Embedding paths and cycles in 3-ary n-cubes with faulty nodes and links

Abstract

The k-ary n-cube, denoted by Q n k , is one of the most important interconnection networks for parallel computing. In this paper, we consider the problem of embedding cycles and paths into faulty 3-ary n-cubes. Let F be a set of faulty nodes and/or edges, and n ⩾ 2 . We show that when | F | ⩽ 2 n - 2 , there exists a cycle of any length from 3 to | V ( Q n 3 - F ) | in Q n 3 - F . We also prove that when | F | ⩽ 2 n - 3 , there exists a path of any length from 2 n - 1 to | V ( Q n 3 - F ) | - 1 between two arbitrary nodes in Q n 3 - F . Since the k-ary n-cube is regular of degree 2 n , the fault-tolerant degrees 2 n - 2 and 2 n - 3 are optimal.