河北师范大学博士学位论文塔特对图论的贡献姓名：王献芬申请学位级别：博士专业：基础数学指导教师：胡作玄;邓明立20100410III 摘 要 摘 要 图论是一门应用广泛的重要数学分支，有着悠久的历史它诞生于1736年，1936年正式成为一门独立学科，从此获得突飞猛进的发展但这近75年的发展史却几乎没人研究作为图论的转折性人物，塔特是20世纪最具国际影响力的图论学家之一他是首屈一指的现代图论先驱，被誉为“图论之父” 他为图论的发展做出了奠基性和开拓性贡献他的许多工作都成为后继者继续发掘和拓展的“金矿” ，至今仍是非常活跃的课题因此， 研究塔特对图论的贡献不仅具有重要的历史意义， 对于现代图论研究也很有价值 目前国内外相关研究极少且不系统，本文在没有更多研究文献可借鉴的情况下，通过认真研读塔特的 4 本专著和几乎全部图论论文， 发现促使塔特广泛而深入研究图论的原动力是四色问题 这样， 按照他尝试求解四色问题的不同角度， 结合相关理论的发展，本文把他的图论研究工作划分为四个主要分支领域（图着色、图因子及其分解、图多项式和拟阵论） ，利用思想史学派的概念分析法，系统研究了塔特对图论的贡献及影响。
在阐述过程中， 叙述他的主要成就， 分析他的思想活动， 这有助于理解数学创造的过程、掌握数学发现的方法、借鉴研究问题的思路和途径全文从以下几个主要方面进行了研究： 1． 按照时间顺序，通过阐述塔特在 6 个不同时期的思想活动、兴趣爱好和学术成就，分析了他从化学转向图论的内因和外因，以及他被载入史册的原因，证明了塔特在图论中的核心地位塔特在图论中的核心地位 2． 塔特在图着色理论上的主要成就塔特在图着色理论上的主要成就在分析塔特对四色问题研究的角度、方法的选择和思想的继承基础上，探讨了他如何否定泰特猜想，并取得可平面图哈密顿问题进展的思维过程，强调了推广原则和退步原则是行之有效的数学方法 3． 塔特对图因子及其分解理论的主要贡献塔特对图因子及其分解理论的主要贡献系统研究了图因子理论的起源及从彼得森到霍尔的发展， 重点分析了塔特如何得到当时图论中最具影响力的 1-因子定理 4． 塔特对图多项式理论的贡献塔特对图多项式理论的贡献全面分析了塔特如何推动色多项式、塔特多项式和流多项式的发展，如何从四色问题中创造了图多项式的理论分支，指出了建立、发展理论和解决问题是数学创造的两个重要方面 5． 塔特对拟阵论的复兴塔特对拟阵论的复兴。
重点分析了他对拟阵的线性表示、结构以及连通性理论的贡献，说明了建立一个新学科不易，拯救濒临衰竭的分支更需要智慧和眼光 IV 关键词： 塔特 拟阵论 图着色 图因子及其分解 色多项式 塔特多项式 处处非零-流 V Abstract As an important branch of mathematics, Graph Theory has extensive applications. It has a long history. Born in 1736, it became an independent discipline in 1936. Since then it has made the rapid development. But there is hardly any research on the history of its developments in this late 75 years. Being a watershed figure in this field, William Thomas Tutte is one of the most internationally influential graph theorists in 20th century. Honored as “the father of graph theory”, he is a pioneering leader in the field of Modern Graph Theory. He has made fundamental and groud-breaking contributions to graph theory. Lots of his work has become the “gold mine” that the successors continue to explore and develop, and the subjects formed from his work are still very active today. Thus this review on Tutte’s contributions to graph theory is of historical significance, and it is also very valuable for the research on modern graph theory. There are few and scattered research literatures on this topic both at home and abroad. Hence, there are no more of them to be used for reference. Based on studying Tutte’s four books and almost all papers on graph theory, this dissertation has found the motive power that induced Tutte to research on graph theory was the four-color problem. According to point of view of his trying to solve it, and the development of relevant theory as well, his work on graph theory could be divided into four main subfields (graph coloring, graph factors and factorization, graph polynomials and matroid theory). By means of concept analysis of the schools of intellectual history, this dissertation discusses Tutte’s contributions to graph theory and his influences on its development, in the process of which his main achievements are described, and his thoughts are analyzed. It may be helpful to understand the process of creating mathematics, and to grasp the method of mathematical discovery, and to borrow ideas and ways of the research. The following several main aspects are discussed: 1. Tutte’s occupying the core status throughout graph theory is demonstrated. According to chronological order, this dissertation analyzes the internal and external reasons that he turned from chemistry to graph theory, and the reasons that he was written into in the history, VI which expatiates upon six different periods of activities, interests and academic achievements. 2. Tutte’s main achievements on graph coloring theory are researched. By analyzing the point of view of his studying the four-color problem, his selection of research, and the inheritance of mathematical ideas, this dissertation discusses how he disproved Tait’s conjecture, and how he made progress in Hamilton problem of planar graphs. The author emphasizes that the extension principle and the reduction principle are two effective methods of mathematics. 3. Tutte’s contributions to graph factors and factorizations theory are reviewed. The origin of graph factors theory and its development from Petersen to Hall are studied systematically, and how Tutte got 1-factor theorem which was the most influential one in graph theory those days are analyzed selectively. 4. Tutte’s contributions to graph polynomial theory are investigated. How Tutte developed the theory of chromatic polynomial, Tutte polynomial and flow polynomial is analyzed, and how he established a branch of graph theory from the four-color theory is discussed. The author point out that establishing and developing the theory and resolving the problem are the two important aspects in mathematical creation. 5. How Tutte renewed matroid theory is explored. His achievements on linear expression of matroids, structure of matroids and connectivity of matroids are reviewed, by which this dissertation explains that creating a new discipline is not easy, bu。