Longest Fault-Free Paths in Hypercubes with both Faulty Nodes and Edges

The hypercube is one of the most versatile and efficient interconnection networks for parallel computation. Let F
_v
(respectively, F
_e
) be the set of faulty vertices (respectively, faulty edges) in an n-dimensional hypercube Q
_n
. In this paper, we show that Q
_n
- F
_v
- F
_e
contains a fault free path with length at least 2
ⁿ
- 2|F
_v
| - 1(or 2
ⁿ
- 2|F
_v
| - 2) between two arbitrary vertices of odd (or even) distance if |F
_v
| + |F
_e
| les n - 2, where n ges 3. Since Q
_n
is bipartite of equal-size partite sets, the path is longest in the worst case. Furthermore, since Q
_n
is regular of vertex-degree n, both the number of faults tolerated and the length of a longest fault-free path obtained are worst-case optimal.