On the genus of pancake network

Both the pancake graph and star graph are Cayley graphs and are especially attractive for parallel processing. They both have sublogarithmic diameter, and are fairly sparse compared to hypercubes. In this paper, we focus on another important property, namely the genus. The genus of a graph is the minimum number of handles needed for drawing the graph on the plane without edges crossing. We will investigate the upper bound and lower bound for the genus of pancake graph and compare these values with the genus of the star graph as well as that of the hypercube .

[1]  Keiichi Kaneko,et al.  Computing the diameters of 14- and 15-pancake graphs , 2005, 8th International Symposium on Parallel Architectures,Algorithms and Networks (ISPAN'05).

[2]  Ivan Hal Sudborough,et al.  On the generalization of the pancake network , 2002, Proceedings International Symposium on Parallel Architectures, Algorithms and Networks. I-SPAN'02.

[3]  Laurie J. Heyer,et al.  Engineering bacteria to solve the Burnt Pancake Problem , 2008, Journal of biological engineering.

[4]  Saïd Bettayeb,et al.  On the Genus of Star Graphs , 1994, IEEE Trans. Computers.

[5]  Mehdi Behzad,et al.  Graphs and Digraphs , 1981, The Mathematical Gazette.

[6]  Ivan Hal Sudborough,et al.  On the Diameter of the Pancake Network , 1997, J. Algorithms.

[7]  Ivan Hal Sudborough,et al.  An (18/11)n upper bound for sorting by prefix reversals , 2009, Theor. Comput. Sci..

[8]  Keiichi Kaneko,et al.  Computing the Diameter of 17-Pancake Graph Using a PC Cluster , 2006, Euro-Par.

[9]  L. Beineke,et al.  The Genus of the n-Cube , 1965, Canadian Journal of Mathematics.

[10]  Sheldon B. Akers,et al.  A Group-Theoretic Model for Symmetric Interconnection Networks , 1989, IEEE Trans. Computers.

[11]  Christos H. Papadimitriou,et al.  Bounds for sorting by prefix reversal , 1979, Discret. Math..