Study For Mixed Anti-ramsey Number Of Cliques

The Ramsey theory is a core issue and plays an important role in graph theory,where the Ramsey number is that an integer k such that there exists a large monochro-matic complete subgraph in the k-edge coloring of complete graph with enough vertices In recent decades,the Ramsey theory has been studied extensively.Besides,as one of the rainbow extension of the Ramsey theory,the anti-Ramsey number of graph is proposed by Erdos et al.[8]in the 1970s,and this parameter is related to the Turan number closely.In this master thesis,we study the mixed anti-Ramsey number of cliques and characterization of extreme edge-coloring.Given two graphs G and H,an edge-coloring of Kn is called a（G,H）-good coloring if there exists neither monochromatic G nor rainbow H.A（G,H）-good coloring with k colors is called a（G,H）-good k-coloring Denote by R（n;G,H）the set of numbers k such that there is a（G,H）-good k-coloring of Kn.The maximum and minimum numbers in the set R（n;G,H）,denoted by max R（n;G,H）and min R（n;G,H）,respectively,are called the mixed anti-Ramsey numbers.The followings are main contentsIn the first chapter,the basic concepts and terminology are introduced.The research background and current status of the mixed anti-Ramsey number are also elaborated in detail,and finally the main results are described brieflyIn the second chapter,we determine the mixed anti-Ramsey numbers min R（n;2K₂,K_s）and max R（n;2K₂,K_s）and show that the number max R（n;2K₂,K_s）is related to the Turan number of K_[s-1/2]closely.In the third chapter,we study the mixed anti-Ramsey number of monochromatic matching and rainbow clique K_s in a general graph G.And we use the Turan number to obtain the lower bound of edge-coloring of G,so that G must contain either a monochromatic matching or a rainbow clique K_s.In the last chapter,we consider the characterization of extreme edge-coloring,which is corresponding to mixed anti-Ramsey number.Moreover,we show that the extreme edge-coloring of the maximum mixed anti-Ramsey number max R（n;2K₂,K_s）is unique.