Cycle embedding in pancake interconnection networks

The ring structure is important for distributed computing, and it is useful to construct a hamiltonian cycle or rings of various length in the network. Kanevsky and Feng [3] proved that all cycles of length l where 6 ≤ l ≤ n!−2 or l = n! can be embedded in the pancake graphs Gn. Later, Senoussi and Lavault [9] presented the embedding of ring of length l, 3 ≤ l ≤ n!, with dilation 2 in the pancake graphs Gn. These results prompt us to explore the possibility of embedding a cycle of length n!−1 into Gn, and to establish some topological properties of the pancake graphs. In this paper, we prove that there exists a hamiltonian path joining any two nodes of the pancake graph Gn. And we show that the pancake graph still has a hamiltonian cycle in the presence of one faulty node. As a consequence, a cycle of length n!−1 can be embedded in Gn. And we expand Kanevsky and Feng’s result as follows: For n ≥ 4, a cycle of length l can be embedded in the pancake graph Gn where 6 ≤ l ≤ n!. ∗Correspondence to: Assistant Professor Jyh-Jian Sheu, Department of Information Management, National Dong Hwa University, Hualien, Taiwan, R.O.C. e-mail: [email protected].