Difference Methods for Double-Change Covering Designs
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866911247113388032 |
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| author | Chafee, Amanda Lynn Stevens, Brett |
| author_facet | Chafee, Amanda Lynn Stevens, Brett |
| contents | A \textbf{double-change covering design} (DCCD) is a $v$-set $V$ and an ordered list $\mathcal{L}$ of $b$ blocks of size $k$ where every pair from $V$ must occur in at least one block and each pair of consecutive blocks differs by exactly two elements. It is \textbf{minimal} if it has the fewest block possible and \textbf{circular} when the first and last blocks also differ by two elements. We give a recursive construction that uses 1-factorizations and expansion sets to construct a DCCD($v+\frac{v+k-2}{k-2},k,b+\frac{v}{k-2}\frac{v+k-2}{2k-4}$) from a DCCD($v,k,b$). We construct circular DCCD($2k-2,k,k-1$) and circular DCCD($2k-3,k,k-2$) from single change covering designs and determine minimal DCCD when $v=2k-2$. We use difference methods to construct five infinite families of minimal circular DCCD($c(4k-6)+1,k,c^2(4k-6)+c$) when $c\leq 5$ for any $k\geq 3$. The recursive construction is then used to build twelve additional minimal DCCD from members of these infinite families. Finally the difference method is used to construct a minimal circular DCCD(61,4,366). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_12275 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Difference Methods for Double-Change Covering Designs Chafee, Amanda Lynn Stevens, Brett Combinatorics 05B10, 05B40, 05C70 A \textbf{double-change covering design} (DCCD) is a $v$-set $V$ and an ordered list $\mathcal{L}$ of $b$ blocks of size $k$ where every pair from $V$ must occur in at least one block and each pair of consecutive blocks differs by exactly two elements. It is \textbf{minimal} if it has the fewest block possible and \textbf{circular} when the first and last blocks also differ by two elements. We give a recursive construction that uses 1-factorizations and expansion sets to construct a DCCD($v+\frac{v+k-2}{k-2},k,b+\frac{v}{k-2}\frac{v+k-2}{2k-4}$) from a DCCD($v,k,b$). We construct circular DCCD($2k-2,k,k-1$) and circular DCCD($2k-3,k,k-2$) from single change covering designs and determine minimal DCCD when $v=2k-2$. We use difference methods to construct five infinite families of minimal circular DCCD($c(4k-6)+1,k,c^2(4k-6)+c$) when $c\leq 5$ for any $k\geq 3$. The recursive construction is then used to build twelve additional minimal DCCD from members of these infinite families. Finally the difference method is used to construct a minimal circular DCCD(61,4,366). |
| title | Difference Methods for Double-Change Covering Designs |
| topic | Combinatorics 05B10, 05B40, 05C70 |
| url | https://arxiv.org/abs/2209.12275 |