On Ramsey Number About Even Wheels

In the 1928, the British mathematician Ramsey put forward a combinatorial theory in his paper. In the future, we called the Ramsey theory.The study of Ramsey number is the famous problem and pop task in the filed of combinatorics,discrete mathematic,graph theory and arithmetic.To validate the Ramsey number is very difficult,to prove the R(G_n, G_m) is the most difficulty especially.But if the one of two graph has a few vertices,we will prove the conclusion easily.For two given graphs G1 and G2, the Ramsey number R{G1. G2) is the smallest positive integer n such that for any graph G of order n, either G contains G1 or the complement of G contains G2.So far we have had the conclusions of R(S_n, W₆) and R(S_n, W_m) ,m is odd.Zhang Kemin has proved those conclusions,but the way he used is not suitable for us to have the other conclusions, we will use the method of constructing the symmetry extremum graph.We will prove the connected and unconnected graph separately, and we will partition the unconnected graph in two situations:has isolate point and dose not have isolate point.In this thesis we shall study the the Ramsey number of the graph,and we will discuss the Ramsey number of the stars versus wheels in particular.In Chapter 1 we first review the history of the Ramsey number,the value in theory and the method of study, then we enumerate the work in this paper.In Chapter 2 we introduce some definitions which used in this article, then sum up some conclusions that have received,such as(S_n denote a star of order n and W_m denote a wheel of order m + 1):Theorem: R{S_n, W₆) = 2n +1, n ≥ 3a,ndR(S_n, W_m) = 3n-2,n≥m-1 ≥2,m odd;At the same time we prove R(S_n, W₆) = 2n + 1, n ≥ 3 use the method of constructing the extremum graph in this article.The Chapter 3 is the most important part in this article ,in this chapter we validate the conclusions of the Ramsey number star versus wheels :Theorem 1: R{S_n, W₈) = 2n + 1, n ≥ 6Theorem 2: R(S_n, W₁0) = 2n + 1, n ≥ 8In Chapter 4 we also prove some Ramsey number of paths versus wheels:Theorem 3: R(P2, Wn) = n + 1, n > 6, Theorem 4:n + 2 n > 9, nn + 2 n = 0,2(mod3),n> 11Theorem 5:...