The Contributions Of William Tutte To Graph Theory

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, 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, but saving one from the endangered failure needs more wisdom and vision.