符号模流与符号圆流

发布时间：2018-06-21 05:10

  本文选题：符号图 + 符号模流　； 参考：《安徽大学》2017年硕士论文

【摘要】：为研究四色问题,Tutte提出了整数流理论.随后,整数流理论逐渐成为图论的经典研究方向.符号图是既含正边又含负边的图,是一般图模型的自然推广.在可定向曲面上,顶点染色的对偶问题是一般图上的整数流;在不可定向曲面上,顶点染色的对偶问题是符号图上的整数流.相比于一般图上的整数流,符号图上的流尚处于起步阶段,有广阔的研究空间,有很多有意义的研究工作值得探讨.模流与圆流是讨论一般图上流问题的经典研究工具.然而在符号图上,模流、圆流与整数流之间的关系更为复杂.本文即讨论了关于符号模流与符号圆流的几个问题.本文第一章首先介绍了符号流理论的研究背景,其次介绍了常用的概念和符号,最后介绍了文章所研究的问题,研究进展及本文所得的主要结果.本文第二章主要是讨论了符号模流中的两个问题.在[17]中,Macajova等刻画了连通的非平衡符号图存在处处非零整数流的等价条件.本章的第二节给出了连通的非平衡符号图存在处处非零模流的等价条件.在[31]中,魏二玲等证明了如果符号6-流猜想在符号立方图是成立的,则符号6-流猜想就是成立的.因此,符号立方图上的流问题非常值得关注.在[18]中,Macajova等刻画了存在处处非零的Z3-流和Z4-流的符号立方图.第二章的第三节,我们刻画了存在处处非零的Z5-流的符号立方图.本文第三章主要讨论了符号圆流的一个问题.设φc(G,σ)和φ(G,σ)分别是(G,σ)的符号圆流数和符号整数流数.2011年,Raspaud和朱绪鼎[20]证明了φ(G,σ)2[φc(G,σ)]-1.2015年,Macajova和Steffen[19]给出一个例子,证明上面上界不能被改进.本章,我们证明了在有流的环欧拉符号图上,φ(G,σ)≤[φ(G,σ)]even.
[Abstract]:In order to study the four-color problem Tutte puts forward the integer flow theory. Subsequently, integer flow theory has gradually become the classical research direction of graph theory. Symbolic graph is a graph with both positive and negative edges. It is a natural generalization of the general graph model. On orientable surfaces, the dual problem of vertex coloring is an integer flow on a general graph, and on an unorientable surface, the dual problem of vertex coloring is an integer flow on a symbolic graph. Compared with the integer flow on the general graph, the flow on the symbol graph is still in its infancy, and there is a wide space for research, and a lot of meaningful research work is worth discussing. Mode flow and circular flow are the classical research tools to discuss the flow problem on a general graph. However, in symbolic graph, the relationship between mode flow, circular flow and integer flow is more complicated. In this paper, some problems about symbolic mode flow and symbolic circular flow are discussed. In the first chapter, the research background of symbolic flow theory is introduced, then the commonly used concepts and symbols are introduced. Finally, the problems studied in this paper, the research progress and the main results obtained in this paper are introduced. In the second chapter, two problems in symbolic mode flow are discussed. In [17], Macajova et al characterizes the equivalent conditions for the existence of everywhere nonzero integer flows in connected nonequilibrium signed graphs. In the second section of this chapter, we give the equivalent conditions for the existence of everywhere nonzero mode flow in a connected nonequilibrium symbolic graph. In [31], Wei Erling and others proved that if the symbolic 6-flow conjecture is true in the symbolic cubic graph, then the symbolic 6-flow conjecture is true. Therefore, the flow problem on symbolic cubic graphs is of great concern. In [18], Macajova et al characterizes the symbolic cubic graphs of Z _ 3-stream and Z _ 4-stream with everywhere nonzero. In the third section of Chapter 2, we characterize the symbolic cubic graphs of Z _ 5-flows with nonzero everywhere. In the third chapter, a problem of symbolic circular flow is discussed. Let 蠁 C G, 蟽) and 蠁 G G, 蟽) be the signed circle flow number and signed integer flow number of G, 蟽, respectively. In 2011, Raspaud and Zhu Xuding [20] proved that 蠁 G, 蟽 n 2 [蠁 C G, 蟽]-1. In 2015, Macajova and Steffen [19] give an example to prove that the upper bound can not be improved. In this chapter, we prove that 蠁 G, 蟽) 鈮，

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