The perfect 1-factorization conjecture for complete graphs

Let K2nK_{2n} be the complete graph on 2n2n vertices. A 1-factorization is a decomposition of its edges into perfect matchings, and it is perfect when the union of any two distinct 1-factors is a Hamiltonian cycle. The perfect 1-factorization conjecture. Every complete graph K2nK_{2n} with an even number of vertices admits a perfect 1-factorization. The conjecture is a strengthening of ordinary 1-factorization for complete graphs; the source attributes it to Ko and does not state whether it has been resolved.

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Primary source

Robert W. Donley, S. James Gates, Tristan Hübsch and Rishi Nath, “A combinatorial introduction to Adinkras”, arXiv:2410.12834 (2024).

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