| Title: | Transforming solutions for the Oberwolfach problem into solutions for the spouse-loving variant |
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| Authors: | ID Lekše, Maruša (Author)
ID Šajna, Mateja (Author) |
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| Files: |  PDF - Presentation file, download (4,34 MB)
MD5: 6E432EE6972DF5B89D1C46133E5D40A8
 URL - Source URL, visit https://onlinelibrary.wiley.com/doi/10.1002/jcd.70020
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | The Oberwolfach problem ${\mathrm {OP}}(F)$, for a $2$-factor $F$ of $K_n$, asks whether there exists a $2$-factorization of $K_n$ (if $n$ is odd) or $K_n-I$ (if $n$ is even) where each $2$-factor is isomorphic to $F$. Here, $I$ denotes any $1$-factor of $K_n$. For even $n$, the problem ${\mathrm {OP}}(F)$ may also be denoted ${\mathrm {OP}}^-(F)$, and has been nicknamed the spouse-avoiding variant. Similarly, the spouse-loving variant is denoted ${\mathrm {OP}}^+(F)$ and asks for a $2$-factorization of $K_n + I$ (the complete graph with the edges of a $1$-factor $I$ duplicated, rather than deleted) in which each $2$-factor is isomorphic to $F$. To date, many more infinite families of cases of ${\mathrm {OP}}$ and ${\mathrm {OP}}^-$ have been solved than of ${\mathrm {OP}}^+(F)$. In this paper, we show how certain solutions to ${\mathrm {OP}}^-(F)$ can be used to construct solutions to ${\mathrm {OP}}^+(F)$; in particular, when the number of odd cycles in the $2$-factor $F$ is not too large. Our technique of setups also allows us to completely solve the two-table ${\mathrm {OP}}^+$; that is, ${\mathrm {OP}}^+(F)$, where $F$ has exactly two components. |
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| Keywords: | 1‐rotational solution, 12‐setup, 2‐factorization, 2‐starter, 6‐setup, complete graph plus a 1‐factor, Oberwolfach problem, spouse‐loving variant |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.08.2026 |
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| Year of publishing: | 2026 |
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| Number of pages: | str. 361-377 |
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| Numbering: | Vol. 34, iss. 8 |
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| PID: | 20.500.12556/DiRROS-31266  |
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| UDC: | 519.17 |
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| ISSN on article: | 1063-8539 |
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| DOI: | 10.1002/jcd.70020 |
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| COBISS.SI-ID: | 285786371 |
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| Publication date in DiRROS: | 23.07.2026 |
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| Views: | 162 |
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| Downloads: | 146 |
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| Metadata: |    |
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