Cycle and path embedding on 5-ary N-cubes

Lin, Tsong-Jie; Hsieh, Sun-Yuan; Huang, Hui-Ling

doi:10.1051/ita:2008004

Lin, Tsong-Jie ; Hsieh, Sun-Yuan ; Huang, Hui-Ling ¹

¹ Department of Information Management, Southern Taiwan University, No. 1, NanTai Street, Tainan 71005, Taiwan;

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 1, pp. 133-144.

Résumé

We study two topological properties of the 5-ary $n$ -cube $Q_{n}^{5}$ . Given two arbitrary distinct nodes $x$ and $y$ in $Q_{n}^{5}$ , we prove that there exists an $x$ - $y$ path of every length ranging from $2 n$ to $5^{n} - 1$ , where $n \geq 2$ . Based on this result, we prove that $Q_{n}^{5}$ is 5-edge-pancyclic by showing that every edge in $Q_{n}^{5}$ lies on a cycle of every length ranging from $5$ to $5^{n}$ .

MR Zbl

DOI : 10.1051/ita:2008004

Classification : 68R10, 68R05, 05C12
Mots clés : graph-theoretic interconnection networks, hypercubes, $k$-ary $n$-cubes, panconnectivity, edge-pancyclicity

Affiliations des auteurs :

Lin, Tsong-Jie ; Hsieh, Sun-Yuan ; Huang, Hui-Ling ¹

¹ Department of Information Management, Southern Taiwan University, No. 1, NanTai Street, Tainan 71005, Taiwan;

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     author = {Lin, Tsong-Jie and Hsieh, Sun-Yuan and Huang, Hui-Ling},
     title = {Cycle and path embedding on 5-ary {N-cubes}},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {133--144},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {1},
     year = {2009},
     doi = {10.1051/ita:2008004},
     mrnumber = {2483447},
     zbl = {1156.68041},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ita:2008004/}
}

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Lin, Tsong-Jie; Hsieh, Sun-Yuan; Huang, Hui-Ling. Cycle and path embedding on 5-ary N-cubes. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 1, pp. 133-144. doi : 10.1051/ita:2008004. http://archive.numdam.org/articles/10.1051/ita:2008004/

Bibliographie
Cité par

[1] S.G. Akl, Parallel Computation: Models and Methods Prentice Hall, NJ (1997).

[2] S. Borkar, R. Cohn, G. Cox, S. Gleason, T. Gross, H.T. Kung, M. Lam, B. Moore, C. Peterson, J. Pieper, L. Rankin, P.S. Tseng, J. Sutton, J. Urbanski and J. Webb, iWarp: an integrated solution to high-speed parallel computing, Proceedings of the 1988 ACM/IEEE conference on Supercomputing (1988) 330-339.

[3] B. Bose, B. Broeg, Y.G. Kwon and Y. Ashir, Lee distance and topological properties of k-ary n-cubes. IEEE Trans. Comput. 44 (1995) 1021-1030. | MR | Zbl

[4] J. Chang, J. Yang and Y. Chang, Panconnectivity, fault-Tolorant Hamiltonicity and Hamiltonian-connectivity in alternating group graphs. Networks 44 (2004) 302-310. | MR | Zbl

[5] K. Day and A.E. Al-Ayyoub, Fault diameter of k-ary n-cube Networks. IEEE Transactions on Parallel and Distributed Systems, 8 (1997) 903-907.

[6] S.A. Ghozati and H.C. Wasserman, The $k$ -ary $n$ -cube network: modeling, topological properties and routing strategies. Comput. Electr. Eng. (2003) 1271-1284.

[7] S.Y. Hsieh, T.J. Lin and H.-L. Huang, Panconnectivity and edge-pancyclicity of 3-ary N-cubes. J. Supercomputing 42 (2007) 225-233.

[8] F.T. Leighton, Introduction to Parallel Algorithms and Architecture: Arrays $\cdot$ Trees $\cdot$ Hypercubes. Morgan Kaufmann, San Mateo, CA (1992). | MR | Zbl

[9] M. Ma and J. M. Xu, Panconnectivity of locally twisted cubes. Appl. Math. Lett. 19 (2006) 673-677. | MR | Zbl

[10] B. Monien and H. Sudborough, Embedding one interconnection network in another. Computing Suppl. 7 (1990) 257-282. | MR | Zbl

[11] W. Oed, Massively parallel processor system CRAY T3D. Technical Report, Cray Research GmbH (1993).

[12] A.L. Rosenberg, Cycles in Networks. Technical Report: UM-CS-1991-020, University of Massachusetts, Amherst, MA, USA (1991).

[13] C.L. Seitz et al., Submicron systems architecture project semi-annual technical report. Technical Report Caltec-CS-TR-88-18, California Institute of Technology (1988).

[14] Y. Sheng, F. Tian and B. Wei, Panconnectivity of locally connected claw-free graphs. Discrete Mathematics 203 (1999) 253-260. | MR | Zbl

[15] Z.M. Song and Y.S. Qin, A new sufficient condition for panconnected graphs. Ars Combinatoria 34 (1992) 161-166. | MR | Zbl

[16] D. Wang, T. An, M. Pan, K. Wang and S. Qu, Hamiltonian-like properies of $k$ -Ary $n$ -Cubes, in Proceedings PDCAT05 of Sixth International Conference on Parallel and Distributed Computing, Applications and Technologies (PDCAT05), IEEE Computer Society Press (2005) pp. 1002-1007.

Cité par Sources :