literature. The presence of 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 interconnection network. The existence of two edge-disjoint Hamiltonian cycles in locally twisted cubes has remained unknown. In this paper, we prove that the locally twisted cube $LT{Q}_n$ with $n⩾4$ contains two edge-disjoint Hamiltonian cycles. Based on the proof of existence, we further provide an $O\left(n{2}^n\right)$-linear time algorithm to construct two edge-disjoint Hamiltonian cycles in an $n$-dimensional locally twisted cube $LT{Q}_n$ with $n⩾4$, where $LT{Q}_n$ contains $2^n$ nodes and $n{2}^{n-1}$ edges.