Optimal fault-tolerant Hamiltonicity of star graphs with conditional edge faults

The star graph is viewed as an attractive alternative to the hypercube. In this paper, we investigate the Hamiltonicity of an n-dimensional star graph. We show that for any n-dimensional star graph (n≥4) with at most 3n−10 faulty edges in which each node is incident with at least two fault-free edges, there exists a fault-free Hamiltonian cycle. Our result improves on the previously best known result for the case where the number of tolerable faulty edges is bounded by 2n−7. We also demonstrate that our result is optimal with respect to the worst case scenario, where every other node of a cycle of length 6 is incident with exactly n−3 faulty noncycle edges.

[1]  Gen-Huey Chen,et al.  Longest Fault-Free Paths in Star Graphs with Edge Faults , 2001, IEEE Trans. Computers.

[2]  Sun-Yuan Hsieh,et al.  Embedding longest fault-free paths onto star graphs with more vertex faults , 2005, Theor. Comput. Sci..

[3]  Chih-Ping Chu,et al.  Multicast communication in wormhole-routed symmetric networks with hamiltonian cycle model , 2005, J. Syst. Archit..

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

[5]  Tzung-Shi Chen,et al.  A dual-hamiltonian-path-based multicasting strategy for wormhole-routed star graph interconnection networks , 2002, J. Parallel Distributed Comput..

[6]  Kyungsook Yoon Lee.  Interconnection networks and compiler algorithms for multiprocessors , 1983 .

[7]  S. Lakshmivarahan,et al.  Embedding of cycles and grids in star graphs , 1990, Proceedings of the Second IEEE Symposium on Parallel and Distributed Processing 1990.

[8]  Selim G. Akl,et al.  Optimal Communication algorithms on Star Graphs Using Spanning Tree Constructions , 1995, J. Parallel Distributed Comput..

[9]  Gen-Huey Chen,et al.  Hamiltonian-laceability of star graphs , 1997, Proceedings of the 1997 International Symposium on Parallel Architectures, Algorithms and Networks (I-SPAN'97).

[10]  Dharma P. Agrawal,et al.  Generalized Hypercube and Hyperbus Structures for a Computer Network , 1984, IEEE Transactions on Computers.

[11]  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.

[12]  Sheldon B. Akers,et al.  The Star Graph: An Attractive Alternative to the n-Cube , 1994, ICPP.

[13]  Yu-Chee Tseng,et al.  Fault-Tolerant Ring Embedding in a Star Graph with Both Link and Node Failures , 1997, IEEE Trans. Parallel Distributed Syst..

[14]  Dilip Sarkar,et al.  Optimal Broadcasting on the Star Graph , 1992, IEEE Trans. Parallel Distributed Syst..

[15]  Tseng-Kuei Li.  Cycle embedding in star graphs with edge faults , 2005, Appl. Math. Comput..

[16]  Gen-Huey Chen,et al.  Hamiltonian‐laceability of star graphs , 2000 .

[17]  Cheng-Kuan Lin,et al.  Mutually independent hamiltonian paths in star networks , 2005, Networks.

[18]  Yi-Ping Hung,et al.  Multicast communication in wormhole-routed 2D torus networks with hamiltonian cycle model , 2009, J. Syst. Archit..

[19]  Gen-Huey Chen,et al.  Longest fault-free paths in star graphs with vertex faults , 2001, Theor. Comput. Sci..

[20]  Norbert Ascheuer,et al.  Hamiltonian path problems in the on-line optimization of flexible manufacturing systems , 1996 .

[21]  Sanjay Ranka,et al.  Embedding meshes on the star graph , 1990, Proceedings SUPERCOMPUTING '90.

[22]  Zevi Miller,et al.  Near Embeddings of Hypercubes into Cayley Graphs on the Symmetric Group , 1994, IEEE Trans. Computers.

[23]  Jean-Claude Bermond.  Interconnection Networks , 1992 .

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

[25]  Jung-Sheng Fu,et al.  Conditional Fault-Tolerant Hamiltonicity of Star Graphs , 2006, 2006 Seventh International Conference on Parallel and Distributed Computing, Applications and Technologies (PDCAT'06).

[26]  Jimmy J. M. Tan,et al.  Hyper hamiltonian laceability on edge fault star graph , 2004, Inf. Sci..

[27]  Selim G. Akl,et al.  On Some Properties and Algorithms for the Star and Pancake Interconnection Networks , 1994, J. Parallel Distributed Comput..

[28]  Charles Delorme,et al.  Table of Large (Delta, D)-Graphs , 1992, Discret. Appl. Math..

[29]  Khaled Day,et al.  A Comparative Study of Topological Properties of Hypercubes and Star Graphs , 1994, IEEE Trans. Parallel Distributed Syst..

[30]  Shahram Latifi,et al.  A Routing and Broadcasting Scheme on Faulty Star Graphs , 1993, IEEE Trans. Computers.

[31]  Selim G. Akl.  Parallel computation: models and methods , 1997 .