Study For Rainbow Numbers Of Cycles

Ramsey theory is one of the core contents and plays important roles in graph theory.The anti-Ramsey problem is an generalization of the Ramsey theory which originated from the anti-Ramsey number proposed by Erdos et al.in 1973.The anti-Ramsey number of H in G denotes the maximum number of colors in an edge-coloring of G without rainbow H.The anti-Ramsey number plus one is called rainbow number.The researchers have carried out extensive study for rainbow numbers for many graphs,including cycles,paths,matchings,cliques and so on.Generally,the host graphs for anti-Ramsey numbers are complete graphs and complete bipartite graphs.During recent years,researchers have gradually been eager to consider anti-Ramsey problems in general graphsIn this thesis,we mainly study the rainbow numbers for cycles.The main contents of this thesis are divided into the following four parts.The first chapter illustrates the research backgrounds and current status of rainbow numbers for cycles.Also,we introduce the necessary basic knowledge and terminologies for the thesis.Meanwhile,main results of the thesis are summarized in this chapterIn the second chapter,we study the upper bound and lower bound of the rainbow number for C4 in plane triangulations.In Section 2.2,we get the lower bound of rainbow numbers of c4.In Section 2.3,we get its upper bound.Our results improve the bounds obtained by M.Hornak et al.In the third chapter,we study the rainbow number of C3 in Kneser graph.In Section 3.2,we give the upper bound and lower bound of rainbow numbers of C3 in graph KG_n,2.In Section 3.3,we give the upper bound and lower bound of rainbow numbers of C3 in graph KG_n,k.In the last chapter,we study the rainbow number of C₃⁺ in Kneser graph.In Section 4.2,we get the upper bound and lower bound of rainbow numbers of C₃⁺ in graph KG_n,2.In Section 3.3,we get the upper bound and lower bound of rainbow numbers of C₃⁺ in graph KG_n,k.