The asymptotic existence of BIBDs having a nesting

Abstract

A $(v,k,\lambda )$-BIBD $(X,\mathcal{B})$ can be nested if there is a mapping $\varphi :\mathcal{B}\to X$ such that $(X,\{B\cup \{\varphi (B)\}\mid B\in \mathcal{B}\})$ is a $(v,k+1,\lambda +1)$-packing. A $(v,k,\lambda )$-BIBD has a (perfect) nesting if and only if its incidence graph has a harmonious (exact) coloring with $v$ colors. This paper shows that given any positive integers $k$ and $\lambda$, if $k\ge 2\lambda +2$, then for any sufficiently large $v$, every $(v,k,\lambda )$-BIBD can be nested into a $(v,k+1,\lambda +1)$-packing; and if $k=2\lambda +1$, then for any sufficiently large $v$ satisfying $v\equiv 1(\mathrm{mod}2k)$, there exists a $(v,k,\lambda )$-BIBD having a perfect nesting. Banff difference families (BDF), as a special kind of difference families (DF), can be used to generate nested designs. This paper shows that if $G$ is a finite abelian group with a large size whose number of $2$-order elements is no more than a given constant, and $k\ge 2\lambda +2$, then a $(G,k,\lambda )$-BDF can be obtained by taking any $(G,k,\lambda )$-DF and then replacing each of its base blocks by a suitable translation. This is a Novák-like theorem. Novák conjectured in 1974 that for any cyclic Steiner triple system of order $v$, it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. Novák's conjecture was generalized to any cyclic $(v,k,\lambda )$-BIBDs by Feng, Horsley and Wang in 2021, who conjectured that given any positive integers $k$ and $\lambda$ such that $k\ge \lambda +1$, there exists an integer $v_0$ such that, for any cyclic $(v,k,\lambda )$-BIBD with $v\ge v_0$, it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. This paper confirms this conjecture for every $k\ge \lambda +2$.