Panconnectivity and edge-pancyclicity of 3-ary N-cubes

Abstract We study two topological properties of the 3-ary n-cube Qn3. Given two arbitrary distinct nodes x and y in Qn3, we prove that there exists an x–y path of every length ranging from d(x,y) to 3n−1, where d(x,y) is the length of a shortest path between x and y. Based on this result, we prove that Qn3 is edge-pancyclic by showing that every edge in Qn3 lies on a cycle of every length ranging from 3 to 3n.

[1]  Jun-Ming Xu,et al.  Edge-fault-tolerant edge-bipancyclicity of hypercubes , 2005, Inf. Process. Lett..

[2]  Jun-Ming Xu,et al.  Panconnectivity and edge-fault-tolerant pancyclicity of augmented cubes , 2007, Parallel Comput..

[3]  Jun-Ming Xu,et al.  Cycles in folded hypercubes , 2006, Appl. Math. Lett..

[4]  Jou-Ming Chang,et al.  Panconnectivity, fault-tolerant hamiltonicity and hamiltonian-connectivity in alternating group graphs , 2004 .

[5]  Deqiang Wang,et al.  Hamiltonian-like Properties of k-Ary n-Cubes , 2005, Sixth International Conference on Parallel and Distributed Computing Applications and Technologies (PDCAT'05).

[6]  Myung M. Bae,et al.  Edge Disjoint Hamiltonian Cycles in k-Ary n-Cubes and Hypercubes , 2003, IEEE Trans. Computers.

[7]  Yaagoub Ashir,et al.  Fault-Tolerant Embeddings of Hamiltonian Circuits in k-ary n-Cubes , 2002, SIAM J. Discret. Math..

[8]  Sun-Yuan Hsieh,et al.  Hamiltonian path embedding and pancyclicity on the Mobius cube with faulty nodes and faulty edges , 2006, IEEE Transactions on Computers.

[9]  Yaagoub Ashir,et al.  Lee Distance and Topological Properties of k-ary n-cubes , 1995, IEEE Trans. Computers.

[10]  Jun-Ming Xu,et al.  Paths in Möbius cubes and crossed cubes , 2006, Inf. Process. Lett..

[11]  Jun-Ming Xu,et al.  Edge-fault-tolerant properties of hypercubes and folded hypercubes , 2006, Australas. J Comb..

[12]  Sajal K. Das,et al.  Book Review: Introduction to Parallel Algorithms and Architectures : Arrays, Trees, Hypercubes by F. T. Leighton (Morgan Kauffman Pub, 1992) , 1992, SIGA.

[13]  Jou-Ming Chang,et al.  Panconnectivity, fault-tolerant hamiltonicity and hamiltonian-connectivity in alternating group graphs , 2004, Networks.

[14]  S. Ghozati,et al.  The k-ary n-cube network : modeling, topological properties and routing strategies , 1999 .

[15]  Charles L. Seitz.  Submicron Systems Architecture Project: Semiannual Technical Report , 1989 .

[16]  Jun-Ming Xu,et al.  Panconnectivity of locally twisted cubes , 2006, Appl. Math. Lett..

[17]  Yukio Shibata,et al.  Pancyclicity of recursive circulant graphs , 2002, Inf. Process. Lett..

[18]  Abdel Elah Al-Ayyoub,et al.  Fault Diameter of k-ary n-cube Networks , 1997, IEEE Trans. Parallel Distributed Syst..

[19]  Yaagoub Ashir,et al.  Embeddings of cycles, meshes and tori in faulty k-ary n-cubes , 1997, Proceedings 1997 International Conference on Parallel and Distributed Systems.

[20]  Jung-Heum Park.  Panconnectivity and edge-pancyclicity of faulty recursive circulant G(2m, 4) , 2008, Theor. Comput. Sci..

[21]  Xiaohua Jia,et al.  Node-pancyclicity and edge-pancyclicity of crossed cubes , 2005, Inf. Process. Lett..

[22]  Sun-Yuan Hsieh,et al.  Embedding cycles and paths in a k-ary n-cube , 2007, 2007 International Conference on Parallel and Distributed Systems.

[23]  S.G. Akl,et al.  On some properties of k-ary n-cubes , 2001, Proceedings. Eighth International Conference on Parallel and Distributed Systems. ICPADS 2001.

[24]  F. Leighton,et al.  Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes , 1991 .

[25]  Jun-Ming Xu,et al.  Edge-pancyclicity and Hamiltonian laceability of the balanced hypercubes , 2007, Appl. Math. Comput..

[26]  Bing Wei,et al.  Panconnectivity of locally connected claw-free graphs , 1999, Discret. Math..

[27]  Jun-Ming Xu,et al.  Edge-pancyclicity of Möbius cubes , 2005, Inf. Process. Lett..

[28]  Mee Yee Chan,et al.  On the Existence of Hamiltonian Circuits in Faulty Hypercubes , 1991, SIAM J. Discret. Math..

[29]  Jun-Ming Xu,et al.  Fault-tolerant pancyclicity of augmented cubes , 2007, Inf. Process. Lett..

[30]  Chun-Hua Chen,et al.  Pancyclicity on Möbius cubes with maximal edge faults , 2004, Parallel Comput..

[31]  Sun-Yuan Hsieh,et al.  Panconnectivity and edge-pancyclicity of k-ary n-cubes , 2009 .

[32]  D. S. SzyId,et al.  Parallel Computation: Models And Methods , 1998, IEEE Concurrency.

[33]  BroegBob,et al.  Lee Distance and Topological Properties of k-ary n-cubes , 1995 .