Some Topics On The Ramsey Number Of Linear Forest And Cycles

Ramsey theory is an important branch of discrete mathematics,and the study of Ramsey number in graph is a major research direction of Ramsey theory.The role of Ramsey number is to quantify some existential theorems in Ramsey theory.Therefore,the study of Ramsey number of graph has theoretical significance,and its research solution is of great significance to the solution of other NP difficult problems.The Ramsey theory introduced by Ramsey of a mathematician of British in 1930.In the study of Ramsey numbers,the bipartite Ramsey numbers,the Gallai-Ramsey number and the size Ramsey number are the most common.As a natural generaliza-tion of the concept of the classical Ramsey theory,the bipartite Ramsey numbers was introduced by Beineke and Schwenk in 1975.The bipartite Ramsey number br(G,H)is the least positive integer p so that any coloring of the edges of K_p,pwith two colors will result in a red copy of G or a blue copy of H.The size Ramsey number (?)r(G,H),introduced by Erd(?)s in 1978.Given graphs F,G,H,the least positive integer m so that any two coloring of the edges of F with|E(F)|=m,F contains either a red copy of G or a blue copy of H.Gallai first studied the structure of rainbow triangle free under the guise of transitive orientations of graphs.Given a graph G and a positive integer k,define the Gallai-Ramsey number gr_k(G:H)to be the minimum number of vertices n such that any k-edge coloring of K_ncontains either a rainbow(all different colored)copy of G or a monochromatic copy of H.The difference between the bipartite Ramsey number,the Gallai-Ramsey number and the size Ramsey number is as follow:the bipartite Ramsey number is the number of one partition of the bipartite graph,we can get this by any coloring the edge of the complete bipartite graph;the size Ramsey number is the size of graph,determined by coloring the edge of graph;the Gallai-Ramsey number is the order of the the complete graph,determined by any coloring the edge of the complete graph.In this thesis,we first study the bipartite Ramsey number related to the path and give some exact values.Next,We study the size Ramsey number of linear forests,the size Ramsey number about 2K₂versus linear forests is given accurately values,involving the size Ramsey number of t K₂(t?3)versus linear forests is given a better upper and lower bounds;Finally,we study the Gallai-Ramsey number of the circle and give the exact values or upper and lower bounds,and generalized to the general situation.