The Bipartite Ramsey Number About Some Sparse Graphs

In the 1928,the British mathematician Frank Ramsey put forward a combinatorial theorem.In the future,we called the Ramsey theorem.It refer to study of partitions of large structures,and if the structure is sufficiently large,there must exists a partition with some properties.Later,the Ramsey theorem developed to Ramsey theory from several seemingly isolated theorems.The Ramsey number is a promotion of Ramsey theory,which has always been remarkable as a famous problem in the graph theory.In this paper,we discussed bipartite Ramsey number,multicolor Ramsey number,and Ramsey number.For given bipartite graphs G1,…,Gk,the bipartite Ramsey number,denoted by br(G1,…,Gk),is defined to be the smallest integer N such that in any k edge coloring of complete bipartite graph KN,N,there exists either a mon-ochromatic Gi.Moreover,for given graphs G1,…,Gk,the multicolor Ramsey number,denoted by R(G1,…,Gk),is defined to be the smallest integer N such that in any edge coloring of complete graph KN by colors i=1,…k,there exists some i,such that Gi is contained in the subgraph induced by all edges in the color i.Especially,if k＝2,it’s called Ramsey number.In this paper,we divided the article into five chapters,the first chapter for the introduction to introduce the background and main results.In chapter 2,which is the main theorem in this article,we obtain the value of bipartite Ramsey number br(H,H),where H is a balanced(β,△)-graph.It is shown that for any fixed 0<γ<1 and integer △≥1,there exist a constant β=β(γ,△)>0 and a natural number n0 such that for every balanced(β,△)-graph H on n≥n0 vertices,then br(H,H)≤<(1+γ/3)n.In particular,br(C2n,C2n)=(2+o(1))n as n→∞.In addition,there is also a similar result holds for br(H,…,H).In chapter 3,we calculate the value of R(Pni,…Pnt,H).We prove that if nt≥4R(Pn1,…Pnt-l,H),then R(Pn1,…,Pnt,H)=(χ(H)-1)(nt-t+∑「ni/2」)+σ(H).Moreo-ver,if Gi is a connected graph and r=R(G1,…,Gn),r’=R(Hp1,…,Hpm),R(G1,…,Gn,K1)=(r-1)(l-1)+1,then R(G1,…,Gn,Hp1,…,Hpm)≤(r-1)(r’-1)+1.The above equality holds if Hp1=Kp1,…Hp＝Kpm.In chapter 4,we determine the bounds about Ramsey number R(Wm,Wn).We prove that 3m+1 ≤R(Wm,Wn)≤8m-3 for odd n,m≥n≥3 and m≥5,2m+1≤R(Wm,Wn)≤7m-2 for even n and m≥n+502.In chapter 5,we make the conclusions of this paper and put forward the future research direction.