Prescribed Hamiltonian Connectedness of 2D Torus

Tori are important fundamental interconnection networks for multiprocessor systems. Hamiltonian paths are important in information communication of multiprocessor systems, and Hamiltonian path embedding capability is an important aspect to determine if a network topology is suitable for a real application. In real systems, some links may have better performance. Therefore, when embedding Hamiltonian path into interconnection networks, it is desirable that these Hamiltonian paths would pass through the links with better performance. Given a two two-dimensional torus T (m, n) with m, n ≥ 5 odd, let L be a linear forest with at most two edges in T (m, n) and let u and v be two distinct vertices in T (m, n) such that none of the paths in L has u or v as internal node or both of them as end nodes. In this paper, we construct a hamiltonian path of T (m, n) between u and v passing through L.