A bipartite graph G = (V, E) is said to be bipancyclic if it contains a cycle of every even length from 4 to | V |. Furthermore, a bipancyclic G is said to be edge-bipancyclic if every edge of G lies on a cycle of every even length. Let Fv (respectively, Fe) be the set of faulty vertices (respectively, faulty edges) in an n-dimensional hypercube Qn. In this paper, we show that every edge of Qn - Fv - Fe lies on a cycle of every even length from 4 to 2n - 2 | Fv | even if | Fv | + | Fe | ≤ n - 2, where n ≥ 3. Since Qn is bipartite of equal-size partite sets and is regular of vertex-degree n, both the number of faults tolerated and the length of a longest fault-free cycle obtained are worst-case optimal.
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics
- Applied Mathematics