Weak and Strong Nestings of BIBDs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917672424308736 |
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| author | Stinson, Douglas R. |
| author_facet | Stinson, Douglas R. |
| contents | 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 $(v,k,λ)$-BIBD in such a way that we end up with a partial $(w,k+1,λ+1)$-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 $(w,k+1,λ+1)$-BIBD. In both cases, the goal is to minimize $w$. We prove lower bounds on $w$ as a function of $v$, $k$ and $λ$ and we find infinite classes of $(v,2,1)$- and $(v,3,2)$-BIBDs that have optimal nestings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_12820 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak and Strong Nestings of BIBDs Stinson, Douglas R. Combinatorics 05B05 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 $(v,k,λ)$-BIBD in such a way that we end up with a partial $(w,k+1,λ+1)$-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 $(w,k+1,λ+1)$-BIBD. In both cases, the goal is to minimize $w$. We prove lower bounds on $w$ as a function of $v$, $k$ and $λ$ and we find infinite classes of $(v,2,1)$- and $(v,3,2)$-BIBDs that have optimal nestings. |
| title | Weak and Strong Nestings of BIBDs |
| topic | Combinatorics 05B05 |
| url | https://arxiv.org/abs/2405.12820 |