Constructions and uses of incomplete pairwise balanced designs

TL;DR

This paper presents explicit constructions for incomplete pairwise balanced designs, enabling new results in related combinatorial designs and demonstrating the versatility of IPBDs as templates for various applications.

Contribution

The paper introduces explicit constructions for IPBDs under specific conditions, expanding the toolkit for combinatorial design theory and related fields.

Findings

Constructed IPBDs for large parameters satisfying divisibility and ratio conditions.

Derived new results for class-uniformly resolvable designs and incomplete Latin squares.

Showcased applications of IPBDs as templates in diverse combinatorial problems.

Abstract

We give explicit constructions for incomplete pairwise balanced designs IPBD$((v;w),K)$, or, equivalently, edge-decompositions of a difference of two cliques $K_v \setminus K_w$ into cliques whose sizes belong to the set $K$. Our constructions produce such designs whenever $v$ and $w$ satisfy the usual divisibility conditions, have ratio $v/w$ bounded away from the smallest value in $K$ minus one, say $v/w > k-1+\epsilon$, for $k =\min K$ and $\epsilon>0$, and are sufficiently large (depending on $K$ and $\epsilon$). As a consequence, some new results are obtained on many related designs, including class-uniformly resolvable designs, incomplete mutually orthogonal latin squares, and group divisible designs. We also include several other applications that illustrate the power of using IPBDs as `templates'.