The Honeymoon Oberwolfach Problem: small cases
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929403848556544 |
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| author | Jerade, Marie Rose Šajna, Mateja |
| author_facet | Jerade, Marie Rose Šajna, Mateja |
| contents | The Honeymoon Oberwolfach Problem HOP$(2m_1,2m_2,\ldots,2m_t)$ asks the following question. Given $n=m_1+m_2+\ldots +m_t$ newlywed couples at a conference and $t$ round tables of sizes $2m_1,2m_2,\ldots,2m_t$, is it possible to arrange the $2n$ participants at these tables for $2n-2$ meals so that each participant sits next to their spouse at every meal, and sits next to every other participant exactly once? A solution to HOP$(2m_1,2m_2,\ldots,2m_t)$ is a decomposition of $K_{2n}+(2n-3)I$, the complete graph $K_{2n}$ with $2n-3$ additional copies of a fixed 1-factor $I$, into 2-factors, each consisting of disjoint $I$-alternating cycles of lengths $2m_1,2m_2,\ldots,2m_t$.
The Honeymoon Oberwolfach Problem was introduced in a 2019 paper by Lepine and Šajna. The authors conjectured that HOP$(2m_1,2m_2,\ldots,$ $2m_t)$ has a solution whenever the obvious necessary conditions are satisfied, and proved the conjecture for several large cases, including the uniform cycle length case $m_1=\ldots=m_t$, and the small cases with $n \le 9$. In the present paper, we extend the latter result to all cases with $n \le 20$ using a computer search. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_00204 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Honeymoon Oberwolfach Problem: small cases Jerade, Marie Rose Šajna, Mateja Combinatorics 05C51 The Honeymoon Oberwolfach Problem HOP$(2m_1,2m_2,\ldots,2m_t)$ asks the following question. Given $n=m_1+m_2+\ldots +m_t$ newlywed couples at a conference and $t$ round tables of sizes $2m_1,2m_2,\ldots,2m_t$, is it possible to arrange the $2n$ participants at these tables for $2n-2$ meals so that each participant sits next to their spouse at every meal, and sits next to every other participant exactly once? A solution to HOP$(2m_1,2m_2,\ldots,2m_t)$ is a decomposition of $K_{2n}+(2n-3)I$, the complete graph $K_{2n}$ with $2n-3$ additional copies of a fixed 1-factor $I$, into 2-factors, each consisting of disjoint $I$-alternating cycles of lengths $2m_1,2m_2,\ldots,2m_t$. The Honeymoon Oberwolfach Problem was introduced in a 2019 paper by Lepine and Šajna. The authors conjectured that HOP$(2m_1,2m_2,\ldots,$ $2m_t)$ has a solution whenever the obvious necessary conditions are satisfied, and proved the conjecture for several large cases, including the uniform cycle length case $m_1=\ldots=m_t$, and the small cases with $n \le 9$. In the present paper, we extend the latter result to all cases with $n \le 20$ using a computer search. |
| title | The Honeymoon Oberwolfach Problem: small cases |
| topic | Combinatorics 05C51 |
| url | https://arxiv.org/abs/2407.00204 |