Algorithmic construction of Hamiltonians in pyramids

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Algorithmic construction of Hamiltonians in pyramids. H. Sarbazi-Azad, M. Ould-Khaoua, L.M. Mackenzie, IPL, 80, 75-79(2001). Previous work. F. Cao, D. F. Hsu, “ Fault Tolerance Properties of Pyramid Networks ”, IEEE Trans. Comput. 48 (1999) 88-93. Connectivity, fault diameter, container.

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### Algorithmic construction of Hamiltonians in pyramids

H. Sarbazi-Azad, M. Ould-Khaoua, L.M. Mackenzie, IPL, 80, 75-79(2001)

Previous work
• F. Cao, D. F. Hsu, “Fault Tolerance Properties of Pyramid Networks”, IEEE Trans. Comput. 48 (1999) 88-93.
• Connectivity, fault diameter, container
Pyramid Pn is not regular
• (P1)=3, ∆(P1)=4
• (P2)=3, ∆(P2)=7
• (Pn)=3, ∆(Pn)=9, for n>=3
result
• Theorem 1. A Pn contains Hamiltonian paths starting with any node x  P = { Pn▲, Pn◤, Pn◣, Pn◥, Pn◢ } and lasting at any node y  P – {x}.
Result(cont.)
• Theorem 2. A pyramid of level n, Pn, is Hamiltonian.
A. Itai, C. Papadimitriou, J. Szwarcfiter, “Hamilton Paths in grid graphs”, SIAM Journal on Computing, 11 (4) (1982) 676-686.
Hamiltonian property of M(m, n)
• In fact, M(m, n) is bipartite.
• M(m,n) is even-size if m*n is even.
• Roughly speaking, for a even-sized M(m, n), there exists a hamiltonian path between any two nodes x, y iff x and y belong to a same partite set.
• There are a few exceptions. (detail)
P1
• 剛剛看過了
Induction
• Case 1. x, y 都在上面 n-1層
Pn is pancyclic
• By induction
Induction
• (1) 3~L
• (2)L+2
• (3)L+3~L+4
• (4)L+5~|V(Pn)|
• (5)L+1