Loop Designs

We show, for \( k = 3, 4, 5 \), that the necessary conditions are sufficient for the existence of graph designs which decompose \( K_v(\lambda, j) \), the complete (multi)graph on \( v \) points with \( \lambda \) multiple edges for each pair of points and \( j \) loops at each vertex, into ordered blocks \( (a_1, a_2, \ldots, a_{k-1}, a_1) \). Each block is the subgraph which contains both the set of unordered edges \( \{a_i, a_j\} \), for each pair of consecutive edges in the ordered list, and also the loop at vertex \( a_1 \).