Abstract
A 4-semiregular 1-factorization is a 1-factorization in which every pair of distinct 1-factors forms a union of 4-cycles. LetK be the complete graphK 2nor the complete bipartite graphK n, n .We prove that there is a 4-semiregular 1-factorization ofK if and only ifn is a power of 2 andn≥2, and 4-semiregular 1-factorizations ofK are isomorphic, and then we determine the symmetry groups. They are known for the case of the complete graphK 2n ,however, we prove them in a different method.
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References
Cameron, P.J.: Parallelisms of Complete Designs. London Math. Soc. Lecture Note Ser. 23. Cambridge: Cambridge Univ. Press 1976
Mendelsohn, E., Rosa, A.: One-Factorizations of the Complete Graph — A Survey. J. Graph Theory9, 43–65 (1985)
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Kobayashi, M., Nakamura, G. On 4-semiregular 1-factorizations of complete graphs and complete bipartite graphs. Graphs and Combinatorics 10, 53–59 (1994). https://doi.org/10.1007/BF01202470
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DOI: https://doi.org/10.1007/BF01202470