1. 1 Fault-tolerant cycle embedding in the hypercube Jung-Sheng Fu Department of Electronics Engineering, National Lien-Ho Institute of Technology Parallel Computing 29 (2003) 821-832

2. 2 Previous results The n-dimensional folded hypercube is (n-1)-link Hamiltonian. The n-dimensional folded hypercube is (n-1)-link Hamiltonian. The n-dimensional star graph is (n-3)-link Hamiltonian. The n-dimensional star graph is (n-3)-link Hamiltonian. The arrangement graph is (k(n-k)-2)-link Hamiltonian. The arrangement graph is (k(n-k)-2)-link Hamiltonian. The WK-recursive network of degree d is (d-3)- link Hamiltonian. The WK-recursive network of degree d is (d-3)- link Hamiltonian. A modification of a d-ary undirected de Bruijn graph is (d-1)-link Hamiltonian. A modification of a d-ary undirected de Bruijn graph is (d-1)-link Hamiltonian.

3. 3 Previous results The n-dimensional hypercube is (n-2)-link Hamiltonian. The n-dimensional hypercube is (n-2)-link Hamiltonian. A fault-free cycle of length of at least A fault-free cycle of length of at least can be embedded in an n-cube with f faulty nodes, where 1<=f<=n-2. can be embedded in an n-cube with f faulty nodes, where 1<=f<=n-2. An n-cube with fe<=n-4 faulty links and fv<=n-1 faulty nodes such that fe+fv<=n-1, a cycle of length of at least [(2^n)-2*fv] can be obtained. An n-cube with fe<=n-4 faulty links and fv<=n-1 faulty nodes such that fe+fv<=n-1, a cycle of length of at least [(2^n)-2*fv] can be obtained.

4. 4 Abstract A fault-free cycle of length of at least (2^n)-2*f can be embedded in an n-cube with f faulty nodes, where n>=3 and 1 =3 and 1<=f<=2n-4.

5. 5 Lemma 1 Let X and Y be two distinct nodes in an n- cube and dH(X,Y)=d, where n>=1. There are X-Y paths in the n-cube whose length are d, d+2, d+4, …,c, where c=(2^n)-1 if d is odd, and c=(2^n)-2 if d is even. Let X and Y be two distinct nodes in an n- cube and dH(X,Y)=d, where n>=1. There are X-Y paths in the n-cube whose length are d, d+2, d+4, …,c, where c=(2^n)-1 if d is odd, and c=(2^n)-2 if d is even. J.S. Fu, G.H. Chen, Hamiltonicity of the hierarchical cubic network, Theory of Computing Systems 35(1)(2002)59- 79. J.S. Fu, G.H. Chen, Hamiltonicity of the hierarchical cubic network, Theory of Computing Systems 35(1)(2002)59- 79.

6. 6

7. 7

8. 8 If f=0, then this lemma can be directly obtained from Lemma 1. If f=0, then this lemma can be directly obtained from Lemma 1. If f=1 If f=1

9. 9

10. 10

11. 11 Case 1 f0=k-1, f1=1 Case 1 f0=k-1, f1=1 Case 1.1 Case 1.1

12. 12 Case 1.2 All nodes in *0 (except X) adjacent to Y are faulty.

13. 13 Case 1.3 Besides Y, at least one other healthy node in *0 is adjacent to X. Case 1.3 Besides Y, at least one other healthy node in *0 is adjacent to X. Case 2. 2<=f0<=k-2 Case 2. 2<=f0<=k-2 Case 3. f0=1, f1=k-1 Case 3. f0=1, f1=k-1

14. 14 Case 1. f0=2k-3, f1=1 Case 1. f0=2k-3, f1=1

15. 15 Case 2. k<=f0<=2k-4, 1<=f1<=k-2 Case 2. k<=f0<=2k-4, 1<=f1<=k-2

16. 16 Case 3. f0=k-1, 1<=f1<=k-1 Case 3. f0=k-1, 1<=f1<=k-1 When f1<=k-2, the discussion is the same as in Case 2. When f1<=k-2, the discussion is the same as in Case 2. Assume that f1=k-1. By assumption, there exists a fault-free cycle C of length of at least [(2^k)-2*f0] in *0. In addition, by Lemma 6, there exists a fault-free U(k+1) ---- V(k+1) path of length of at least [(2^k)-2*f1-1] in *1.

17. 17 Discussion and conclusion It is not easy to prove that a fault-free cycle of length of at least (2^n)-2*f cannot be embedded in an n-cube with f faulty nodes, where n>=5 and f>=2n-3. It is not easy to prove that a fault-free cycle of length of at least (2^n)-2*f cannot be embedded in an n-cube with f faulty nodes, where n>=5 and f>=2n-3. This paper reveals that a bipartite graph may tolerate faulty nodes more than its degree without affecting the embedding results. This paper reveals that a bipartite graph may tolerate faulty nodes more than its degree without affecting the embedding results.

18. 18

19. 19 Discussion and conclusion To embed fault-free cycles in the star graph, the butterfly graph, and the other bipartite graph with more faulty nodes tolerable is one of the further projects. To embed fault-free cycles in the star graph, the butterfly graph, and the other bipartite graph with more faulty nodes tolerable is one of the further projects.