Some Researches On The Anti-Ramsey Number Of Cycles

The study of the anti-Ramsey number of graphs is one of the frontier topics in graph theory research,and it is closely related to the core issues of graph theory such as extremal graph theory and Ramsey theory.Different from the classic Ramsey theory,the research object of the anti-Ramsey number of graphs is the rainbow graph.This problem is also regarded as one of the generalizations of Ramsey theory,and has gradually become the hot topics in graph theory research.Its ideas are increasingly permeating to many fields such as algebra,combinatorics,and number theory.Anti-Ramsey number refers to the maximum number of colors for a given graph G and H such that there is no rainbow subgraph H in the edge-colored graph G.It was proposed by Erdos et al.in the 1970s,and pointed out that there is a close relation-ship between the anti-Ramsey number and the Turan number of the graph.This thesis studies the problem of anti-Ramsey number of cycles in edge-colored graphs,and mainly considers the anti-Ramsey number of some short cycles and disjoint cycles in complete(multipartite)graphs.The followings are the main structures and research results.In the first chapter,it mainly introduces the basic concepts and terminologies of graph theory involved in this thesis,elaborates the research background and research status of anti-Ramsey numbers of cycles in detail,and briefly describes the main results of this thesis.In the second chapter,it mainly studies the anti-Ramsey numbers of C3 and C3+ in a complete multipartite graph,and describes the relationship between the anti-Ramsey numbers of C3 and C3+in a complete multipartite graph and the number of vertices in each partite set of the complete multipartite graph.The complete multipartite graph not only contains the complete graph,but also the complete split graph.The results obtained in this thesis cover the relevant results of the complete graph and the complete split graph.In the third chapter,it mainly studies the anti-Ramsey number of C4 in a complete multipartite graph.We first obtain the lower bound of the anti-Ramsey number of C4 in a complete multipartite graph by constructing extremal coloring,and then use induction and contradiction methods to finally derive the contradiction by continuously character-izing the structure of the graph,thereby proving the upper bound.Similarly,the result of the anti-Ramsey number of C4 in the complete multipartite graph covers the relevant results in the complete graph and the complete split graph in this thesis.In the fourth chapter,it mainly studies the anti-Ramsey number of the disjoint cycles in the complete graph.We have determined the anti-Ramsey numbers of 2C3 and ?'2 in the complete graph successively,and described the relationship between the anti-Ramsey numbers of 2C2 and ?'2 in the complete graph and the number of vertices.