Orthogonal Decomposition and Packing of Complete Graphs

An H-decomposition of a graph G is a partition of the edge-set of G into subsets, where each subset induces a copy of the graph H. A k-orthogonal H-decomposition of a graph G is a set of k H-decompositions of G, such that any two copies of H in distinct H-decompositions intersect in at most one edge. In case G=Kn and H=Kr, a k-orthogonal Kr-decomposition of Kn is called an (n, r, k) completely reducible super-simple design. We prove that for any two fixed integers r and k, there exists N=N(k, r) such that for every n>N, if Kn has a Kr-decomposition, then Kn also has an (n, r, k) completely-reducible super-simple design. If Kn does not have a Kr-decomposition, we show how to obtain a k-orthogonal optimalKr-packing of Kn. Complexity issues of k-orthogonal H-decompositions are also treated.