Sampling complete designs

Suppose ${\Gamma }^{\prime }$ to be a subgraph of a graph $\Gamma$. We define a sampling of a $\Gamma$-design $\mathfrak{B}=(V,B)$ into a ${\Gamma }^{\prime }$-design ${\mathfrak{B}}^{\prime }=(V,B^{\prime })$ as a surjective map $\xi :B\to B^{\prime }$ 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 $B^{\prime }$ under $\xi$ 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 $\xi$ might induce several embeddings of the design ${\mathfrak{B}}^{\prime }$ into $\mathfrak{B}$, although the converse is not true in general. In the present paper, we study in more detail the behaviour of samplings of $\Gamma$-complete designs of order $n$ into ${\Gamma }^{\prime }$-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.