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

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

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

Basic Definition

A B CD E ABCDEA ACDEBA ADEBCA AEBCDA

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

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 , 2011.

Basic Definition

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

Basic Definition ??? IHC(G)=2

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

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 , Known Results -examples

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 , Known Results -examples

P4P S4S Known Results -examples

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 , 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 , Y.K. Shih, J.J.M. Tan, and L.H. Hsu Mutually independent bipanconnected property of hypercube, Applied Mathematics and Computation, Vol. 217 pp , 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 , 2011.

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 , 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, 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 ,, Hsun Su, Shih-Yan Chen and Shin-Shin Kao* Mutually independent Hamiltonian cycles in Alternating Group Graphs, J. Supercomputing, in press, 2011.

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

Known Results -theories

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.

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

P4P4 S4S4

Lemma 2 Known Results -theories Theorem 1

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

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

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.

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

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

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

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

Proof of LEM2’ Case 2. deg(x) =3.

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

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.

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 G is Hamiltonian. It violates Dirac’s Thm, but satisfies Ore’s Thm.

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. ^_^

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.

~ the End~ Thank you very much!