New results on path-decompositions and their down-links

In a recent paper the concept of \emph{down-link} from a
$(K_v,\Gamma)$-design $\cB$ to a $(K_n,\Gamma')$-design $\cB'$ has been
introduced. In the present paper the spectrum problems for
$\Gamma'=P_4$ are studied. General results on the existence of path-decompositions
and embeddings between path-decompositions playing a fundamental role for the construction of down-links
are also presented.