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
Basic Definition
A
Basic Definition
E
A
B
C
D
E
A
A
C
D
E
B
A
A
D
E
B
C
A
A
E
B
C
D
A
B
D
C
Basic Definition
A
B
C
D
E
A
A
C
D
E
B
A
A
D
E
B
C
A
A
E
B
C
D
A
A=Airport
B=Beautiful mountains
C=Craft museum
D=Delicious food
E=Extraordinary 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. 319-331, 2011.
Basic Definition
A
Basic Definition
A
B
C
D
E
A
A
E
B
C
D
A
A
D
E
B
C
A
A
C
D
E
B
A
K5 is vertex symmetric,
E
B
D
C
1
Basic Definition
5
1
2
5
6
4
3
1
1
3
2
5
6
4
1
1
5
6
4
3
2
1
1
4
?
?
?
5
1
6
5
2
3
1
4
6
6
4
3
1
2
5
6
2
6
4
3
IHC(G)=2
Outline
Basic Definition
 Known Results -examples
 Known Results -theories
 Current work

Known Results -examples
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
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
1234
P4
4321
1234
S4
4231
3241 2431
3214 2134
3421 2341 3214 2134
2314 3124
2431 3241 2314 3124
4231
1324
2341 3421
4132 1342
2413
3412
4213 1423 4312 1432
2413
4213 1423
1432 4312
3412
1243 4123 1342 4132
3142
2143
1243 4123
2143
1324
3142
4321
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. 27-52, 2009.
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 .
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. 5474-5483, 2009.
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.
Outline
Basic Definition
 Known Results -examples
 Known Results -theories
 Current work

Known Results -theories
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.
Known Results -theories
Lemma 1
A
A B C D E A
A E B C D A
E
B
A D E B C A
A C D E B A
D
C
P4
S4
Known Results -theories
Lemma 2
Theorem 1
Outline
Basic Definition
 Known Results -examples
 Known Results -theories
 Current work

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.
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.
x
y
…
 Proof.
1
2
3
4
j
j+1
n-3 n-2
 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)<=n-3, and deg(y)=d>=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.
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.
x
y
1
2
3
4
j
j+1
n-3 n-2
Ci  x, i, i  1,..., j , y, j  1, j  2,..., n  2,1,2,..., i  1, x for 1  i  j;
Ci  x, i, i  1,..., n  2,1,2,..., j , y, j  1, j  2,..., i  1, x for j  2  i  n  2.
Totally n-3 MIHCs, n-3=2(n-2)-n+1=2deg(x)-1.
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
…
1
2
3
4
j
j+1
n-3 n-2
Let {i1 , i2 , i3 ,..., id } be the neighbors of y, where
i1  1  i2  i3  ...  id .
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.
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.
Suppose that Node (n-2) is not adjacent to
i j  1 for any 2  j  d . 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
A contradiction! Thus Node (n-2) must be adjacent to
Node i j  1 for some 2  j  d . Let it be i j *  k .
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
…
1
2
3
4 k-1 k
n-3
n-2
Ci  x, i, i  1,..., k  1, n  2, n  3,..., k , y,1,2,..., i  1, x , for 2  i  k  1;
Ci  x, i, i  1,..., k , y,1,2,..., k  1, n  2, n  3,..., i  1, x , for k  i  n  2.
Totally n-3 MIHCs, n-3=2(n-2)-n+1=2deg(x)-1.
Proof of LEM2’
Case
2. deg(x)<=n-3,
and deg(y)=d >=3.
Current work
THM 1.
(Dirac’s type)
Can we rewrite the main theorems into
the Ore-typed results?
Current work
THM 1.
(Dirac’s type)
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 IHC (G )  2 (G )  n  1.
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 IHC (G )  2 (G )  n  1.
1
5
2
G is Hamiltonian.
It violates Dirac’s Thm,
but satisfies Ore’s Thm.
2 (G )  n  1 is negative!
6
4
3
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 IHC (G )  min{deg( x)  deg( y ) |
x, y are nonadjacen t.}  n  1.
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 1vertex hamiltonian or G belongs to one of the three
families G1,G2 and G3.
~ the End~
Thank you very much!