On Ramsey Numbers Of Some Graphs

Graph theory is a branch of mathematics, especially an important branch of discrete mathematics. It has been applied in many different fields in the modern world, such as physiccs, chemistry, astronomy, geography, biology, as well as in computer science and engineering.This thesis discuss the Ramsey numbers of graphs. For given two graphs G and H, the Ramsey number R(G, H) is defined to be the smallest positive integer n such that every graph F of order n must contain G as a subgraph or F must contain H as a subgraph. We discuss the Ramsey numbers for trees versus wheels. The structure of this thesis is arranged as follows:We recite briefly the devolopment of the the Ramsey numbers of graphs, and introduce the basic concepts of graph theory and terms in Chapter 1.For the speciality of star graph, the Ramsey numbers of stars versus other graphs have been widely considered. In chapter 2, we study the Ramsey numbers of stars versus wheels. In this chapter, we firstly give and prove some results about the Ramsey numbers of graph;then we present and discuss some results of the Ramsey numbers stars versus wheels making use of references [26] and [32].Path is also a special graph. Many people working on mathematics researched on the Ramsey numbers of paths or wheels. In chapter 3, we study the Ramsey numbers of paths versus wheels.. Faudree and others considered the Ramsey numbers for all path-cycle pairs. Surahmat et al obtained the Ramsey numbbers of a path versus W₄ or W₅. Yaojun Chen evaluated the Ramsey numbers of paths versus wheels in a more general situation and gave the following result: R(P_n, W_m) = 3n — 2, for m odd and n ≥m — 1 ≥ 2;R(P_n, W)_m) = 2n — 1, form even and n≥ m — 1 ≥3. So we give the Ramsey numbers for paths versus wheels for other cases in section 2:Stars and paths are special tree graphs. So we study the Ramsey numbers of trees versus wheels in chapter 4. E.T.Baskoro gave the Ramsey numbers of trees versus W4 or W5: R(Tn, W4) = 2n - 1, where n > 4;R(Tn, W5) = 3n - 2, where n > 5. So E.T.Baskoro guessed the following result: R(Tn, Wm) = 3n — 2 for n > m > 7 and m is odd. In the section 2 of this chapter, we prove the following results: R(Tn, Wm) = 3n — 2 for m is odd and m > 7, where n = m, m + 1, m + 2.