The intersection problem for small G-designs

A G-design of order n is a pair (P, B) where P is the vertex set of the complete graph Kn and B is an edge-disjoint decomposition of Kn into isomorphic copies of the simple graph G. Following terminology, we call these copies "blocks". Given a particular the intersection problem asks for which k is it possible to find two (P, Bl) and (P, B 2) of order n, with IBI n B21 = k, that is, with k common blocks. Here we complete the solution of this intersection problem for several G-designs where G is "small", so that now it is solved for all connected graphs G with at most four vertices or at most four edges. and Let G be a simple graph which is some subgraph of K n , the complete undirected graph on n vertices. A G-design of order n is a pair (V, where V is the vertex set of and B is an edge-disjoint decomposition of Kn into copies of the simple graph G. Following design terminology, we refer to these copies of G as blocks. Thus, for example, a Steiner triple system is a K3-design and a balanced incomplete block design with block size four and index A = 1 is a K4-design. The number of blocks,