1. Some Interesting Properties of Interconnection Networks
Lih-Hsing Hsu

6. R.S. Chou and L.H. Hsu (1994), "1-Edge Fault Tolerant Design for Meshes," Parallel Processing Letters, Vol. 4, pp

10. Y. C. Chuang, C. H. Chang, and L. H
Y.C. Chuang, C.H. Chang, and L.H. Hsu (2002), "Optimal 1-Edge Fault-Tolerant Designs for Ladders," Information Processing Letters, Vol. 84, pp

11. Good Graph Theorist
Good-Graph Theorist

13. A k-regular Hamiltonian and Hamiltonian connected graph G is super fault-tolerant Hamiltonian if G remains Hamiltonian after removing at most k-2 nodes and/or edges and remains Hamiltonian connected after removing at most k-3 nodes and/or edges.

14. Introduction
The fault-tolerant Hamiltonicity, Hf(G), is defined to be the maximum integer l such that G-F remains Hamiltonian for every F  V(G)  E(G) with |F|  l if G is Hamiltonian, and undefined if otherwise.
Obviously, Hf(G)  (G)-2, where (G) = min{deg(v) | v  V(G)}.
A regular graph G is optimal fault-tolerant Hamiltonian if Hf(G) = (G)-2.

15. Introduction
The fault-tolerant Hamiltonian connectivity, Hkf(G), is defined to be the maximum integer l such that G-F remains Hamiltonian connected for every F  V(G)  E(G) with |F|  l if G is Hamiltonian connected, and undefined if otherwise.
Obviously, Hf(G)  (G)-3, where (G) = min{deg(v) | v  V(G)}.
A regular graph G is optimal fault-tolerant Hamiltonian connected if Hf(G) = (G)-3.

16. Preliminaries Preliminaries
The hypercube is a popular network because of its attractive properties, including regularity, symmetry, small diameter, strong connectivity, recursive construction, partitionability, and relatively low link complexity.

17. Preliminaries Hypercube Q3 Hypercube Q4 000 010 001 011 110 100 101
111
Hypercube Q3
Hypercube Q4

18. Preliminaries
There are some variations of the hypercube appearing in literature; such as Twisted-cubes, Crossed-cubes, Möbius cubes, and so on. These variations preserve most of the good topological properties of the hypercube, and even better.
We generalize these cubes and maintain its fault tolerance.

19. Definition of Twisted-cube TQn
The Twisted n-cube TQn is defined for odd values of n. The vertex set of the twisted n-cube TQn is the set of all binary strings of length n. Let u = un-1un-2… u1u0 be any vertex in TQn.
A twisted 1-cube, TQ1, is a complete graph with two vertices 0 and 1.

20. Twisted-cube
For 0  i  n-1, let the i-th parity function be Pi(u) = ui  ui-1  …  u0, where  is the exclusive-or operation.
Suppose that n  3. We can decompose the vertices of TQn into four sets, TQn-2, TQn-2, TQn-2, and TQn-2 where TQn-2 consists of those vertices u with un-1 = i and un-2 = j.
For each (i,j)  {(0,0), (0,1), (1,0), (1,1)}, the induced subgraph of TQn-2 in TQn is isomorphic to TQn-2.

21. Twisted-cube
The edges that connect these four subtwisted cubes can be described as follows: Any vertex un-1un-2… u1u0 with Pn-3(u) = 0 is connected to ūn-1ūn-2… u1u0 and ūn-1un-2… u1u0; and to un-1ūn-2… u1u0 and ūn-1un-2… u1u0 if Pn-3(u) = 1.

22. Twisted-cube

23. Twisted-cube

24. W. T. Huang, J. M. Tan, C. N. Hung, and L. H
W.T. Huang, J.M. Tan, C.N. Hung, and L.H. Hsu (2002), "Fault-Tolerant Hamiltonicity of Twisted Cubes," Journal of Parallel and Distributed Computing, Vo.l 62, pp

25. Definition of Crossed-cube CQn
Two two-digit binary strings x = x1x0 and y = y1y0 are pair related, denoted by x ~ y, if and only if (x,y)  {(00,00), (10,10), (01,11), (11,01)}.
CQ1 is a complete graph with two vertices labeled by 0 and 1.
CQn consists of two identical (n-1)-dimension crossed cubes, CQn-10 and CQn-11.

26. Crossed-cube
The vertex u = 0un-2…u0  V(CQn-10) and vertex v = 1vn-2…v0  V(CQn-11) are adjacent in CQn if and only if
(1) un-2 = vn-2 if n is even; and
(2) for 0  i < (n-1)/2, u2i+1u2i ~ v2i+1v2i.

27. Crossed-cube

28. W. T. Huang, Y. C. Chuang, L. H. Hsu, and J. M
W.T. Huang, Y.C. Chuang, L.H. Hsu, and J.M. Tan (2002), "On the Fault-Tolerant Hamiltonicity of Crossed Cubes," IEICE Transaction on Fundamentals, Vol. E85-A, pp

29. Definition of Möbius cube MQn
The Möbius cube, MQn = (V,E), has 2n vertices.
Each vertex is labeled by a unique n-bit binary string as its address and has connections to n other distinct vertices.
The vertex with address X = xn-1xn-2…x0 connects to n other vertices Yi, 0  i  n-1, where the address of Yi satisfies
(1) Yi = (xn-1…xi+1xi…x0) if xi+1=0; or
(2) Yi = (xn-1…xi+1xi…x0) if xi+1=1.

30. Möbius cube
X connects to Yi by complementing the bit Xi if xi+1=0, or by complementing all bits of xi...x0 if xi+1=1.
For the connection between X and Yn-1, we can assume that the unspecified xn is either 0 or 1, which gives slightly different topologies.
If xn is 0, we call the network generated the ``0-möbius cube", denoted by 0-MQn; and if xn is 1, we call the network generated the ``1-möbius cube", denoted by 1-MQn.

31. Möbius cube
0-MQ4
1-MQ4

32. W. T. Huang, Y. C. Chuang, L. H. Hsu, and J. M
W.T. Huang, Y.C. Chuang, L.H. Hsu, and J.M. Tan (2000), "Fault-Free Hamiltonian Cycle in Faulty Mobius Cube," J. Computing and Sys, Vol. 4, pp

33. One construction scheme
construction schemes
One construction scheme
Graph G(G1,G2 ; M) G G
a perfect 1 2
matching M

34. Y. C. Chen, C. H. Tsai, L. H. Hsu, and Jimmy J. M
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

35. Theorem
Assume that G1 and G2 are k-regular super fault-tolerant Hamiltonian where k  5 and |V(G1)| = |V(G2)|. Then G(G1, G2 ; M) is (k+1)-regular super fault-tolerant Hamiltonian.

37. Y. C. Chen, L. H. Hsu, and Jimmy J. M
Y.C. Chen, L.H. Hsu, and Jimmy J.M. Tan (2006), "A Recursively Construction Scheme for Super Fault-Tolerant Hamiltonian Graphs," Applied Mathematics and Computation, Vol. 177, pp (k5)
T.L. Kueng, C.K. Lin, T. Liang, J.J.M Tan, and L.H. Hsu (2008), "Fault-tolerant Hamiltonian Connectedness of Cycle Composition Networks," Applied Mathematics and Computation, Vol. 196 pp (4)

38. C.K. Lin, T.Y. Ho, J.M. Tan, and L.H. Hsu ``Fault-Tolerant Hamiltonicity and Fault-Tolerant Hamiltonian Connectivity of the Folded Petersen Cube Networks", accepted by International Journal of Computer Mathematics.
PkQn

43. Preliminaries
Let G1 and G2 be two k-regular super fault-tolerant Hamiltonian graphs with the same number of nodes, and let M be an arbitrary perfect matching. Then G(G1,G2 ; M) is (k+1)-regular. We expect that its fault-tolerant Hamiltonicity Hf(G) and fault-tolerant Hamiltonian connectivity Hf(G) are also increased by 1.

44. Lemma 1
Let G be a k-Hamiltonian graph, FG be a set of faults in G with |FG|  k, and u be a healthy node in G. Then there are at least k- fG +2 edges incident to node u, such that each one of them is on some Hamiltonian cycle in G - FG.

45. Deleting edge e, G-FG-{e} still contains a Hamiltonian cycle.
We know that G is k-Hamiltonian, and
There are fG faults in G.
Repeating this process k-fG times, we find k-fG+2 edges incident to node u and each one of them is on some Hamiltonian cycle in G-FG.
Suppose fG < k. Let HC be a Hamiltonian cycle in G-FG, and let e be an edge on HC and incident to node u.
Deleting edge e, G-FG-{e} still contains a Hamiltonian cycle.
Hence, G-FG is still Hamiltonian even if we add k-fG more faults to G-FG. u
fG faults e e
Graph G

46. Lemma 2
Let G be a k-Hamiltonian connected graph, FG be a set of faults in G with |FG|  k, and {x,y,u} be three distinct health nodes in G. Then there are at least k-fG+2 edges incident to node u, such that each one of them is on some x,y-Hamiltonian path in G-FG.

47. Deleting edge e, G-FG-{e} still contains an x,y-Hamiltonian path.
It is known that G is k-Hamiltonian connected, and there are fG faults in G.
Thus, G-FG is still Hamiltonian connected even if we add fG more faults to G-FG.
Repeating this process k-fG times, we find k-FG+2 edges incident to node u and each one of them is on some x,y-Hamiltonian path in G-FG.
Suppose fG < k. Let HP be an x,y-Hamiltonian path in G-FG, and let e be an edge on HP and incident to node u.
Deleting edge e, G-FG-{e} still contains an x,y-Hamiltonian path. x u
fG faults e y
Graph G

48. Lemma 3
Let Gr and Gs be two k-regular graphs with the same number of nodes. If the total number of faults in G(Gr ,Gs;M) is no greater than k, there exists at least one healthy matching edge between Gr and Gs.
k+1 vertices

49. Lemma 4
Let Gr and Gs be two k-regular graphs with the same number of nodes, and let x and y be two healthy nodes in G(Gr, Gs;M). If the total number of faults in G(Gr,Gs;M) is no greater than k-2, there exists at least one healthy matching edge between Gr and Gs whose endpoints are neither x nor y. y
k+1 vertices x

50. Main Result
Theorem 1
Assume k  4. Let G1 and G2 be two k-regular super fault-tolerant Hamiltonian graphs and |V(G1)| = |V(G2)|. Then graph G(G1, G2 ; M) is (k-1)-Hamiltonian.

51. Proof of Theorem 1
Assume k  4. Let G1 and G2 be two k-regular super fault-tolerant Hamiltonian graphs and |V(G1)| = |V(G2)|. Then graph G(G1, G2 ; M) is (k-1)-Hamiltonian.

52. Case 1: All k-1 faults are in the same component.
Assume that all faults are in G1.
Let (u1,u2) and (v1,v2) be two matching edges between G1 and G2.
Since G1 is (k-2)-Hamiltonian and f1=k-1, G1-F1 has a Hamiltonian path <u1,HP1,v1>. u 2 v u 1 v G G 1 2
In G2, there exists a u2,v2-Hamiltonian path <u2,HP2,v2> since f2=0

53. Case 2: Not all k-1 faults are in the same component.
By Lemma 1, we can find a node v1 incident to u1 such that (u1,v1) is on a Hamiltonian cycle in G1-F1, and the matching edge (v1,v2) incident to v1 is healthy.
We may w.l.o.g. assume that f2  f1  k-2 and G2-F2 is Hamiltonian connected if k  4.
By Lemma 3, there exists a healthy matching edge between G1 and G2, say (u1,u2).
G2-F2 is Hamiltonian connected if k  4. u 1 2 v 1 v 2 G G 1 2

54. Main Result
Theorem 2
Assume k  5. Let G1 and G2 be two k-regular super fault-tolerant Hamiltonian graphs and |V(G1)| = |V(G2)|. Then graph G(G1, G2 ; M) is (k-2)-Hamiltonian connected.

55. Proof of Theorem 2. Case 1: x and y are not in the same component.
Subcase 1-1: All k-2 faults are in the same component.
Subcase 1-2: Not all k-2 faults are in the same component.
Case 2: x and y are in the same component.
Subcase 2-1: All k-2 faults are in G1.
Subcase 2-2: All k-2 faults are in G2.
Subcase 2-2-1: At least one of x2 and y2 is healthy.
Subcase 2-2-2: Both x2 and y2 are faulty.
Subcase 2-3: Neither all k-2 faults are in G1 nor all k-2 faults are in G2.

56. Subcase 1-1: All k-2 faults are in the same component.
On the Hamiltonian cycle of G1 -F1, there are two nodes incident to x.
One of these two nodes is not matched with y, say u1. G G 1 2 y u 1 x u 2
In G2, there is a u2,y-Hamiltonian path because G2 is Hamiltonian connected.

57. Subcase 1-2: Not all k-2 faults are in the same component.
We may assume that f2  f1  k-3.
By Lemma 4, we can find a healthy matching edge (u1,u2) between G1 and G2, where u1  x and u2  y. G G 1 2 y x u 1 2
There is one x,u1-Hamiltonian path in G1-F1 and one u2,y-Hamiltonian path in G2-F2.

58. Subcase 2-1: All k-2 faults are in G1.
Let g be a faulty edge or a faulty node.
In G1-(F1-{g}), there is a Hamiltonian path joining x and y. Removing the fault g, this Hamiltonian path is separated into two subpaths, which cover all the nodes of G1-F1. G G 1 2
In G2, there exists a u2,v2-Hamiltonian path since f2=0. x u 1 v u 2 y g v 2

59. Subcase 2-2-1: At least one of x2 and y2 is healthy.
In G2-F2, there exists a Hamiltonian cycle since f2=k-2.
W.l.o.g., we may assume y2 is healthy.
We add the matching edge (y,y2).
In G1–{y} ,we claim that there exists a fault free x, u1-Hamiltonian path. Suppose not, then k-3 < 1, so k < 4. It is a contradiction.
On this fault free cycle, there are two nodes incident to node y2. At least one of these two nodes is not adjacent to x, say u2. G G 1 2 y y x 2 u 2 u 1 u

60. Subcase 2-2-2: Both x2 and y2 are faulty.
In G2-F2, there exists a Hamiltonian cycle since f2=k-2.
Let v2 be a node on this cycle incident to u2.
Node v2 is not matched with x and y since x2 and y2 are faulty in this subcase.
We can find a healthy node u1 incident to y such that u1x and u2 is healthy, where u2 is the matching node of u1 in G2.
In this subcase, we claim that G1-{u1,y} has a fault free x, v1-Hamiltonian path for k5. Suppose not, k-3 < 2, and k<5. It is a contradiction. G G 1 2 v 1 v 2 u u u x 1 2 y

61. Subcase 2-3: Neither all k-2 faults are in G1 nor all k-2 faults are in G2.
Both G1-F1 and G2-F2 are Hamiltonian connected.
In G1-F1, by Lemma 2, there are at least (k-3)-f1+2=k-1-f1 edges incident to node u1, such that each one of them is on some x,y-Hamiltonian path in G1-F1.
Among these k-1-f1 edges, we claim that there is at least one, say (u1,v1), such that v1,v2, and (v1,v2) are healthy.
By Lemma 4, there is at least one healthy matching edge between G1 and G2, say (u1,u2), such that u1 {x,y}. G G 1 2
In G2-F2, there is a u2,v2-Hamiltonian path as a result of f2k-3. x u u 1 2 y v 2 v 1