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Perfect 1-factorisations of K18 with a non-trivial automorphism group

  • Andrei V. Ivanov
  • Leonid M. Tertitski
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Abstract

A perfect 1-factorisation (P1F) of K2n is a partition of its edges into 2n−1 perfect matchings, any two of which form a Hamiltonian cycle. Exhaustive classification is known only up to K16. We present a method for the exhaustive generation, up to isomorphism, of perfect 1-factorisations with a non-trivial automorphism group, which reduces the search tree far enough to make K18 and beyond feasible. It combines two ideas. For automorphism groups of order greater than 2, classical results of Ihrig on the automorphisms of P1Fs are used as tools that reject almost all prescribed cycle types without search. For the order-2 case, which dominates, candidates are recalculated from inherited compatibility lists, in the spirit of Gill and Wanless, with an outer loop over two further rows that keeps those lists small, and Ihrig's results restrict the first factor. As the numerical proof of the method we compute all 10,710 P1Fs of K18 with a non-trivial automorphism group; before, only six were known.

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Cite

BibTeXRIS
BibTeX
@misc{ivanov2026perfect,
  title = {{Perfect 1-factorisations of $K_18$ with a non-trivial automorphism group}},
  author = {Andrei V. Ivanov and Leonid M. Tertitski},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01807},
  eprint = {2611.01807},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01807},
}
RIS
TY  - UNPB
TI  - Perfect 1-factorisations of K_18 with a non-trivial automorphism group
AU  - Andrei V. Ivanov
AU  - Leonid M. Tertitski
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01807
UR  - https://arxiv.org/abs/2611.01807
AB  - A perfect 1-factorisation (P1F) of K_2n is a partition of its edges into 2n−1 perfect matchings, any two of which form a Hamiltonian cycle. Exhaustive classification is known only up to K_16. We present a method for the exhaustive generation, up to isomorphism, of perfect 1-factorisations with a non-trivial automorphism group, which reduces the search tree far enough to make K_18 and beyond feasible. It combines two ideas. For automorphism groups of order greater than 2, classical results of Ihrig on the automorphisms of P1Fs are used as tools that reject almost all prescribed cycle types without search. For the order-2 case, which dominates, candidates are recalculated from inherited compatibility lists, in the spirit of Gill and Wanless, with an outer loop over two further rows that keeps those lists small, and Ihrig's results restrict the first factor. As the numerical proof of the method we compute all 10,710 P1Fs of K_18 with a non-trivial automorphism group; before, only six were known.
ER  -