The existence of non-trivial hyperfactorizations ofK 2n

Abstract

A λ-hyperfactorization ofK 2n is a collection of 1-factors ofK 2n for which each pair of disjoint edges appears in precisely λ of the 1-factors. We call a λ-hyperfactorizationtrivial if it contains each 1-factor ofK 2n with the same multiplicity γ (then λ=γ(2n−5)!!). A λ-hyperfactorization is calledsimple if each 1-factor ofK 2n appears at most once. Prior to this paper, the only known non-trivial λ-hyperfactorizations had one of the following parameters (or were multipliers of such an example)

  1. (i)

    2n=2a+2, λ=1 (for alla≥3); cf. Cameron [3];

  2. (ii)

    2n=12, λ=15 or 2n=24, λ=495; cf. Jungnickel and Vanstone [8].

In the present paper we show the existence of non-trivial simple λ-hyperfactorizations ofK 2n for alln≥5.

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This is the revised version of an earlier preprint bearing the same title (Research Report CORR 87-51, Dept. of Combinatorics and Optimization, University of Waterloo).

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Boros, E., Jungnickel, D. & Vanstone, S.A. The existence of non-trivial hyperfactorizations ofK 2n . Combinatorica 11, 9–15 (1991). https://doi.org/10.1007/BF01375468

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AMS subject classification (1980)

  • 05 B 30 (primary)
  • 05 C 99 (secondary)