1. Some New Directions about Interconnection Networks

2. Purpose of this talk A story of research How to do research as a student How to do research as a professor

3. Interconnection Networks

4. Hypercubes 0 1 00 01 10 11 000 001 010 011 100 101 110 111 Q1Q1 Q2Q2 Q3Q3

5. Let u = u n−1 u n−2...u 1 u 0 and v = v n−1 v n−2...v 1 v 0 be two n-bit binary strings. The Hamming distance h(u, v) between two vertices u and v is the number of different bits in the corresponding strings of both vertices. The n-dimensional hypercube consists of all n- bit binary strings as its vertices and two vertices u and v are adjacent if and only if h(u, v) = 1.

6. Hypercubic Like graphs Twisted cubes Cross cubes Mobius cubes Locally twisted cubes n regular graph with 2 n vertices

7. Bypartite Hypercubic Like Graphs

8. Other Cubic Graphs Folded hypercubes (bipartite or nonbipartite) Enhance hypercubes (bipartite or nonbipartite) Augmented cubes

9. Other Families Star graphs (bipartite) Pancake graphs (n,k)-star graphs Arrangement graphs Butterfly (bipartite or nonbipartite) Recursive circulant graphs Cubic family (honeycomb torus, Christmas tree, honeycomb disk, spider web, brother tree) etc

10. S. Latifi, S. Zheng, N. Bagherzadeh, Optimal ring embedding in hypercubes with faulty links, in: Fault-Tolerant Computing Symp., 1992, pp. 178–184. Y.C. Tseng, S.H. Chang, and J.P. Sheu, Fault- tolerant ring embedding in a star graph with both link and node failures, IEEE Trans Parallel Distrib Syst 8 (1997), 1185–1195.

11. Another story F. Harary, J.P. Hayes, Edge fault tolerance in graphs, Networks 23 (1993) 135–142. F. Harary, J.P. Hayes, Node fault tolerance in graphs, Networks 27 (1996) 19–23. Diameter about n/4

12. K.Mukhopadhyaya, B.P. Sinha, Hamiltonian graphs with minimum number of edges for fault- tolerant topologies, Inform. Process. Lett. 44 (1992) 95–99. Diameter about n/6

15. Diameter about O(n 1/2 )Diameter about O(log n)

16. Fault Hamiltonian and Fault Hamiltonian Connected

17. Home

18. Fault Hamiltonian and Fault Hamiltonian Connected n-2 fault tolerant hamiltonian and n-3 fault tolerant hamiltonian connected d-2 fault tolerant hamiltonian and d-3 fault tolerant hamiltonian connected

19. General Rules Y. C. Chen, C. H. Tsai, L. H. Hsu, and Jimmy J. M. Tan (2004), "On some super fault-tolerant Hamiltonian graphs," Applied Mathematics and Computation, Vol. 148, pp. 729-741.

20. Other families of Interconnection Networks C.H. Tsai, J.M. Tan, Y.C. Chen, and L.H. Hsu, (2002) "Hamiltonian Properties of Faulty Recursive Circulant Graphs," Journal of Interconnection Networks, Vol 3, Nos, 3&4, pp. 273-289. C.N. Hung, H. C. Hsu, K. Y. Liang, and L. H. Hsu, (2003) "Ring Embedding in Faulty Pancake Graphs," Information Processing Letters, Vol 86, pp. 271-275.

21. H.C. Hsu, Y.L. Hsieh, J.M. Tan, and L.H. Hsu, (2003) " Fault Hamiltonicity and Fault Hamiltonian Connectivity of the (n,k)-star Graphs," Networks, Vol 42(4), pp. 189-201. H.C. Hsu, T.K. Li, J.M. Tan, and L.H. Hsu (2004). "Fault Hamiltonicity and Fault Hamiltonian Connectivity of the Arrangement Graphs," IEEE Trans. on Computers, Vol. 53 (1), pp. 39-53.

22. C.H. Tsai, J.M. Tan, T. Liang, and L.H. Hsu (2002), ``Fault-Tolerant Hamiltonian Laceability of Hypercubes", Information Processing Letters, Vol. 83, pp. 301-306. H.C. Hsu, L.C.Chiang, Jimmy J.M. Tan, L.H. Hsu (2005), `` Fault hamiltonicity of augmented cubes", Parallel Computing, Vol. 31, pp.131-145. Y.H. Teng, Jimmy J.M. Tan, L.H. Hsu (2005), ``Honeycomb rectangular disks", Parallel Computing, Vol. 31, pp.371-388.

23. How about bipartite graphs Hamiltonian laceable

24. Edge fault tolerance hamiltonian laceable

25. Edge fault tolerance strong hamiltonian laceable

26. Hyper hamiltonian laceable

28. pancyclic

30. Panconnected A lot of people work on pancyclic and panconnected recently.

31. Globally 3*-connected Graphs M. Albert, E.R.L. Aldred, D. Holton, and J. Sheehan, On globally 3*-connected graphs, Australasian Journal of Combinatorics, 24, 2001, 193-207.

35. Global 3*-connected graph

39. Mutually independent hamiltonian cycles and hamiltonian paths

40. 出入 口

41. 出入口出入口

42. Mutually independent hamiltonian cycles and hamiltonian paths

43. Star and cycle (independent hamiltonian cycles) (n,k)-star graph (independent hamiltonian paths) Folded hypercubes (KBJ) Other families of graphs and math works Independent paths and cycles

44. Panpositionable Hamiltonian 出入口 1 0 1 1 1 2 2 2 3 2 2 2

46. Panpositionable Hamiltonian

47. Diagnosability

48. Thanks