Perfect $1$-factorisations of $K_{11,11}$

[2506.02455] Perfect $1$-factorisations of $K_{11,11}$ Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2506.02455 (math) [Submitted on 3 Jun 2025 ( v1 ), last revised 9 Apr 2026 (this version, v2)] Title: Perfect $1$-factorisations of $K_{11,11}$ Authors: Jack Allsop , Ian M. Wanless View a PDF of the paper titled Perfect $1$-factorisations of $K_{11,11}$, by Jack Allsop and Ian M. Wanless View PDF HTML (experimental) Abstract: A perfect $1$-factorisation of a graph is a decomposition of that graph into $1$-factors such that the union of any two $1$-factors is a Hamiltonian cycle. A Latin square of order $n$ is row-Hamiltonian if for every pair $(r,s)$ of distinct rows, the permutation mapping $r$ to $s$ has a single cycle of length $n$. We report the results of a computer enumeration of the perfect $1$-factorisations of the complete bipartite graph $K_{11,11}$. This also allows us to find all row-Hamiltonian Latin squares of order $11$. Finally, we plug a gap in the literature regarding how many row-Hamiltonian Latin squares are associated with the classical families of perfect $1$-factorisations of complete graphs. Subjects: Combinatorics (math.CO) MSC classes: 05C70, 05B15 Cite as: arXiv:2506.02455 [math.CO] (or arXiv:2506.02455v2 [math.CO] for this version) https://doi.org/10.48550/arXiv.2506.02455 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Australasian Journal of Combinatorics 95 (2026), 114-130 Submission history From: Ian Wanless [ view email ] [v1] Tue, 3 Jun 2025 05:20:54 UTC (19 KB) [v2] Thu, 9 Apr 2026 02:15:09 UTC (17 KB) Full-text links: Access Paper: View a PDF of the paper titled Perfect $1$-factorisations of $K_{11,11}$, by Jack Allsop and Ian M. Wanless View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2025-06 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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