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 L T 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 L T Q n is that the diameter is only about half of the diameter of Q n . Recently, some interesting properties of L T 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 L T Q n with n ⩾ 4 contains two edge-disjoint Hamiltonian cycles. Based on the proof of existence, we further provide an O ( n 2 n ) -linear time algorithm to construct two edge-disjoint Hamiltonian cycles in an n -dimensional locally twisted cube L T Q n with n ⩾ 4 , where L T Q n contains 2 n nodes and n 2 n − 1 edges.