Weak and Strong Nestings of BIBDs

TL;DR

This paper investigates two nesting types of BIBDs, establishing lower bounds on the number of added points and identifying classes with optimal nestings, advancing combinatorial design theory.

Contribution

It introduces and analyzes weak and strong nestings of BIBDs, providing lower bounds and constructing optimal examples for specific parameters.

Findings

Lower bounds on the number of new points in nestings.

Infinite classes of BIBDs with optimal nestings identified.

Characterization of weak and strong nesting properties.

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 $w \geq v$. 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.