Path embedding in faulty hypercubes

Abstract

There are some interesting results concerning longest paths or even cycles embedding in faulty hypercubes. This paper considers the embeddings of paths of all possible lengths between any two fault-free vertices in faulty hypercubes. Let f v (respectively, f e ) denote the number of faulty vertices (respectively, edges) in an n-dimensional hypercube Q n . We prove that there exists a fault-free path of length l between any two distinct fault-free vertices u and v in Q n with f v + f e ⩽ n - 2 for each l satisfying d Q n ( u , v ) + 2 ⩽ l ⩽ 2 n - 2 f v - 1 and 2 | ( l - d Q n ( u , v ) ) . The bounds on path length l and faulty set size f v + f e for a successful embedding are tight. That is, the result does not hold if l < d Q n ( u , v ) + 2 or l > 2 n - 2 f v - 1 or f v + f e > n - 2 . Moreover, our result improves some known results.