On the cycle embedding of pancake graphs

Abstract

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≦1≦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: A cycle of length l can be embedded in the pancake graph Gn, n≧4, if and only if a≦l≦n!.