On c-Bhaskar Rao Designs and tight embeddings for path designs

Under the right conditions it is possible for the ordered blocks of a path design PATH(v, k, ) to be considered as unordered blocks and thereby create a BIBD(v, k, ).We call this a tight embedding.We show here that, for any triple system TS(v, 3), there is always such an embedding and that the problem is equivalent to the existence of a (−1)-BRD(v, 3, 3), i.e., a c-Bhaskar Rao Design. That is, we also prove the incidence matrix of any triple system TS(v, 3) can always be signed to create a (−1)-BRD(v, 3, 3) and, moreover, the signing determines a natural partition of the blocks of the triple system making it a nested design. © 2007 Elsevier B.V. All rights reserved.