Embedding Two Edge-Disjoint Hamiltonian Cycles and Two Equal Node-Disjoint Cycles into Twisted Cubes Ruo-Wei Hung ‡§ , Shang-Ju Chan ‡ , and Chien-Chih Liao ‡ Abstract—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 network. Edge-disjoint Hamiltonian cycles also provide the edge-fault tolerant Hamiltonicity of an interconnection network. Two node-disjoint cycles in a network are called equal if the number of nodes in the two cycles are the same and every node appears in one cycle exactly once. The presence of two equal node-disjoint cycles provides algorithms that require a ring structure to be preformed in the network simultaneously. The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. The n-dimensional twisted cube, an important variation of the hypercube, possesses some properties superior to the hypercube. In this paper, we present linear time algorithms to construct two edge-disjoint Hamiltonian cycles and two equal node-disjoint cycles in an n-dimensional twisted cube. Index Terms—edge-disjoint Hamiltonian cycles, equal node- disjoint cycles, twisted cubes, parallel computing, inductive construction I. I NTRODUCTION P ARALLEL computing is important for speeding up computation. The topology design of an interconnection network is the first thing to be considered. Many topologies have been proposed in the literature [4], [6], [8], [9], [10], [13], [18], and the desirable properties of an interconnection network include symmetry, relatively small degree, small diameter, embedding capabilities, scalability, robustness, and efficient routing. Among those proposed interconnection net- works, the hypercube is a popular interconnection network with many attractive properties such as regularity, symmetry, small diameter, strong connectivity, recursive construction, partition ability, and relatively low link complexity [24]. The topology of an interconnection network is usually modeled by a graph, where nodes represent the processing elements and edges represent the communication links. In this paper, we will use graphs and networks interchangeably. The n-dimensional twisted cube TQ n , an important vari- ation of the hypercube, was first proposed by Hilbers et al. [13] and possesses some properties superior to the hypercube. The twisted cube is derived from the hypercube by twisting some edges. Due to these twisted edges, the diameter, wide diameter, and fault diameter of TQ n are about half of those of the comparable hypercube [5]. An n-dimensional twisted cube is (n - 3)-Hamiltonian connected [16] and (n - 2)- pancyclic [22], whereas the hypercube is not. Moreover, its ‡ Department of Computer Science and Information Engineering, Chaoyang University of Technology, Wufeng, Taichung 41349, Taiwan § Corresponding author’s e-mail: rwhung@cyut.edu.tw performance is better than that of the hypercube even if it is asymmetric [1]. Recently, some interesting properties, such as conditional link faults, of the twisted cube TQ n were in- vestigated. Yang et al. [27] showed that, with n e +n v  n-2, a faulty TQ n still contains a cycle of length l for every 4  l  |V (TQ n )|- n v , where n e and n v are the numbers of faulty edges and faulty nodes in TQ n , respectively, and |V (TQ n )| denotes the number of nodes in TQ n . In [12], Fu showed that TQ n can tolerate up to 2n - 5 edge faults, while retaining a fault-free Hamiltonian cycle. Fan et al. [11] showed that the twisted cube TQ n , with n  3, is edge-pancyclic and provided an O(l log l + n 2 + nl)-time algorithm to find a cycle of length l containing a given edge of the twisted cube. In [11], the author also asked if TQ n is edge-pancyclic with (n - 3) faults for n  3. Yang [28] answered the question and showed that TQ n is not edge- pancyclic with only one faulty edge for any n  3, and that TQ n is node-pancyclic with (⌊ n 2 ⌋- 1) faulty edges for every n  3. Lai et al. [20] embedded a family of 2-dimensional meshes into a twisted cube. A Hamiltonian cycle in a graph is a simple cycle that passes through every node of the graph exactly once. The ring structure is important for distributed computing, and its benefits can be found in [19]. Two Hamiltonian cycles in a graph are said to be edge-disjoint if there exists no common edge in them. The edge-disjoint Hamiltonian cycles can provide an advantage for algorithms that make use of a ring structure [25]. Consider the problem of all-to-all broadcasting in which each node sends an identical message to all other nodes in the network. There is a simple solution for the problem using an n-node ring that requires n - 1 steps, i.e., at each step, every node receives a new message from its ring predecessor and passes the previous message to its ring successor. If the network admits edge-disjoint rings, then messages can be divided and the parts broadcast along different rings without any edge (link) contention. If the network can be decomposed into edge-disjoint Hamiltonian cycles, then the message traffic will be evenly distributed across all communication links. Edge-disjoint Hamiltonian cycles also form the basis of an efficient all-to-all broad- casting algorithm for networks that employ wormhole or cut-through routing [21]. Further, edge-disjoint Hamiltonian cycles also provide the edge-fault tolerant Hamiltonicity of an interconnected network; that is, when a Hamiltonian cycle of an interconnected network contains one faulty edge, then the other edge-disjoint Hamiltonian cycle can be used to replace it for transmission. The existence of a Hamiltonian cycle in twisted cubes has been verified [16]. However, there has been little work reported so far on edge-disjoint Proceedings of the International MultiConference of Engineers and Computer Scientists 2012 Vol I, IMECS 2012, March 14 - 16, 2012, Hong Kong ISBN: 978-988-19251-1-4 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) IMECS 2012