Panconnectivity and edge-pancyclicity of faulty recursive circulant G(2m, 4)

In this paper, we investigate a problem on embedding paths into recursive circulant G(2^m,4) with faulty elements (vertices and/or edges) and show that each pair of vertices in recursive circulant G(2^m,4), m>=3, are joined by a fault-free path of every length from m+1 to |V(G(2^m,4)@?F)|-1 inclusive for any fault set F with |F|@?m-3. The bound m-3 on the number of acceptable faulty elements is the maximum possible. Moreover, recursive circulant G(2^m,4) has a fault-free cycle of every length from 4 to |V(G(2^m,4)@?F)| inclusive excluding 5 passing through an arbitrary fault-free edge for any fault set F with |F|@?m-3.

[1]  Jung-Heum Park.  Hamiltonian Decomposition of Recursive Circulants , 1998, ISAAC.

[2]  Jimmy J. M. Tan,et al.  Fault-tolerant cycle-embedding of crossed cubes , 2003, Inf. Process. Lett..

[3]  André Raspaud,et al.  Routing in Recursive Circulant Graphs: Edge Forwarding Index and Hamiltonian Decomposition , 1998, WG.

[4]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

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

[6]  Kyung-Yong Chwa,et al.  Embedding Trees in Recursive Circulants , 1996, Discret. Appl. Math..

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

[8]  Jeongmi Park.  Strong Hamiltonicity of Recursive Circulants , 2001 .

[9]  Kyung-Yong Chwa,et al.  Embedding Trees into Recursive Circulants , 1996 .

[10]  Toru Araki,et al.  Edge-pancyclicity of recursive circulants , 2003, Inf. Process. Lett..

[11]  Daniel K. Biss.  Hamiltonian decomposition of recursive circulant graphs , 2000, Discret. Math..

[12]  Kyung-Yong Chwa,et al.  Recursive circulants and their embeddings among hypercubes , 2000, Theor. Comput. Sci..

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

[14]  Xiaohua Jia,et al.  Optimal Embeddings of Paths with Various Lengths in Twisted Cubes , 2007, IEEE Transactions on Parallel and Distributed Systems.

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

[16]  Hyeong-Seok Lim,et al.  Fault-Hamiltonicity of hypercube-like interconnection networks , 2005, 19th IEEE International Parallel and Distributed Processing Symposium.

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

[18]  Elwood S. Buffa,et al.  Graph Theory with Applications , 1977 .

[19]  박정흠.  Cycle Embedding of Faulty Recursive Circulants , 2004 .

[20]  Hyeong-Seok Lim,et al.  Panconnectivity and Pancyclicity of Hypercube-Like Interconnection Networks with Faulty Elements , 2006, ISPA Workshops.

[21]  Rynson W. H. Lau,et al.  Edge-Pancyclicity of Twisted Cubes , 2005, ISAAC.

[22]  C. C. Chen,et al.  On strongly hamiltonian abelian group graphs , 1981 .

[23]  Lih-Hsing Hsu,et al.  On embedding cycles into faulty twisted cubes , 2006, Inf. Sci..

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

[25]  Sun-Yuan Hsieh,et al.  Cycle Embedding on the M ¨ obius Cube with Both Faulty Nodes and Faulty Edges , 2005, 11th International Conference on Parallel and Distributed Systems (ICPADS'05).

[26]  Abhijit Sengupta.  On Ring Embedding in Hypercubes with Faulty Nodes and Links , 1998, Inf. Process. Lett..