Edge-disjoint Hamiltonian cycles in two-dimensional torus

The torus is one of the popular topologies for the interconnecting processors to build high-performance multicomputers. This paper presents methods to generate edge-disjoint Hamil-tonian cycles in 2D tori. 1. Introduction. A multicomputer system consists of multiple nodes that communicate by exchanging messages through an interconnection network. At a minimum, each node normally has one or more processing elements, a local memory, and a communication module. A popular topology for the interconnection network is the torus. Also called a wrap-around mesh or a toroidal mesh, this topology includes the k-ary n-cube which is an n-dimensional torus with the restriction that each dimension is of the same size, k, and the hypercube, which is a k-ary n-cube with k = 2; a mesh is a subgraph of a torus. Several parallel machines, both commercial and experimental, have been designed with a toroidal interconnection network. Included among these machines are the following: the iWarp (torus) [5], Cray T3D and T3E (3D torus) [13], the Mosaic (k-ary n-cube) [14], and the Tera parallel computer (torus) [2]. Some topological properties of torus and k-ary n-cubes based on Lee distance are given in [6, 7]. The existence of disjoint Hamiltonian cycles in the cross-product of various graphs has been discussed in [1, 4, 8, 9, 10, 11, 15]; however, a straightforward way of generating such cycles was not known until the results in [3], where some simple ways of generating edge-disjoint Hamiltonian cycles in k-ary n-cubes are presented. In this paper, some simple solutions to this problem are described for 2D torus. For example, Figure 1.1 gives two edge-disjoint cycles in C 3 × C 4. The rest of the paper is organized as follows. Section 2 gives some preliminaries about the definition of torus. Section 3 discusses the results on edge-disjoint Hamil-tonian cycles on the 2D torus. Section 4 is the conclusion of this paper.