Sampling complete designs

Suppose Γ′ to be of 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 any block of B into one of its subgraphs. A sampling will be called regular when the number of preimages of any 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 [28]. 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 generalizations.