# 臺灣博碩士論文加值系統

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 網路理論是現代通訊和網路系統的基礎，其中有許多不同的分析方法去研究和傳輸資訊相關的各種問題。這篇論文是專注在於網路流的問題，網路流的問題不僅僅只是通訊，還包括標籤、著色、週期涵蓋以及圖形理論的各種主題分解。本論文涉及經典的網路流量問題，每個頂點都滿足基爾霍夫流量定律，包含有向圖內的無處零流以及更普遍的雙向圖內無處零流，更確切來說，我們處理與零和流動問題，作為一個雙向導向無處零流量的特殊情況，和他們的相關的無向圖最小流量數字。第一章，簡要介紹無處零流的背景和近期理論發展和著名的猜想。第二章，專門研究流動數字圖表，像是定期圖表、迷圖、輪圖。第三章，利用第一章的零合流的觀念延伸出常數流。第四章，為第二、三章代數進一步延伸阿貝爾群流的觀念。第五章，歸納出來的公開問題。
 Network theory is the foundation of modern communication andinternet systems, in which there are many different analytic ways tostudy various kinds of problems related to the data transmission.This thesis is dedicated to certain network flow problem. Thenetwork flow problem is closely related to not just communications,but also to labeling, coloring, cycle covers, and factorizationsamong various topics in graph theory.This thesis concerns classic network flow problems including thenowhere-zero flow on directed graphs and more generally thenowhere-zero flow on bi-directed graphs, for which the Kirchhoffcurrent law type of flow condition is satisfied along each vertex.More precisely, we deal with the zero-sum flow problems, as aspecial case of the bi-directed nowhere-zero flows, and theirassociated minimum flow numbers over undirected graphs.To begin with, Chapter one gives a brief introduction about thebackground of nowhere-zero flows, also the recent theoreticaldevelopment and the well-known conjectures. Chapter two is a studyof flow numbers on special classes of graphs, such as regulargraphs, fans graphs, and wheels graphs. On the study of zero-sumflow in Chapter two, the concept of constant-sum is used. Thus wedefine constant-sum flow and study them in Chapter three. Chapterfour is the outcome of further algebraic extension of Chapter twoand Chapter three by extending the integer flows to Abelian groupflows. We conclude with some open problems in the last Chapter five.
 1 Nowhere-Zero Flows 11.1 Network Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Nowhere-Zero Flows on Directed Graph . . . . . . . . . . . . . 21.3 Nowhere-Zero Flows on Bidirected Graphs . . . . . . . . . . . 52 Zero-Sum Flows and Zero-Sum Flow Numbers 92.1 Survey of Previous Results . . . . . . . . . . . . . . . . . . . . 92.2 Zero-Sum Flow Numbers for General Graphs . . . . . . . . . . 122.3 Zero-Sum Flow Numbers for Regular Graphs . . . . . . . . . . 142.3.1 Even Regular Graphs . . . . . . . . . . . . . . . . . . . 142.3.2 2-Edge-Connected Odd Regular Graphs . . . . . . . . . 162.4 More Examples of Flow Numbers . . . . . . . . . . . . . . . . 192.4.1 Cartesian Product of Regular Graphs With Paths andCycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.2 Fan Graphs and Wheel Graphs . . . . . . . . . . . . . 232.4.3 Two Infinite Families of Examples of Zero-Sum FlowNumbers 5 and 6 . . . . . . . . . . . . . . . . . . . . . 293 Constant-Sum Flows 333.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Constant-Sum Flows of Regular Graphs . . . . . . . . . . . . 343.2.1 Odd Regular Graphs . . . . . . . . . . . . . . . . . . . 343.2.2 Even Regular Graphs . . . . . . . . . . . . . . . . . . . 353.3 Constant-Sum Flows of Fan Graphs and Wheel Graphs . . . . 373.3.1 Fan Graphs . . . . . . . . . . . . . . . . . . . . . . . . 383.3.2 Wheel Graphs . . . . . . . . . . . . . . . . . . . . . . . 404 Constant-Sum Group Flows 434.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Basic Properties and Mini-Survey . . . . . . . . . . . . . . . . 444.3 4r-Regular Graphs with Constant-Sum Z4-Flows . . . . . . . . 475 Conclusion and Further Studies 485.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . 485.2 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 49
 [1] S. Akbari, N. Ghareghani, G. B. Khosrovshahi, S. Zare, A note onzero-sum 5-flows in regular graphs, the Electronic Journal of Combinatorics19(2) (2012), # P7.[2] S. Akbari, G.B. Khosrovshahi and A. Mofidi, Zero-Sum Flows inDesigns, Journal of Combinatorial Designs 19 (2011), 355-364.[3] S. Akbari, A. Daemi, O. Hatami, A. Javanmard and A. Mehrabian,Zero-Sum Flows in Regular Graphs, Graphs and Combinatorics 26(2010), 603-615.[4] S. Akbari, G.B. Khosrovshahi, N. Ghareghani and A. Mahmoody, OnZero-Sum 6-Flows of Graphs, linear Algebra and its Applications430 (2009), 3047-3052.[5] S. Akbari, G.B. Khosrovshahi, N. Ghareghani and H. Maimani, TheKernels of The Incidence Matrices of Graphs Revisited, LinearAlgebra and its Applications 414 (2006), 617-625.[6] J. Akiyama and M. Kano, Factors and Factorizations of Graphs-asurvey, Journal of Graph Theory 9 (1985), 1-42.[7] A. Bouchet, Nowhere-Zero Integral Flows on a BidirectedGraph, J. Combin. Theory, Ser. B 34 (1983), 279-292.[8] R. Diestel, Graph Theory, Third Ed., Springer, Heidelberg, 2005.[9] M. DeVos, Flows on Bidirected Graphs, Princeton University Press,Princeton, N.J., 2004.[10] L. R. Ford and D. R. Fulkerson, Flows in Networks, Princeton UniversityPress, Princeton, N.J., 1962.[11] A. Khelladi, Nowhere-zero integer chains and flows in bidirectedgraphs, J. Combin. Theory, Ser. B 43 (1987), 95-115.[12] E. Lˆe, Graph Flows and the Zero-Sum Conjecture, Pomona College,Claremont, C.A., 2010.[13] J. Petersen, Die Theorie der regularen graphs, Acta Math(15)(1891), 193-220.[14] P.D. Seymour, Nowhere-Zero 6-Flows, J. Combin. Theory, Ser. B 30(1981), 130-135.[15] T.M. Wang and S.W. Hu, On Zero-Sum Minimum Flow Numbers,Submitted, 2012.[16] T.M. Wang and S.W. Hu, Flow Number of Zero-Sum Flow, the 6thInternational Frontiers of Algorithmics Workshop (FAW 2012, Beijing,China), Lecture Notes in Computer Science(LNCS) 7285, pp. 269-278,2012 (EI).[17] T.M.Wang and S.W. Hu, Constant Sum Flows in Regular Graphs,the 5th International Frontiers of Algorithmics Workshop (FAW 2011,Jinhua, China), Lecture Notes in Computer Science(LNCS) 6681, pp.168-175, 2011 (EI).[18] T.M. Wang and T.B. Reynolds, Group Magic Sums of RegularGraphs, submitted, 2010.[19] T.M. Wang and C.M. Lin, Magic Sum Spectra of Group MagicGraphs, India-Taiwan Conference on Discrete Mathematics (2009).[20] T.M. Wang and C.M. Lin, On Zero Magic Sums of Integer MagicGraphs, accepted by Ars Combinatoria, 2009.[21] D. B. West, Introduction to Graph Theory, 2nd edn, Prentice Hall,Englewood Cliffs, 2001.[22] R. Xu and C. Q. Zhang, On flows in bidirected graphs, DiscreteMathematics 299 (2005),335-343.[23] C.Q. Zhang, Circular Flows of Nearly Eulerian Graphs andVertex-Splitting, Journal of Graph Theory 40 (2002), 147-161.
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