Petersen’s Contribution To The Theory Of Graph Factors

Since 1736, graph theory has had more than 200 years’ development history.Mathematicians researched graph theory around the famous “ four color problem ”. Since the20 th century, with the intersection of graph theory and other disciplines, many new disciplines have appeared, such as algebraic graph theory, probability graph theory, random graph theory and so on. The theory of graph factors, which belongs to algebraic graph theory mainly studies how to describe the general graphs having a graph factor. Danish mathematician Petersen’ s originality of thinking and interdisciplinary ideas led him to create the first essay which contains the basic conclusion of Graph Theory(the theory of graph factors), and the title of the essay is “ Die Theorie der Regul?ren Graphs ”, Petersen’ s contributions to the graph factors are regarded as groundbreaking work.On the basis of sorting, studying and analyzing literatures, and attaching the history of graph factors, this paper focuses on Petersen’ s three essays about Graph Theory and systematically studies and analyzes Petersen’ s innovative thinking and interdisciplinary ideas.The main results of this paper are as follows:1. This paper introduces the life of Petersen. Petersen is one of the leaders of Danish mathematics in the second half of the 19 th century. He had a hard life, but he did not forget to study mathematics. His unique creativity and elegance teaching attitude, makes him get respected and enjoy international honors.2. This paper summarizes the academic background in the formation period.(1) Tait’ s research on the “ four color problem ” made 1-factors stand out.(2) Sylvester applied the visual “ map ” to study invariant theory, Sylvester’ s method provided the source of creative thinking for Petersen.3. This paper explores Petersen’ s research work in the theory of graph factors. With the core of Petersen’ s three essays about Graph Theory, this paper carries out a detailed study.This paper analyzes the creative mind, the framework and the main conclusions of “ Die Theorie der Regul?ren Graphs ”. Then this paper analyzes Petersen’s two theorems,meanwhile the paper gives a brief analysis of Petersen Graph which is the counterexample of Tait “ theorem ” and Petersen’ s work in the 3-degree regular graphs.4. This paper expounds the impact of Petersen during the development period of the theory of graph factors. In order to reveal Petersen’ s significant impact in the follow-up development period of the theory of graph factors, this paper analyzes the work of K?nig,Menger, Hall, Tutte on the theory of graph factors, which were influenced by Petersen.Thereby this paper makes the pioneering position of Petersen’ s work in the theory of graph factors more explicit.