Edge Disjoint Hamiltonian Cycles in k-Ary n-Cubes and Hypercube

sMyung M. Bae and Bella Bose, IEEE Tran. Computers, vol. 52, no.

10, 2003, pp. 1271-1284

Abstract

• The k-ary n-cube has n edge-disjoint hamiltonian cycles.

• The hypercube has edge-disjoint hamiltonian cycles.

2

n

Outline

• Definitions

• EDHC of hypercube

• EDHC of k-ary n-cube

K-ary n-cube

• Vertex: an-1 an-2…a1a0, 0 ai k.

• Edge: an-1 an-2…a1a0 and bn-1 bn-2…b1b0 are adjacent if ai=bi+-1 for some i, and aj = bj for all j i

Cross product of graphs

• G1=(V1, E1) and G2=(V2, E2)

• G1xG2:

V={(u,v)|uV1, v V2}, and

E={ ( (u1, v1), (u2, v2) )| (u1, u2) E1 and v1=v2), or (u1=u2 and (v1, v2) E2 }.

• Cmk=CkxCkX…Ck

• Tk1,2k,…kn = Ck1xCk2x…Ckn

Edge Disjoint Hamiltonian Cycles in hypercubes

• Case 1: n=2m

• Qn = Q2m

= Q2xQ2x…xQ2

= C4xC4x…XC4

= Cm4

Edge Disjoint Hamiltonian Cycles in hypercubes(cont.)

• Case 2: n=2m+1

• Qn = Q2m+1

= Q1xQ2m

Edge Disjoint Hamiltonian Cycles in k-ary n-cube (k>2)

• 1. k-ary 2-cube

• 2. Tkr,k

• 3. k-ary 3-cube

• 4. k-ary n-cube (n>3)

k-ary 2-cube

Tkr,k

k-ary 3-cube

k-ary 3-cube (k is odd)

k-ary 3-cube (cont.)

k-ary 3-cube (cont.)

k-ary n-cube (n>3)

• Case 1: n=2m and m2

k-ary n-cube (n>3) (cont.)

• Case 2: n=2m+1, m 1

The end