給定一個圖形，去找到這個圖形存在所有長度的圓圈(cycle)是一個令人關注的泛圓性問題。如果一個圖包含所有長度的圓圈，那麼我們就說這個圖具有泛圓性(pancyclic)這個特性。而現在，令我們感興趣的是點泛圓性問題。我們說一個圖形具有點泛圓性，那就表示針對這圖形上的任何一個點，我們都可以找到所有長度的圓圈，使得點被包含在圓圈裡。在這篇論文中，我們的目的是在證明雙扭超方體(twisted cube)具有點泛圓性。

A graph G = (V,E) is said to be pancyclic if it contains cycles of all lengths from 4 to V in G. Now, we are interesting in node-pancyclic problem. We say that a graph G is node-pancyclic if for every node u, G contains cycles C of all lengths such that u is in C. The twisted cube is an alternative to the popular hypercube network. In this paper, we prove that the twisted cube is node-pancyclic.

[1] S. Abraham and K. Padmanabhan,"The twisted cube topology multiprocessors:
a study in network asymmetry",Journal of Parallel and Distributed
Computing, vol. 13, issue 1, pp. 104-110, 1991.

[2] S. G. Akl,Parallel Computation:Models and Methods,Prentice Hall, 1997.

[3] Toru Araki and Yukio Shibata,"Pancyclicity of recursive circulant
graphs",Information Processing Letters, vol. 81, issue 4, pp. 187-190,
2002.

[4] L. Bhuyan and D. P. Agrawal,"Generalized hypercubes and hyperbus structure
for a computer network",IEEE Transactions on Computers, c. 33, pp. 323-
333, 1984.

[5] E. van Blanken, J. van den Heuvel, and H. J. Veldman,"Pancyclicity of
hamiltonian line graphs",Discrete Mathematics, vol. 138, issue 1-3, pp.
379-385, 1995.

[6] J. A. Bondy,"Pancyclic graphs",Journal of Combinatorial Theory-Series B,
vol. 11, issue 1, pp. 80--84, 1971.

[7] C. P. Chang, J. N. Wang, and L. H. Hsu,"Topological properties of twisted
cubes",Information Sciences, vol. 113, issue 1-2, pp. 147-167, 1999.

[8] J. W. McGeea and C. A. Rodger,"Embedding coverings of 2-paths with 3-
paths",Discrete Mathematics, vol. 284, issue 1-3, pp.217-223, 2004.

[9] P. E. Haxell and T. uczak,"Embedding trees into graphs of large girth ,
Discrete Mathematics, vol. 216, issue 1-3, pp. 273-278, 2000.

[10] Nader Bagherzadeh, Martin Dowd and Nayla Nassif,"Embedding an arbitrary
binary tree into the star graph",IEEE Transactions on Computers, vol. 45,
no. 4, pp. 475-481, 1996.

[11] J. Fan, X. Lin, and X. Jia, Node-pancyclicity and edge-pancyclicity of
crossed cubes",Information Processing Letters, vol. 93, issue 3, pp. 133
138, 2005.

[12] Jun-Ming Xu, Zheng-Zhong Du and Min Xu,"Edge-fault-tolerant edge-
bipancyclicity of hypercubes",Information Processing Letters, vol. 96,
issue 4, pp. 146-150, 2005.

[13] Ajay K. Gupta and Susanne E. Hambrusch,"Embedding Complete Binary Trees
into Butterfly Networks",IEEE Transactions on Computers, vol. 40, no. 1,
pp. 853--863, 1991.

[14] P. A. J. Hilbers, M. R. J. Koopman, and J. L. A. van de Snepscheut,"The
twisted cube",Parallel Architectures and Languages Europe, Volume I:
Parallel Architectures, Eindhoven, The Netherlands, June 15-19, 1987,
Proceedings, pp. 152--159, 1987.

[15] S. Y. Hsieh and C. H. Chen,"Pancyclicity on Mobius Cubes with Maximal
Edge Faults",Parallel Computing, vol. 30, no. 3, pp. 407--421, 2004.

[16] W. T. Huang, J. M. Tan, C. N. Hung, and L. H. Hsu,"Fault-tolerant
hamiltonicity of twisted cubes",Journal of Parallel and Distributed
Computing, vol. 62, issue 4, pp. 591-604, 2002.

[17] M. Kouider and A. Marczyk,"On pancyclism in hamiltonian graphs",Discrete
Mathematics, vol. 251, issue 1-3, pp. 119-127, 2002.

[18] Jung-Sheng Fu,"Fault-tolerant cycle embedding in the hypercube",Parallel
Computing, vol. 29, no. 6, pp. 821--832, 2003.

[19] F. T. Leighton,Introduction to Parallel Algorithms and Architecture:
Arrays.Trees.Hypercubes. Morgan Kaufmann, San Mateo, CA, 1992.

[20] B. Randerath, I. Schiermeyer, M. Tewes, and L. Volkmann,"Vertex pancyclic
graphs",Discrete Applied Mathematics, vol. 120, issue 1-3, pp. 219-237,
2002.

[21] G. Tel,Topics in distributed algorithms. Cambridge Int'l Series on
Parallel Computation, Cambridge Universitry Press, 1991.

[22] E. Abuelrub and S. Bettayeb,"Embedding of complete binary trees in
twisted hypercubes",Proceedings of International Conference on Computer
Applications in Design, Simulation, and Analysis, pp. 1–4, 1993.

[23] Arkady Kanevsky and Chao Feng,"On the embedding of cycles in pancake
graphs",Parallel Computing, vol. 21, no. 6, pp. 923-936, 1995.

[24] K.Day and A. Tripathi ,"Embedding of Cycles in Arrangement Graphs",IEEE
Transactions on Computers, vol. 42, no. 8, pp. 1002-1006, 1993.

[25] H. C. Keh and J. C. Lin,"On fault-tolerant embedding of Hamiltonian
cycles, linear arrays and rings in a Flexible Hypercube",Parallel
Computing, vol. 26, no. 6, pp. 769--781, 2000.

[26] Y. C. Tseng, S. H. Chang and J. P. Sheu,"Fault-tolerant ring embedding in
star graphs with bothlink and node failures",IEEE Transactions on
Parallel and Distributed Systems, vol. 8, no. 12, pp. 1185-1195, 1997.