Study Of Heterochromatic Subgraphs In The Edge Colored Graphs

An r-edge-coloring of a graph G is a surjective assignment of r colors to the edges of G. An edge-colored graph G is called heterochromatic if any two edges (if it has) of it have different colors.Anti-Ramsey number of the graphs were introduced by Erdos et al. in 1973. They proved that the anti-Ramsey number was closely related to Turan number. Given a pos-itive integer n and let (?) be a family of graphs. The anti-Ramsey number R*(n,(?)) of (?) is the maximum number of colors in an edge coloring of Kn that has no heterochro-matic copy of any graph in (?). The early results of anti-Ramsey were approximate. In recent years, researchers considered the exact formulas for the anti-Ramsey number of some special graphs, such as trees, paths, cycles, stars, cliques and small bipartite graphs etc. Also, the bipartite version of the anti-Ramsey number were consideredIn this paper, we consider the heterochromatic subgraphs in edge colored graphs. The main results obtained in this dissertation may be summarized as follows:In the first part, we study the bipartite version of the anti-Ramsey number of (?)k, where (?)k is a family of trees of k edges. Firstly we described the exact expression. This result shows that the number is closely related to the Turan numberIn the second part, we generalized the definition of the anti-Ramsey number into common graphs. We study the heterochromatic subgraphs of the family (?)k in edge colored girds. In the third part, we study the edge colorings of the complete bipartite graph that avoids the heterochromatic path P4, P5 and P6. For the case of heterochromatic path Pk,k≥7, it is too hard to describe clearly.