Nestings of BIBDs with block size four
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913297518821376 |
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| author | Buratti, Marco Kreher, Donald L. Stinson, Douglas R. |
| author_facet | Buratti, Marco Kreher, Donald L. Stinson, Douglas R. |
| contents | In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the \emph{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 \emph{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 $v \equiv 1 \bmod 6$, 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 $v \equiv 1 \text{ or } 4 \bmod 12$, $v \geq 13$. This is accomplished by a variety of direct and recursive constructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02666 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nestings of BIBDs with block size four Buratti, Marco Kreher, Donald L. Stinson, Douglas R. Combinatorics 05B05 In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the \emph{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 \emph{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 $v \equiv 1 \bmod 6$, 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 $v \equiv 1 \text{ or } 4 \bmod 12$, $v \geq 13$. This is accomplished by a variety of direct and recursive constructions. |
| title | Nestings of BIBDs with block size four |
| topic | Combinatorics 05B05 |
| url | https://arxiv.org/abs/2404.02666 |