On the spouse-loving variant of the Oberwolfach problem
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910548894941184 |
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| author | Bolohan, Noah Buchanan, Iona Burgess, Andrea Šajna, Mateja Van Snick, Ryan |
| author_facet | Bolohan, Noah Buchanan, Iona Burgess, Andrea Šajna, Mateja Van Snick, Ryan |
| contents | We prove that $K_n+I$, the complete graph of an even order with a $1$-factor duplicated, admits a decomposition into $2$-factors, each a disjoint union of cycles of length $m \geq 5$ if and only if $m \mid n$, except possibly when $m$ is odd and $n=4m$. In addition, we show that $K_n+I$ admits a decomposition into $2$-factors, each a disjoint union of cycles of lengths $m_1, \ldots, m_t$, whenever $m_1, \ldots, m_t$ are all even. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21745 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the spouse-loving variant of the Oberwolfach problem Bolohan, Noah Buchanan, Iona Burgess, Andrea Šajna, Mateja Van Snick, Ryan Combinatorics 05B30, 05C51, 05C70 We prove that $K_n+I$, the complete graph of an even order with a $1$-factor duplicated, admits a decomposition into $2$-factors, each a disjoint union of cycles of length $m \geq 5$ if and only if $m \mid n$, except possibly when $m$ is odd and $n=4m$. In addition, we show that $K_n+I$ admits a decomposition into $2$-factors, each a disjoint union of cycles of lengths $m_1, \ldots, m_t$, whenever $m_1, \ldots, m_t$ are all even. |
| title | On the spouse-loving variant of the Oberwolfach problem |
| topic | Combinatorics 05B30, 05C51, 05C70 |
| url | https://arxiv.org/abs/2407.21745 |