Sampling complete designs

Suppose Γ′Γ′ to be a subgraph of a graph ΓΓ. We define a sampling   of a ΓΓ-design B=(V,B)B=(V,B) into a Γ′Γ′-design B′=(V,B′)B′=(V,B′) as a surjective map ξ:B→B′ξ:B→B′ mapping each block of BB into one of its subgraphs. A sampling will be called regular   when the number of preimages of each block of B′B′ under ξξ is a constant. This new concept is closely related with the classical notion of embedding, which has been extensively studied, for many classes of graphs, by several authors; see, for example, the survey by Quattrocchi (2001) [29]. Actually, a sampling ξξ might induce several embeddings of the design B′B′ into BB, although the converse is not true in general. In the present paper, we study in more detail the behaviour of samplings of ΓΓ-complete designs of order nn into Γ′Γ′-complete designs of the same order and show how the natural necessary condition for the existence of a regular sampling is actually sufficient. We also provide some explicit constructions of samplings, as well as propose further generalisations.