行政院國家科學委員會專題研究計畫 期中進度報告
組合學的應用(1/3)
計畫類別： 個別型計畫 計畫編號： NSC91-2115-M-009-011-執行期間： 91 年 08 月 01 日至 92 年 07 月 31 日 執行單位： 國立交通大學應用數學系 計畫主持人： 傅恆霖 報告類型： 精簡報告 報告附件： 出席國際會議研究心得報告及發表論文 處理方式： 本計畫可公開查詢中
華
民
國 92 年 5 月 15 日
In the first year of this 3-year term proposal, we have obtained several important results in cycle systems and graph decompositions which extend the study of my long term research. Also, the study of Applied Combinatorics gets start with finishing writing two joint papers, one with F. Hwang and the other one with Y. Mutoh and M. Jimbo. In coming summer, I have arranged to attend a workshop in Kyoto and then visit M. Jimbo in Keio University and R. Fuji-Hara in Tsukuba University to collaborate the research in the applications of combinatorial designs.
Through the first year, we have finished more than 10 papers which can be seen from my updated vita on publications. Here, we simply report 5 of them.
1. On 3-stage Clos networks with different nonblocking requirements to two types of calls (Joint work with F. Hwang). We prove that 3-stage Clos network C(n, n, 4, r, 2n) satisfies the requirement： strictly nonblocking for point-to-point calls among the 2-cast traffic, and is rearrangeable for genuine 2-cast calls.
2. A resolvable r×c grid-block packing and its application to DNA library screening (Joint work with Y. Mutoh and M. Jimbo). We give some constructions of a resolvable r×c grid-block packing and give a brief survey of its application to DNA library screening.
3. Packing graphs with graph of size three (Joint work with Chun-Cheng Chen and Kuo-Ching Huang). We mainly prove a conjecture of G. Chartrand et al that any graph of size 3m can be decomposed into isomorphic copies of graphs of size 3.
4. Maximal sets of Hamilton cycles in K2p-F (Joint work with S. L. Logan and C. A.
Rodger). We determine the spectrum of the size of maximal set of Hamilton
cycles in K2p-F with two different proofs.
5. Cyclic m-cycle systems of the complete graph (Joint work with S. L. Wu). We obtain the necessary and sufficient conditions for the existence of a cycle m-cycle system of the complete graph for each m≦32.
期中自評報告(第一年)
在第一年的研究中，延續以前的研究，在圈系及圖的分割方面分別獲得一些 成果；在應用方面則分別在 3-stage Clos Network 及 DNA Screening 方面完成論 文的寫作。在今年暑假，我將前往日本開會及訪問，開始與 Jimbo 及 Fuji-Hara 合作組合設計應用的研究。
雖然在這一年中我們完成的論文將近十篇，以下就其中的五篇敘述研究成果 的摘要。
1. On 3-stage Clos networks with different nonblocking requirements to two types of calls (Joint work with F. Hwang).
在這篇論文中，我們證明了 3-級 Clos 網路 C(n, n, 4, r, 2n)在點對點 2-cast 的 通話方面絕對不阻塞，而最原始的點對點 2-cast 通話可以重排。
2. A resolvable r×c grid-block packing and its application to DNA library screening (Joint work with Y. Mutoh and M. Jimbo).
在這篇論文中，我們提供一些可分解 r×c grid-block 的裝填方法，有了足夠 多的 Resolution Classes，就可以應用到 DNA library screening.
3. Packing graphs with graph of size three (Joint work with Chun-Cheng Chen and Kuo-Ching Huang).
在這篇論文中，我們證明了 G. Chartrand 等人的一個猜測：任意邊數為 3m 的 圖都可以分割成 m 個同構的 3 邊圖。為了達到這個目的，我們用裝填的概念 得到一個更強的結果。
4. Maximal sets of Hamilton cycles in K2p-F (Joint work with S. L. Logan and C. A.
Rodger).
這個結果提供了極多哈米爾頓圈存在的可能性；由於建構方法有其獨到之 處，所以這篇論文基本上有兩個不同的證明方法。
5. Cyclic m-cycle systems of the complete graph (Joint work with S. L. Wu).
在這篇論文中，我們以極詳細的方式證明只要 m≦32，循環 m-圈系存在的充 要條件可以完全決定。
1. [1] C. Clos, A study of nonblocking switching networks, Bell Syst. Tech. J. 32(1953), 406-424.
[2] D. Z. Du and H. Q. Ngo, An extension of DHH-Erdös conjecture on cycle-plus-triangle graphs, Taiwan J. Math., to appear.
[3] F. K. Hwang, S-C. Liaw and L. D. Tong, Strictly nonblocking 3-stage Clos networks with some rearrangeable multicast capability, IEEE Trans. Commun., to appear.
[4] F. K. Hwang and C. H. Lin, Broadcasting in a three-stage point-to-point nonblocking network, Inter. J. Rel. Qual. Safety Eng. 2(1995), 299-307.
[5] G. M. Masson and B. W. Jordan, Generalized multi-stage connection networks, Networks 2(1972), 191-209.
2. [1] E. Barillot, B. Lacroix and D. Cohen (1991), Theoretical analysis of library screening using a N-dimensional pooling strategy, Nucleic Acids Research 19, 6241-6247, Oxford University Press.
[2] T. Berger, J. W. Mandell and P. Subrahmanya (2000), Maximally efficient two-stage screening, Biometrics, 56, 833-840.
[3] H. L. Fu, F. K. Hwang, M. Jimbo, Y. Mutoh and C. L. Shiue (2001), Decomposing complete graphs into Kr ×Kc's, accepted to J.S.P.I.
[4] E. Knill, W. J. Bruno and D. C. Torney (1998), Non-adaptive group testing in the presence of errors, Discrete Applied Mathematics, 88, 261-290.
[5] Y. Mutoh, T. Morihara, M. Jimbo and H. L. Fu (2001), The existence of 2×4 grid-block designs and its applications, accepted to SIAM. J. Discrete Mathematics.
3. [1] G. Chartrand, A. D. Polimeni and M. J. Stewart, The existence of 1-factors in lin graphs, squares, and total graphs, Indag. Math. 35(1973), 228-232.
[2] G. Chartrand, F. Saba and C. M. Mynhardt, Prime graphs, prime-connected graphs and prime divisors of graphs, Utilitas Mathematica 46(1994), 179-191. 4. [1] D. E. Bryant, S. El-Zanati, and C. A. Rodger, Maximal sets of Hamilton cycles
on Kn,n, J. Graph Theory, 33 (2000), 25-31.
[2] M. D. Daven, J. A. MacDougall, and C. A. Rodger, Maximal sets of Hamilton cycles in complete multipartite graphs, J. Graph Theory, to appear.
[3] D. G. Hoffman, C. A. Rodger, and A. Rosa, Maximal sets of 2-factors and Hamilton cycles, J. Combin. Theory (B), 57(1993), 69-76.
[4] C. D. Leach and C. A. Rodger, Nondisconnecting disentanglements of amalgamated 2-factorizations of complete multipartite graphs, J. Combin. Designs, 9(2001), 460-467.
5. [1] B. Alspach and H. Gavlas, Cycle decompositions of K and n Kn −I, J. Combin. Theory, Ser. B 81(2001), 77-99
[2] M. Buratti and A. Del Fra, Existence of cyclic k -cycle systems of the
complete graph, Discrete Math. 261(2003), 113-125.
[3] H. L. Fu and S. L. Wu, Cyclically decomposing the complete graph into cycles, in preprint.
[4] A. Rosa, On cyclic decompositions of the complete graph into (4m+2)-gons, Mat. Fyz. Casopis Sloven. Akad. Vied 16(1966), 349-352.
[5] A. Rosa, On cyclic decompositions of the complete graph into polygons with odd number of edges (Slovak), Casopis Pest. Mat. 91(1966), 53-63.
