Sampling Complete Designs

Suppose $\Gamma'$ to be a subgraph of a graph $\Gamma$. We define a \emph{sampling} of a $\Gamma$-design ${\fB}=(V,B)$ into a $\Gamma'$-design ${\fB'}=(V,B')$ as a surjective map $\xi:B\to B'$ mapping each block of $B$ into one of its subgraphs. A sampling will be called {\it regular} when the number of preimages of each block of $B'$ under $\xi$ is a constant. This new concept is closely related with the classical notion of \emph{embedding}, which has been extensively studied, for many classes of graphs, by several authors; see, for example, the survey \cite{Q2002a}. Actually, a sampling $\xi$ might induce several embeddings of the design $\fB'$ into $\fB$, 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 $\Gamma'$--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.