Loop Designs

Abstract

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 $(a_1,a_2,\dots ,a_{k-1},a_1)$. 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 $a_1$.