1. Mutually Independent Hamiltonian Cycles on various interconnection networks- examples and theorems 海峽兩岸圖論與組合數學研討會 06/30/2011 高欣欣 中原大學應用數學系

2. Outline Basic Definition Known Results -examples Known Results -theories Current work

3. Outline Basic Definition Known Results -examples Known Results -theories Current work

4. Basic Definition

6. A B CD E ABCDEA ACDEBA ADEBCA AEBCDA

7. ABCDEA ACDEBA ADEBCA AEBCDA A=A irport B=B eautiful mountains C=C raft museum D=D elicious food E=E xtraordinary local scenery

8. Basic Definition Hsun Su, Jing-Ling Pan and Shin-Shin Kao* Mutually independent Hamiltonian cycles in k-ary n-cubes when k is even, Computers and Electrical Engineering, Vol. 37, Issue 3, pp. 319-331, 2011.

9. Basic Definition

10. A B CD E ABCDEA AEBCDA ADEBCA ACDEBA K 5 is vertex symmetric,

11. Basic Definition 1 1256431 1325641 1564321 14???51 2 34 5 6 6523146 6431256 IHC(G)=2

12. Outline Basic Definition Known Results -examples Known Results -theories Current work

13. C.-M. Sun, C.-K. Lin, H.-M. Huang, and L.-H. Hsu, “Mutually Independent Hamiltonian Cycles in Hypercubes,” Journal of Interconnection Networks 7, pp. 235-255, 2006. Known Results -examples

14. C.-K. Lin, H.-M. Huang, J. J. M. Tan and L.-H. Hsu, “Mutually Independent Hamiltonian Cycles of Pancake Networks and the Star Networks,” Discrete Mathematics, Vol. 309, pp. 5474-5483, 2009. Known Results -examples

15. P4P4 1234 4321 3214 2314 2134 3124 1324 3421 2431 4231 2341 3241 3142 4132 1432 3412 1342 4312 2413 1423 4123 4213 1243 2143 S4S4 1234 4231 3214 2314 2134 3124 1324 3241 2341 4321 2431 3421 3412 4312 1342 3142 1432 4132 2413 1423 4123 4213 1243 2143 Known Results -examples

16. Selina Y.P. Chang, Justie S.T. Juan, C.K. Lin, Jimmy J.M. Tan, and L.H. Hsu Mutually Independent Hamiltonian Connectivity of (n,k)-Star Graphs, Annals of Combinatorics, Vol. 13 pp. 27-52, 2009. Known Results -examples Y.K. Shih, C.K. Lin, D. Frank Hsu, J.J.M. Tan and L.H. Hsu The Construction of Mutually Independent Hamiltonian Cycles in Bubble-Sort Graphs, Int’l Journal of Computer Mathematics, Vol. 87, pp.2212-2225, 2010. Y.K. Shih, J.J.M. Tan, and L.H. Hsu Mutually independent bipanconnected property of hypercube, Applied Mathematics and Computation, Vol. 217 pp. 4017-4023, 2010. T.L. Kung, C.K. Lin, T. Liang, J.J.M. Tan, and L.H. Hsu Fault-free mutually independent Hamiltonian cycles of faulty star graphs, Int’l Journal of Computer Mathematics, Vol. 88 pp. 731-746, 2011.

17. C.-K. Lin, H.-M. Huang, J. J. M. Tan and L.-H. Hsu Mutually Independent Hamiltonian Cycles of Pancake Networks and the Star Networks, Discrete Mathematics, Vol. 309, pp. 5474-5483, 2009. Known Results -examples Yuan-Kang Shih, Hui-Chun Chuang, Shin-Shin Kao* and Jimmy J.M. Tan Mutually independent Hamiltonian cycles in dual-cubes, J. Supercomputing, Vol.54, p.239 － 251, 2010. Hsun Su, Jing-Ling Pan and Shin-Shin Kao* Mutually independent Hamiltonian cycles in k-ary n-cubes when k is even, Computers and Electrical Engineering, Vol. 37, Issue 3, pp. 319-331,, 2011. Hsun Su, Shih-Yan Chen and Shin-Shin Kao* Mutually independent Hamiltonian cycles in Alternating Group Graphs, J. Supercomputing, in press, 2011.

18. Outline Basic Definition Known Results -examples Known Results -theories Current work

19. Known Results -theories

20. Yuan-Kang Shih, Cheng-Kuan Lin, Jimmy J. M. Tan and Lih-Hsing Hsu Mutually Independent Hamiltonian Cycles in Some graphs Ars Combinatonia, accepted, 2008.

21. Lemma 1 A B CD E ABCDEA AEBCDA ADEBCA ACDEBA Known Results -theories

22. P4P4 S4S4

23. Lemma 2 Known Results -theories Theorem 1

24. Outline Basic Definition Known Results -examples Known Results -theories Current work

25. Can we rewrite the theorems above into the Ore-typed results? LEM2. LEM2’. Let x, y be two nonadjacent vertices of G such that deg(x)>=deg(y), deg(x) + deg(y)>=n and G-{x, y} is hamiltonian. Then there exists at least 2deg(x)-n+1 MIHC’s in G beginning with x. Current work

26. Proof. Case 1. deg(x)=n-2, and deg(y)=d>=2. Case 1.1. y is adjacent to j and j+1 for some j. Case 1.2. y is NOT adjacent to j and j+1 for any j. Case 2. deg(x) =3. Case 2.1. y is adjacent to j and j+1 for some j. Case 2.2. y is NOT adjacent to j and j+1 for any j. x y jj+1n-3321n-24 … LEM2’. Let x, y be two nonadjacent vertices of G such that deg(x)>=deg(y), deg(x) + deg(y)>=n and G-{x, y} is hamiltonian. Then there exists at least 2deg(x)-n+1 MIHC’s in G beginning with x.

27. Proof of LEM2’ Case 1. deg(x)=n-2, and deg(y)=d ≥ 2. Case 1.1. y is adjacent to j and j+1 for some 1 ≤ j ≤ n-2. Totally n-3 MIHCs, n-3=2(n-2)-n+1=2deg(x)-1. x y jj+1n-3321n-24

28. Proof of LEM2’ Case 1. deg(x)=n-2, and deg(y)=d >=2. Case 1.2. y is not adjacent to j and j+1 for any 1<=j<=n-2. WLOG, suppose that y is adjacent to node 1 and j with 3<=j<=n-3. x y jj+1n-3321n-24 … Let be the neighbors of y, where

29. Proof of LEM2’ Case 1. deg(x) =2. Case 1.2. y is not adjacent to j and j+1 for any 1<=j<=n-2. WLOG, suppose that y is adjacent to node 1 and j with 3<=j<=n-3. Suppose that Node (n-2) is not adjacent to Then deg(n-2)<= (n-2)-(d-1)=n-d-1 總點數減去自己和 y, 再減去前述 (d-1) 個點 Note that y is not adjacent to Node (n-2). Since deg(y)+deg(n-2)>=n, Node (n-2) is adjacent to at least (n-d) nodes. Let it be A contradiction! Thus Node (n-2) must be adjacent to Node

30. Case 1. deg(x)=n-2, and deg(y)=d >=2. Case 1.2. y is not adjacent to j and j+1 for any 1≤ j≤ n-2. WLOG, suppose that y is adjacent to node 1 and j with 3≤ j ≤ n-3. Proof of LEM2’ Totally n-3 MIHCs, n-3=2(n-2)-n+1=2deg(x)-1. x y n-3321 n-2 4 … k-1k

31. Proof of LEM2’ Case 2. deg(x) =3.

32. Can we rewrite the main theorems into the Ore-typed results? Current work THM 1. (Dirac’s type)

33. THM 1’.(Ore-typed, 1st version) Current work THM 1. (Dirac’s type) Assume that G is a graph with n=|V(G)|>=3, and deg(x)+deg(y)>=n for any nonadjacent pair {x,y}. Then Achieved by LEM2’, with the construction of MIHCs beginning with y.

34. Current work THM 1’.(Ore-typed, 1st version) Assume that G is a graph with n=|V(G)|>=3, and deg(x)+deg(y)>=n for any nonadjacent pair {x,y}. Then 1 2 34 5 6 G is Hamiltonian. It violates Dirac’s Thm, but satisfies Ore’s Thm.

35. Current work THM 1’.(Ore-typed, 2nd version) Assume that G is a graph with n=|V(G)|>=3, and deg(x)+deg(y)>=n for any nonadjacent pair {x,y}. Then PS. We are working on it. ^_^

36. Current work- an extra result THM. Let G=(V,E) be a graph with |G|=|V|=n >=3. Suppose that deg(u)+deg(v) >= n holds for any nonadjacent pair { u, v } of V, then either G is 1- vertex hamiltonian or G belongs to one of the three families G1,G2 and G3.

37. ~ the End~ Thank you very much!