The property of edge-disjoint Hamiltonian cycles in transposition networks and hypercube-like networks

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. In this paper, we first study the property of edge-disjoint Hamiltonian cycles in transposition networks which belong to a subclass of Cayley graphs. The transposition networks include other famous network topologies as their subgraphs, such as meshes, hypercubes, star graphs, and bubble-sort graphs. We first give a novel decomposition of transposition networks. Using the proposed decomposition, we show that n-dimensional transposition network with n > 5 contains four edge-disjoint Hamiltonian cycles. By using the similar approach, we present linear time algorithms to construct two edge-disjoint Hamiltonian cycles on two variants of hypercubes, including twisted cubes and crossed cubes.