We show, for k = 3,4,5, that the necessary conditions are sufficient for the existence of graph designs which decompose Kv(λj), the complete (multi)graph on v points with λ multiple edges for each pair of points and j loops at each vertex, into ordered blocks (a1, a 2⋯, ak-1 a1)- Each block is the subgraph which contains both the set of unordered edges {ai, aj}, for each pair of consecutive edges in the ordered list, and also the loop at vertex a1.
