1. Edge Disjoint Hamiltonian Cycles in k-Ary n-Cubes and Hypercubes Myung M. Bae and Bella Bose, IEEE Tran. Computers, vol. 52, no. 10, 2003, pp. 1271-1284

2. Abstract The k-ary n-cube has n edge-disjoint hamiltonian cycles. The hypercube has edge-disjoint hamiltonian cycles.

3. Outline Definitions EDHC of hypercube EDHC of k-ary n-cube

4. K-ary n-cube Vertex: a n-1 a n-2 …a 1 a 0, 0  a i  k. Edge: a n-1 a n-2 …a 1 a 0 and b n-1 b n-2 …b 1 b 0 are adjacent if a i =b i +-1 for some i, and a j = b j for all j  i

5. Cross product of graphs G 1 =(V 1, E 1 ) and G 2 =(V 2, E 2 ) G 1 xG 2 : V={(u,v)|u  V1, v  V2}, and E={ ( (u1, v1), (u2, v2) )| (u1, u2)  E1 and v1=v2), or (u1=u2 and (v1, v2)  E2 }.

6. C m k =C k xC k X…C k T k1,2k,…kn = C k1 xC k2 x…C kn

7. Edge Disjoint Hamiltonian Cycles in hypercubes Case 1: n=2m Qn = Q 2m = Q 2 xQ 2 x…xQ 2 = C 4 xC 4 x…XC 4 = C m 4

8. Edge Disjoint Hamiltonian Cycles in hypercubes(cont.) Case 2: n=2m+1 Q n = Q 2m+1 = Q 1 xQ 2m

9. Edge Disjoint Hamiltonian Cycles in k-ary n-cube (k>2) 1. k-ary 2-cube 2. T k r,k 3. k-ary 3-cube 4. k-ary n-cube (n>3)

10. k-ary 2-cube

11. T k r,k

12. k-ary 3-cube

13. k-ary 3-cube (k is odd)

14. k-ary 3-cube (cont.)

16. k-ary n-cube (n>3) Case 1: n=2m and m  2

17. k-ary n-cube (n>3) (cont.) Case 2: n=2m+1, m  1

18. The end