Geodesic
Cycles of lentgth~9 in the Pancake graph
Diskretnyj analiz i issledovanie operacij, Tome 18 (2011) no. 6, pp. 33-60.

Voir la notice de l'article provenant de la source Math-Net.Ru

A cycle $C_l$ of length $l$, where $6\le l\le n!$, can be embedded in the Pancake graph $P_n$, $n\ge3$, that is the Cayley graph on the symmetric group with the generating set of all prefix-reversals. The algebraic characterization of cycles of length six and seven via products of generating elements is known. We continue to study odd cycles. The explicit description of cycles of length nine by means of 10 canonical forms is given. It is also proved that each of vertices of $P_n$, $n\ge4,$ belongs to $\frac{8n^3-45n^2+61n-12}2$ cycles of length nine. In general, there are $O(n!\,n^3)$ different cycles of length nine in the graph. Ill. 5, tab. 1, bibliogr. 10.
Keywords: admissible subgraph, indicator of subgraph's quality, Pareto optimal subgraph. 
@article{DA_2011_18_6_a2,
     author = {E. V. Konstantinova and A. N. Medvedev},
     title = {Cycles of lentgth~9 in the {Pancake} graph},
     journal = {Diskretnyj analiz i issledovanie operacij},
     pages = {33--60},
     publisher = {mathdoc},
     volume = {18},
     number = {6},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DA_2011_18_6_a2/}
}
TY  - JOUR
AU  - E. V. Konstantinova
AU  - A. N. Medvedev
TI  - Cycles of lentgth~9 in the Pancake graph
JO  - Diskretnyj analiz i issledovanie operacij
PY  - 2011
SP  - 33
EP  - 60
VL  - 18
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DA_2011_18_6_a2/
LA  - ru
ID  - DA_2011_18_6_a2
ER  - 
%0 Journal Article
%A E. V. Konstantinova
%A A. N. Medvedev
%T Cycles of lentgth~9 in the Pancake graph
%J Diskretnyj analiz i issledovanie operacij
%D 2011
%P 33-60
%V 18
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DA_2011_18_6_a2/
%G ru
%F DA_2011_18_6_a2
E. V. Konstantinova; A. N. Medvedev. Cycles of lentgth~9 in the Pancake graph. Diskretnyj analiz i issledovanie operacij, Tome 18 (2011) no. 6, pp. 33-60. http://geodesic.mathdoc.fr/item/DA_2011_18_6_a2/

[1] Konstantinova E. V., Medvedev A. N., “Tsikly dliny sem v Pancake grafe”, Diskret. analiz i issled. operatsii, 17:5 (2010), 46–55 | MR

[2] Asai S., Kounoike Y., Shinano Y., Kaneko K., “Computing the diameter of 17-pancake graph using a PC cluster”, Euro-Par 2006 Parallel Processing, Proc. 12th Int. Euro–Par Conf. (Dresden, Germany, August 28 – September 1, 2006), Lect. Notes Comput. Sci., 4128, Springer-Verl., Berlin–Heidelberg, 2006, 1114–1124

[3] Cibulka J., “On average and highest number of flips in pancake sorting”, Theor. Comput. Sci., 412 (2011), 822–834 | DOI | MR | Zbl

[4] Dweighter H., “E 2569 in elementary problems and solutions”, Amer. Math. Monthly, 82:1 (1975), 1010 | DOI | MR

[5] Gates W. H., Papadimitriou C. H., “Bounds for sorting by prefix-reversal”, Discrete Math., 27 (1979), 47–57 | DOI | MR

[6] Hyedari M. H., Sudborough I. H., “On the diameter of the pancake network”, J. Algorithms, 25:1 (1997), 67–94 | DOI | MR

[7] Kanevsky A., Feng C., “On the embedding of cycles in pancake graphs”, Parall. Comput., 21 (1995), 923–936 | DOI | MR | Zbl

[8] Ke Qiu, “Optimal broadcasting algorithms for multiple messages on the star and pancake graphs using minimum dominating sets”, Congr. Numerantium, 181 (2006), 33–39 | MR | Zbl

[9] Sheu J. J., Tan J. J. M., Chu K. T., “Cycle embedding in pancake interconnection networks”, Proc. 23rd Workshop Comb. Math. Comput. Theory (Taiwan, 2006), Da-Yeh Univ. Press, Changhua, Taiwan, 2006, 85–92

[10] Zaks S., “A new algorithm for generation of permutations”, BIT, 24 (1984), 196–204 | DOI | MR | Zbl