Some Researches Of Rainbow Numbers In Some Graphs

The Ramsey theory is one of the important research areas in graph theory.It has great theoretical significance and applicable value.During the recent decades,many researchers were interested in generalizations of the Ramsey theory.One of the rainbow generalizations of the Ramsey theory is the anti-Ramsey number which is introduced by Erdos et al.in 1973.Usually,the anti-Ramsey number plus one is called rainbow number.Note that the rainbow number is closely related to the Turan number.The rainbow numbers for several special graph classes in complete graphs or complete bipartite graphs have been determined.Recently,researchers began to focus on the rainbow numbers in more general host graphs.In this master thesis,we study the rainbow numbers of matchings and cycles.The thesis consists of four chapters.The first chapter is devoted to introduction the research backgrounds,recent devel-opments,basic definitions and notations in the area of rainbow numbers.Also,we state the main results obtained.In the second chapter,we study the rainbow numbers of matchings in the complete split graphs and determine the formula.In the section 2.1,we present some necessary preliminaries,including the Turan results and the canonical decomposition of graphs.In the section 2.2,we present the outline of our main results.And we complete the proof of our main results in the section 2.3.Since the complete split graph is one generalizations of the complete graph,the results in our thesis cover the previous results about the rainbow numbers of matchings in the complete graph.In the third chapter,we study the rainbow numbers of small cycles in the complete split graphs.The section 3.1 is devoted to introducing some prerequisite knowledge.In the section 3.2,we study the rainbow numbers of C4 in the complete split graphs and determine its formula.Then we consider the rainbow numbers of C5 in the complete split graphs and give its upper bound and lower bound.We study the rainbow numbers of matchings in hypergraphs in the last chapter.In the section 4.1,we introduce some necessary preliminaries.In the section 4.2,we deter-mine the exact value of the rainbow numbers of 2K2 in 3-partite 3-uniform hypergraphs at first.And then we give the upper bound of the rainbow numbers of matchings in the complete s-uniform hypergraphs.