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A complete multigraph with order v and index A, denoted by Kv, is an undirected graph with v vertices, where any two distinct vertices x and y are joined by edges (x,y). Let H be a finite simple graph. A graph design H-GD(v) is a pair (X, B), where X is the vertex set of Kv and B is a partition of all edges in Kv, such that each member (block) of B is a subgraph of Kv, isomophic to H.In this thesis, we discuss the existence problems of graph design for eight graphs with six vertices and nine edges.In Chapter 1, some concepts are introduced, such as H-design, holey H-design and incomplete H-design. And, the necessary condition to exist H-design for the eight graphs H is listed. As well as, two basic lemmas are rewrited.In Chapter 2, construct some holey H-designs using quasigroups, directed product automorphism group as well as other methods.In Chapter 3, list the arrangement to complete the graph designs for each graph, respectively.In Chapter 4, prove that Hk-GD(9)(2 < k < 8) do not exist. As well as, give the necessary and sufficient conditions of the existences of Hk-GD(9)(k = 4,5,6,7,8),H2-GD(6), H2-GD(10) and H7-GD(6). Construct all GDs listed in Chapter 3.In Chapter 5, construct all incomplete H-designs listed in Chapter 3.In Chapter 6, construct the graph designs for the graphs Km+2 \ Km by using Skolem sequences .In Chapter 7, obtain our main conclusions.