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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
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Abstract
• The k-ary n-cube has n edge-disjoint hamiltonian cycles.
• The hypercube has edge-disjoint hamiltonian cycles.
2
n
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Outline
• Definitions
• EDHC of hypercube
• EDHC of k-ary n-cube
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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
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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 }.
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• Cmk=CkxCkX…Ck
• Tk1,2k,…kn = Ck1xCk2x…Ckn
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Edge Disjoint Hamiltonian Cycles in hypercubes
• Case 1: n=2m
• Qn = Q2m
= Q2xQ2x…xQ2
= C4xC4x…XC4
= Cm4
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Edge Disjoint Hamiltonian Cycles in hypercubes(cont.)
• Case 2: n=2m+1
• Qn = Q2m+1
= Q1xQ2m
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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)
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k-ary 2-cube
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Tkr,k
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k-ary 3-cube
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k-ary 3-cube (k is odd)
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k-ary 3-cube (cont.)
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k-ary 3-cube (cont.)
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k-ary n-cube (n>3)
• Case 1: n=2m and m2
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k-ary n-cube (n>3) (cont.)
• Case 2: n=2m+1, m 1
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The end