輪動系統指的是一個擁有並行處理器網路的自我穩定系統，它的特點是相連兩個處理器不能在同一時間進入臨界區間；而一個公平的輪動系統指的是當系統內的處理器必須等到其他處理器都進入過臨界區間一次之後，才能再次進入臨界區間。若能讓處理器在最短時間內可以再一次進入臨界區間，則這樣的系統設計是效能最佳的，而這個問題相當於圖論中的著色問題。
訊息傳遞的簡單性以及良好的容錯能力，是迪布恩網路的吸引力。我們提出兩個著色方法來解決在二維k元迪布恩圖上的著色問題，第一個方法需要 2logk+1 個顏色，而第二個方法只需要p+1個顏色，C(p-1, (p-1)/2) < k <= C(p, p/2)，同時我們也證明當 k = C(p, p/2) 時，第二個方法是最佳的。另外，我們也提出一個方法，使得這些著色方法可以從二維的k元迪布恩圖擴展到任意維度的k元迪布恩圖，而不會增加顏色的數量。

An alternator is a self-stabilizing system which consists of a network of concurrent processors. One of its properties is that any two processors of an alternator system cannot execute the critical step at the same time if
they are adjacent. This exclusion property transforms the alternator design problem into the coloring problem.
And an alternator is said to be 1-fair if no processor executes the critical step twice when one or more other processors have not executed the critical step yet. The simplicity of routing message and the capability of fault
tolerance of de Bruijn networks attract us to design 1-fair alternator on them.
In this thesis, two algorithms are proposed to solve the coloring problem on the de Bruijn network. The first one uses $2ceil{log_2k}+1$ colors to color the $k$-ary de Bruijn graph with two digits, while the second one uses $p+1$ only colors, where ${{p-1}choose{floor{(p-1)/2}}} < k leq {pchoose{floor{p/2}}}$. We also prove that the second coloring method is optimal when $k = {pchoose{floor{p/2}}}$. In other words, the chromatic number of the
$k$-ary de Bruijn graph with two digits is $p+1$, where
$k = {pchoose{floor{p/2}}}$. Furthermore, the extension of our coloring method can be applied to the $k$-ary de Bruijn graph with three or more digits.

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2. Preliminaries and Previous Works . . . . . . . . . . . . . . 4
Chapter 3. Coloring Methods for de Bruijn Graphs . . . . . . . . . . 6
3.1 A Coloring Method with 2 log2 k + 1 Colors . . . . . . . . . . . . . . 6
3.2 An Improved Coloring Method . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Extension to dB(k,m) . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Chapter 4. Conclusions and Future Works . . . . . . . . . . . . . . . . 20
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

[1] M. Beale and S. M. S. Lau, “Complexity and autocorrelation properties of a class of de Bruijn sequence,” Electronics Letters, Vol. 22, No. 20, pp. 1046–1047, May 1986.
[2] J. C. Bermond and P. Fraigniaud, “Broadcasting and gossiping in de Bruijn networks,” SIAM Journal on Computing, Vol. 23, No. 1, pp. 212–225, Feb. 1994.
[3] J. Bruck, R. Cypher, and C. T. Ho, “Fault-fault de Bruijn and shuffle-exchange networks,” IEEE Transactions on Parallel and Distributed Systems, Vol. 5, No. 5, pp. 548–553, May 1994.
[4] N. D. de Bruijn, “A combinatorial problem,”Koninklijke Netherlands: Academe Van Wetenschappen, Vol. 49, pp. 758–764, 1946.
[5] E. W. Dijkstra, “Self-stabilizing systems in spite of distributed control,” Communications of the ACM, Vol. 17, No. 11, pp. 643–644, 1974.
[6] D. Du and F. K. Hwang, “Generalized de Bruijn digraphs,” Networks, Vol. 18, No. 1, pp. 27–38, 1988.
[7] K. Engel, Sperner theory. Cambridge University Press, 1997.
[8] A. H. Esfahanian and S. L. Hakimi, “Fault-tolerant routing in de Bruijn communication networks,”IEEE Transactions on Computers, Vol. C-34, No. 9, pp. 777–788, Sept. 1985.
[9] M. G. Gouda and F. F. Haddix, “The alternator,” ICDCS ’99: Workshop on Self-stabilizing Systems, Washington, DC, USA, pp. 48–53, IEEE Computer
Society, 1999.
[10] S.-T. Huang and B.-W. Chen, “Optimal 1-fair alternators,”Information Processing Letters, Vol. 80, No. 3, pp. 159–163, 2001.
[11] S.-T. Huang, Y.-S. Huang, and S.-S. Hung, “Alternators on uniform rings of odd size,” Distributed Computing, Vol. 16, No. 4, pp. 263–268, 2003.
[12] J.-W. Mao, The Coloring and Routing Problems on de Bruijn Interconnection Networks. Ph.D. Dissertation, National Sun Yat-Sen University, Kaohsiung, Taiwan, July 2003.
[13] J.-W. Mao and C.-B. Yang, “Shortest path routing and fault-tolerant routing on de Bruijn networks,” Networks, Vol. 35, No. 3, pp. 207–215, 2000.
[14] J.-W. Mao and C.-B. Yang, “A design for node coloring and 1-fair alternator on de Bruijn networks.,” Proceedings of the International Conference on Parallel and Distributed Processing Techniques and Applications, Vol. 4, Las Vegas,
Nevada, USA, pp. 1872–1878, 2003.
[15] G. Mayhew and S.W. Golomb, “Linear spans of modified de Bruijn sequences,”IEEE Transactions on Information Theory, Vol. 36, No. 5, pp. 1166–1167, 1990.
[16] D. K. Pradhan and S. M. Reddy, “A fault-tolerant communication architecture for distributed systems,” IEEE Transactions on Computers, Vol. 31, No. 9, pp. 863–870, Sept. 1982.
[17] E. Sopena, “Oriented graph coloring,” Discrete Mathematics, Vol. 229, No. 1-3, pp. 359–369, 2001.
[18] E. Sperner, “Ein satz uber untermengen einer endlichen menge,” Mathematische Zeitschrift 27, pp. 544–548, 1928.
[19] M. A. Sridhar, “The undirected de Bruijn graph: Fault tolerance and routing algorithms,” IEEE Transactions on Circuits and Systems I-Fundamental Theory and Applications, Vol. 39, No. 1, pp. 45–48, 1992.
[20] M. A. Sridhar and C. S. Raghavendra, “Fault-tolerant networks based on the de Bruijn graph,” IEEE Transactions on Computers, Vol. 40, No. 10, pp. 1167–1174, Oct. 1991.