Weak and Strong Nestings of BIBDs

Abstract

We study two types of nestings of balanced incomplete block designs (BIBDs). In both types of nesting, we wish to add a point (the nested point) to every block of a $(v,k,\lambda )$-BIBD in such a way that we end up with a partial $(w,k+1,\lambda +1)$-BIBD for some $w\ge v$. In the case where $w>v$, we are introducing $w-v$ new points. This is called a weak nesting. A strong nesting satisfies the stronger property that no pair containing a new point occurs more than once in the partial $(w,k+1,\lambda +1)$-BIBD. In both cases, the goal is to minimize $w$. We prove lower bounds on $w$ as a function of $v$, $k$ and $\lambda$ and we find infinite classes of $(v,2,1)$- and $(v,3,2)$-BIBDs that have optimal nestings.