Subdesigns in Complementary Path Decompositions and Incomplete Two-fold Designs with Block Size Four

Abstract

We give a complete solution to the existence problem for subdesigns in complementary ${\mathrm{P}}_3$-decompositions, where ${\mathrm{P}}_3$ denotes the path of length three. As a corollary we obtain the spectrum for incomplete designs with block size four and $\lambda =2$, having one hole.