Paths and Cycles in d-Dimensional Tori with Faults

This paper is concerned with the paths and the cycles in d-dimensional tori with faulty vertices and/or edge. Let fv be the number of faulty vertices and fe be the number of faulty edges. It is shown that in any non-bipartite d-dimensional k1 × k2 × · · · kd torus with ki ≥ 3 for each 1 ≤ i ≤ d, (a) if fv + fe ≤ 2d − 3, there is a fault-free spanning path between any pair of non-faulty vertices, and (b) if fv + fe ≤ 2d− 2, there is a fault-free spanning cycle. It is also shown that in any bipartite d-dimensional k1 × k2 × · · · kd torus with ki ≥ 4 for each 1 ≤ i ≤ d, (a) if fv + fe ≤ 2d − 2, there is a fault-free path of length at least N − 2fv − 1 between any pair of non-faulty vertices which belong to the different partite sets, and there is a fault-free path of length at least N − 2fv − 2 between any pair of non-faulty vertices which belong to the same partite set, and (b) if fv + fe ≤ 2d− 1 and fe ≤ 2d− 2, there is a fault-free cycle of length at least N − 2fv, and if fv = 0 and fe = 2d − 1 and all the fault edges are not incident to a common vertex, there is a fault-free spanning cycle, and if fv = 0 and fe = 2d− 1 and all the fault edges are incident to a common vertex, there is a fault-free cycle of length N − 2 where N is the number of vertices.