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 = K_n and H = K_r, a k-orthogonal K_r-decomposition of K_n 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 K_n has a K_r-decomposition, then K_n also has an (n, r, k) completely-reducible super-simple design. If K_n does not have a K_r-decomposition, we show how to obtain a k-orthogonal optimal K_r-packing of K_n . Complexity issues of k-orthogonal H-decompositions are also treated.