G-designs without blocking sets, Note

For an undirected graph G and a natural number n, a G-design of order n is an edge partition of the complete graph Kn with n vertices into subgraphs G1, G2, . . . , each isomorphic to G. A set T ⊂ V (Kn) is called a blocking set if it meets the vertex set V (Gi) of each Gi in the decomposition, but contains none of them. In a previous paper [J. Combin. Designs 4 (1996), 135–142] the first and third authors proved that if G is a cycle, then there exists a G-design without blocking sets. Here we extend this theorem for all graphs G, moreover we prove that for every G and every integer k ≥ 2 there exists a non-k-colorable G-design.