Some Results On Anti-Ramsey Numbers In Several Graphs

Anti-Ramsey problem is an extension of the classical Ramsey problem,and the main thing is the study of anti-Ramsey number.The anti-Ramsey number ar(G,F)is the max-imum number of colors in an edge-coloring of G such that G doesn't contain any rainbow subgraphs isomorphic to F.Originally,the host graph in anti-Ramsey problems is complete graph.Later,the host graphs have transformed into some other special graphs.The paper mainly studies some results on anti-Ramsey numbers when complete k-partite graphs,several special join graphs and bisplit graphs as the host graphs respectively.This paper is divided into four chapters.In the first chapter,we mainly introduce the background and advance of anti-Ramsey problems.Basic definitions and concepts are cov-ered in this paper.Then,we list some main results in this paper.In the second chapter,we study the anti-Ramsey number in complete k-partite graphs.We present the upper bound and lower bound of ar(K_p1,p2,...,pk,T_n),the lower bound of ar(K_p1,p2,...,pk,C)and the lower bound of ar(K_p1,p2,...,pk,M).Furthermore,if p₁=p₂=...=p_k?2,we give the exact value of ar(K_p1,p2,...,pk,T_n)and ar(K_p1,p2,...,pk,C).In the third chapter,we study the anti-Ramsey numbers when the host graphs are C_n?(?),P_n?(?),W_n?(?) and F_n?(?) respectively.Here,the subgraph are C₃,C₄ and C₃⁺.In the fourth chapter,we present the anti-Ramsey numbers when bisplit graph as the host graph and C₃,C₄ and C₃⁺ as the subgraphs in this chapter.