如果一个边染色图的每条边的颜色均不相同，我们则称其是彩虹的。如果一 个边染色图的任意两条相邻边颜色均不相同，我们则称其是正常染色的。图H在图G中的反拉姆齐数，记作ar(G,H)，是边染色的图G中使得其不含彩虹子图 H 所能用的最多的颜色数。图H在图G中的广义反拉姆齐数，记作pr(G,H)，是边染色的图G中使得其不含正常染色子图H所能用的最多的颜色数。反拉姆齐数最早是由Erdos等人于1973年提出的，此后，研究学者们对完全图中各式各样的子 图的反拉姆齐数进行了广泛的研究，并从不同的角度对反拉姆齐数进行了推广。在这篇论文中，我们主要研究边染色的完全图和完全r-部图中存在彩虹(或正常染色)子图的颜色数条件，边染色的完全r-部r-一致超图中存在彩虹匹配的颜色数条件以及r-部r-一致超图和3-一致二部超图中存在完美匹配的充分性条件。本文的主要结果如下:1、对充分大的n，我们分别给出了完全图K_n中星森林和双星图的反拉姆齐数的精确值以及线性森林的反拉姆齐数的近似值;2、我们分别给出了完全r-部图中C_3和C_4的反拉姆齐数的精确值;3、我们揭示了广义反拉姆齐数与Turan 数之间的关系，对于充分大的l和n，我们给出了完全图K_n中路 P_l的广义反拉姆齐数的精确值，我们还分别给出了完 全图中 C_5、C_6 和 K^{-}_{4}

An edge-colored graph is called rainbow, if all of its edges have different colors. While an edge-colored graph is called properly colored, if any two adjacent edges of it have different colors. The anti-Ramsey number of a graph H in a graph G, denoted by ar(G, H), is the maximum number of colors in an edge-colored G with no rainbow copy of H. The generalized anti-Ramsey number of a graph H in a graph G, denoted by pr(G, H), is the maximum number of colors in an edge-colored G with no properly colored copy of H. The anti-Ramsey number was first studied by Erdos et al. in 1973. Since then, the researchers carried out extensive study for anti-Ramsey numbers of many graphs in the complete graphs. Also, the anti-Ramsey numbers have been generalized to various types.In this thesis, we will study the color number conditions for the existences of some rainbow (or properly colored) graphs in edge-colored complete graphs and complete r- partite graphs respectively, and the color number conditions for rainbow matchings in edge-colored complete r-partite r-uniform hypergraph and some sufficient conditions for the existence of a perfect matching in r-partite r-uniform hypergraphs and 3-uniform hm-bipartite hypergraphs respectively.The main results of this thesis are as follows:1. For sufficiently large n, we determine the exact value of the anti-Ramsey num- bers of star forests and double stars in K_n and the approximate value of the anti-Ramsey numbers of linear forests in the complete graphs respectively.2. We determine the exact value of the anti-Ramsey numbers of C_3 and C_4 in the complete r-partite graphs respectively.3. We show the relationship between the generalized anti-Ramsey numbers and Turan numbers. For sufficiently large l and n, we determine the exact values of the generalized anti-Ramsey numbers of P_l in K_n. Furthermore, we determine the exact value of the generalized anti-Ramsey numbers of C_5, C_6 and K^{-}_{4}