Nestings of BIBDs with block size four
arxiv(2024)
摘要
In a nesting of a balanced incomplete block design (or BIBD), we wish to add
a point (the nested point) to every block of a (v,k,λ)-BIBD in
such a way that we end up with a partial (v,k+1,λ+1)-BIBD. In the case
where the partial (v,k+1,λ+1)-BIBD is in fact a
(v,k+1,λ+1)-BIBD, we have a perfect nesting. We show that a
nesting is perfect if and only if k = 2 λ + 1.
Perfect nestings were previously known to exist in the case of Steiner triple
systems (i.e., (v,3,1)-BIBDs) when v ≡ 1 6, as well as for some
symmetric BIBDs. Here we study nestings of (v,4,1)-BIBDs, which are not
perfect nestings. We prove that there is a nested (v,4,1)-BIBD if and only if
v ≡ 1 or 4 12, v ≥ 13. This is accomplished by a
variety of direct and recursive constructions.
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