Sampling complete designs

Suppose Γ ′ to be a subgraph of a graph Γ . We define a sampling of a Γ -design B = ( V , B ) into a Γ ′ -design B ′ = ( V , B ′ ) as a surjective map ξ : B → B ′ mapping each block of B into one of its subgraphs. A sampling will be called regular when the number of preimages of each block of B ′ under ξ 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 ξ might induce several embeddings of the design B ′ into B , although the converse is not true in general. In the present paper, we study in more detail the behaviour of samplings of Γ -complete designs of order n into 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.